274:
1741:
5934:
6315:
1332:
6620:
6052:
1069:
1825:, one physical interpretation of this problem is as follows: fix the temperature on the boundary of the domain according to the given specification of the boundary condition. Allow heat to flow until a stationary state is reached in which the temperature at each point on the domain does not change anymore. The temperature distribution in the interior will then be given by the solution to the corresponding Dirichlet problem.
6367:
53:
5776:
1018:
8850:
6310:{\displaystyle \nabla ^{2}f={\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial f}{\partial r}}\right)+{\frac {1}{r^{2}\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial f}{\partial \theta }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}}=0.}
1327:{\displaystyle \nabla ^{2}f={\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial f}{\partial r}}\right)+{\frac {1}{r^{2}\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial f}{\partial \theta }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}}=0.}
6615:{\displaystyle {\frac {1}{R}}{\frac {d}{dr}}\left(r^{2}{\frac {dR}{dr}}\right)=\lambda ,\qquad {\frac {1}{Y}}{\frac {1}{\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial Y}{\partial \theta }}\right)+{\frac {1}{Y}}{\frac {1}{\sin ^{2}\theta }}{\frac {\partial ^{2}Y}{\partial \varphi ^{2}}}=-\lambda .}
9409:
1707:
1518:
5516:
835:
822:
6853:
8695:
7715:
2006:
9107:
1523:
4333:
1348:
5880:
7333:
1869:
within the domain where the equation is satisfied. If any two functions are solutions to
Laplace's equation (or any linear homogeneous differential equation), their sum (or any linear combination) is also a solution. This property, called the
4764:
680:
6734:
5771:{\displaystyle u(P)={\frac {1}{4\pi }}a^{3}\left(1-{\frac {\rho ^{2}}{a^{2}}}\right)\int _{0}^{2\pi }\int _{0}^{\pi }{\frac {g(\theta ',\varphi ')\sin \theta '}{(a^{2}+\rho ^{2}-2a\rho \cos \Theta )^{\frac {3}{2}}}}d\theta '\,d\varphi '}
7841:
5105:
7209:
5502:
denotes the angle with the vertical axis, which is contrary to the usual
American mathematical notation, but agrees with standard European and physical practice. Then the solution of the Laplace equation with Dirichlet boundary values
4062:
3670:
7573:
5260:
1013:{\displaystyle \nabla ^{2}f={\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial f}{\partial r}}\right)+{\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial \phi ^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}=0.}
8303:
1885:
8113:
6730:
4179:
3184:
9080:
4829:
5918:
at the center of the sphere is the mean value of its values on the sphere. This mean value property immediately implies that a non-constant harmonic function cannot assume its maximum value at an interior point.
5447:
2520:
8407:
3546:
Thus every analytic function corresponds to a steady incompressible, irrotational, inviscid fluid flow in the plane. The real part is the velocity potential, and the imaginary part is the stream function.
2347:
8845:{\displaystyle {\begin{aligned}\mathbf {g} &=-\nabla V,\\\nabla \cdot \mathbf {g} &=\nabla \cdot (-\nabla V)=-\nabla ^{2}V,\\\implies \nabla ^{2}V&=-\nabla \cdot \mathbf {g} .\end{aligned}}}
4406:
2595:
4097:). It is common to take a different sign convention for this equation than one typically does when defining fundamental solutions. This choice of sign is often convenient to work with because −Δ is a
3329:
4160:
449:
8688:
8349:
8208:
7214:
4093:
rather than a function; but it can be thought of as a limit of functions whose integrals over space are unity, and whose support (the region where the function is non-zero) shrinks to a point (see
3846:
3544:
5367:
2888:
8700:
3904:
7511:
9404:{\displaystyle R(r)=(-1)^{l}{\frac {(l!)^{2}r_{s}^{l}}{(2l)!}}P_{l}\left(1-{\frac {2r}{r_{s}}}\right)+(-1)^{l+1}{\frac {2(2l+1)!}{(l)!^{2}r_{s}^{l+1}}}Q_{l}\left(1-{\frac {2r}{r_{s}}}\right).}
1702:{\displaystyle \nabla ^{2}f={\frac {1}{\sqrt {|g|}}}{\frac {\partial }{\partial \xi ^{i}}}\!\left({\sqrt {|g|}}g^{ij}{\frac {\partial f}{\partial \xi ^{j}}}\right)=0,\qquad (g=\det\{g_{ij}\})}
4660:
5781:
3460:
3222:
be the horizontal and vertical components of the velocity field of a steady incompressible, irrotational flow in two dimensions. The continuity condition for an incompressible flow is that
2447:
2208:
1513:{\displaystyle \nabla ^{2}f={\frac {\partial }{\partial \xi ^{j}}}\left({\frac {\partial f}{\partial \xi ^{k}}}g^{kj}\right)+{\frac {\partial f}{\partial \xi ^{j}}}g^{jm}\Gamma _{mn}^{n}=0,}
4534:
2778:
4164:
The
Laplace equation is unchanged under a rotation of coordinates, and hence we can expect that a fundamental solution may be obtained among solutions that only depend upon the distance
8900:
4882:
3717:
8172:
3778:
4454:
2656:
9580:
7745:
4911:
2789:
The close connection between the
Laplace equation and analytic functions implies that any solution of the Laplace equation has derivatives of all orders, and can be expanded in a
2108:
1064:
7113:
3387:
2947:
4173:
5963:(left to right). Zonal, sectoral, and tesseral harmonics are depicted along the left-most column, the main diagonal, and elsewhere, respectively. (The negative order harmonics
8058:
2713:
372:
8968:
8506:
8482:
5996:
3269:
7981:
6037:
3574:
1852:
alone. For the example of the heat equation it amounts to prescribing the heat flux through the boundary. In particular, at an adiabatic boundary, the normal derivative of
476:
8599:
8230:
8011:
621:
404:
4665:
7919:
589:
545:
8929:
8435:
8237:
500:
7869:
8619:
8526:
8031:
8067:
7949:
6658:
2952:
2668:
implies that the value of the line integral connecting two points is independent of the path. The resulting pair of solutions of the
Laplace equation are called
8639:
8566:
8546:
8458:
8135:
7889:
3930:
9001:
4768:
5146:
2452:
817:{\displaystyle \nabla ^{2}f={\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}=0.}
6848:{\displaystyle \lambda \sin ^{2}\theta +{\frac {\sin \theta }{\Theta }}{\frac {d}{d\theta }}\left(\sin \theta {\frac {d\Theta }{d\theta }}\right)=m^{2}}
304:
8356:
5384:
2232:
10014:
4340:
2531:
3274:
4110:
8644:
8307:
7710:{\displaystyle f(r,\theta ,\varphi )=\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }f_{\ell }^{m}r^{\ell }Y_{\ell }^{m}(\theta ,\varphi ),}
9974:
3789:
3487:
2817:
3851:
3193:
7449:
2001:{\displaystyle {\frac {\partial ^{2}\psi }{\partial x^{2}}}+{\frac {\partial ^{2}\psi }{\partial y^{2}}}\equiv \psi _{xx}+\psi _{yy}=0.}
5323:
4328:{\displaystyle -1=\iiint _{V}\nabla \cdot \nabla u\,dV=\iint _{S}{\frac {du}{dr}}\,dS=\left.4\pi a^{2}{\frac {du}{dr}}\right|_{r=a}.}
3406:
2375:
2140:
9644:. 8th edition / ed., Brooks/Cole, Cengage Learning, 2013. Chapter 12: Boundary-value Problems in Rectangular Coordinates. p. 462.
4481:
2718:
547:
is a twice-differentiable real-valued function. The
Laplace operator therefore maps a scalar function to another scalar function.
10019:
9465:
8571:
A potential that does not satisfy
Laplace's equation together with the boundary condition is an invalid electrostatic potential.
8692:
The gravitational field is conservative and can therefore be expressed as the negative gradient of the gravitational potential:
8857:
4846:
3675:
3394:
3205:
630:
297:
1846:. Physically, this corresponds to the construction of a potential for a vector field whose effect is known at the boundary of
409:
9934:
9906:
9739:
9718:
9677:
9649:
9628:
8177:
8140:
3910:. The Laplace equation can be used in three-dimensional problems in electrostatics and fluid flow just as in two dimensions.
3728:
4415:
2612:
5928:
2361:
also satisfies the
Laplace equation. Conversely, given a harmonic function, it is the real part of an analytic function,
2036:
3389:
then the continuity condition is the integrability condition for this differential: the resulting function is called the
3340:
2893:
2672:. This construction is only valid locally, or provided that the path does not loop around a singularity. For example, if
10000:
Find out how boundary value problems governed by
Laplace's equation may be solved numerically by boundary element method
4591:
2683:
663:. In general, Laplace's equation describes situations of equilibrium, or those that do not depend explicitly on time.
340:
9880:
8934:
290:
155:
3225:
4561:
9668:
9619:
6966:
629:, a generalization of Laplace's equation. Laplace's equation and Poisson's equation are the simplest examples of
594:
377:
6653:. Applying separation of variables again to the second equation gives way to the pair of differential equations
7364:
7328:{\displaystyle r^{2}\nabla ^{2}Y_{\ell }^{m}(\theta ,\varphi )=-\ell (\ell +1)Y_{\ell }^{m}(\theta ,\varphi ).}
6988:
20:
2793:, at least inside a circle that does not enclose a singularity. This is in sharp contrast to solutions of the
10034:
9961:
1874:, is very useful. For example, solutions to complex problems can be constructed by summing simple solutions.
330:
7546:
6984:
2134:
160:
150:
122:
9536:
1031:
10024:
9956:
6930:
6044:
5875:{\displaystyle \cos \Theta =\cos \theta \cos \theta '+\sin \theta \sin \theta '\cos(\varphi -\varphi ')}
9533:
The delta symbol, Δ, is also commonly used to represent a finite change in some quantity, for example,
4090:
1829:
9951:
8036:
3725:
is the charge density. The first
Maxwell equation is the integrability condition for the differential
10029:
6970:
6359:
214:
107:
98:
8487:
8463:
5966:
7954:
6007:
5472:
458:
8582:
8213:
7994:
1871:
1338:
828:
136:
648:, which are important in multiple branches of physics, notably electrostatics, gravitation, and
7898:
7542:
7103:
3556:
1792:
553:
509:
9504:
8908:
8414:
7836:{\displaystyle r<R={\frac {1}{\limsup _{\ell \to \infty }|f_{\ell }^{m}|^{{1}/{\ell }}}}.}
5489:
5100:{\displaystyle \iiint _{V}\left\,dV=\iiint _{V}\nabla \cdot \left\,dV=\iint _{S}\left\,dS.\,}
4903:
1745:
1024:
673:
626:
485:
230:
32:
8117:
Now, the electric field can be expressed as the negative gradient of the electric potential
7848:
7204:{\displaystyle Y_{\ell }^{m}(\theta ,\varphi )=Ne^{im\varphi }P_{\ell }^{m}(\cos {\theta })}
9813:
9515:
9507:
uses the Laplace equation to show that stable static ferromagnetic suspension is impossible
9461:
9445:
8979:
8604:
8511:
8016:
4065:
3924:
334:
240:
221:
175:
117:
8:
9968:
9755:
Chicone, C.; Mashhoon, B. (2011-11-20). "Nonlocal Gravity: Modified Poisson's Equation".
9623:. 7th ed., Brooks/Cole, Cengage Learning, 2012. Chapter 14: Partial Derivatives. p. 908.
9101:
7928:
4577:
4565:
1882:
Laplace's equation in two independent variables in rectangular coordinates has the form
1729:
127:
89:
9817:
9782:
9764:
9484:
9479:
9473:
9432:
8624:
8551:
8531:
8443:
8353:
Plugging this relation into Gauss's law, we obtain Poisson's equation for electricity,
8120:
7891:
are chosen instead. In that case, one needs to expand the solution of known regions in
7874:
7740:
7561:
7514:
7107:
4471:
3481:
2665:
634:
278:
185:
9985:
3665:{\displaystyle \nabla \times (u,v,0)=(v_{x}-u_{y}){\hat {\mathbf {k} }}=\mathbf {0} ,}
9982:
9930:
9902:
9876:
9826:
9801:
9786:
9735:
9714:
9673:
9645:
9624:
9499:
8641:
the gravitational constant. Then Gauss's law for gravitation in differential form is
6874:
4545:
4098:
2016:
1866:
1862:
1843:
1798:
1740:
645:
273:
200:
112:
84:
4101:. The definition of the fundamental solution thus implies that, if the Laplacian of
9821:
9774:
9510:
9489:
4759:{\displaystyle \nabla \cdot \nabla G=-\delta (x-x',y-y',z-z')\qquad {\text{in }}V,}
4580:
is a fundamental solution that also satisfies a suitable condition on the boundary
4475:
3907:
641:
452:
260:
255:
245:
74:
44:
9845:
8437:
and Poisson's equation reduces to Laplace's equation for the electric potential.
7736:
7560:
The general solution to Laplace's equation in a ball centered at the origin is a
3390:
653:
180:
145:
9999:
9591:
Physical applications often take the solution that vanishes at infinity, making
8298:{\displaystyle \nabla \cdot \mathbf {E} =\nabla \cdot (-\nabla V)=-\nabla ^{2}V}
2890:
with suitably defined coefficients whose real and imaginary parts are given by
644:. The twice continuously differentiable solutions of Laplace's equation are the
9494:
8061:
7892:
7564:
of the spherical harmonic functions multiplied by the appropriate scale factor
5933:
4557:
4553:
4467:
2801:
649:
235:
170:
165:
60:
9927:
Handbook of Linear Partial Differential Equations for Engineers and Scientists
10008:
9849:
7922:
4094:
2794:
1822:
1721:
660:
195:
190:
79:
8108:{\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho }{\varepsilon _{0}}}.}
6725:{\displaystyle {\frac {1}{\Phi }}{\frac {d^{2}\Phi }{d\varphi ^{2}}}=-m^{2}}
5314:
from the center of the sphere is reflected along its radial line to a point
3179:{\displaystyle f(z)=\sum _{n=0}^{\infty }\left+i\sum _{n=1}^{\infty }\left,}
2790:
1821:
is equal to some given function. Since the Laplace operator appears in the
657:
9582:. Its use to represent the Laplacian should not be confused with this use.
318:
250:
69:
9598:. This does not affect the angular portion of the spherical harmonics.
8508:
is surrounded by a conducting material with a specified charge density
8064:
for electricity (Maxwell's first equation) in differential form states
7382:
4057:{\displaystyle \Delta u=u_{xx}+u_{yy}+u_{zz}=-\delta (x-x',y-y',z-z'),}
479:
9778:
9075:{\displaystyle \Psi (r,\theta ,\varphi )=R(r)Y_{l}(\theta ,\varphi ),}
4824:{\displaystyle G=0\quad {\text{if}}\quad (x,y,z)\qquad {\text{on }}S.}
4168:
from the source point. If we choose the volume to be a ball of radius
9990:
7406:
7381:
represent colatitude and longitude, respectively. In particular, the
5255:{\displaystyle u(x',y',z')=\iiint _{V}Gf\,dV+\iint _{S}G_{n}g\,dS.\,}
4549:
4463:
2515:{\displaystyle \psi _{x}=-\varphi _{y},\quad \psi _{y}=\varphi _{x}.}
6362:, two differential equations result by imposing Laplace's equation:
5470:′. A consequence of this expression for the Green's function is the
2786:
is single-valued only in a region that does not enclose the origin.
9969:
Laplace Equation (particular solutions and boundary value problems)
8854:
Using the differential form of Gauss's law of gravitation, we have
4107:
is integrated over any volume that encloses the source point, then
3192:. These trigonometric functions can themselves be expanded, using
2664:
may be defined by a line integral. The integrability condition and
503:
9769:
5381:
will be outside the sphere. The Green's function is then given by
640:
The general theory of solutions to Laplace's equation is known as
9802:"The Laplace and poisson equations in Schwarzschild's space-time"
7413:
6929:
of the second equation at the boundary points of the domain is a
5298:, the Green's function may be obtained by means of a reflection (
4459:
2355:
satisfies the Laplace equation. A similar calculation shows that
322:
9873:
Introduction to Partial Differential Equations with Applications
8402:{\displaystyle \nabla ^{2}V=-{\frac {\rho }{\varepsilon _{0}}}.}
6624:
The second equation can be simplified under the assumption that
5442:{\displaystyle {\frac {1}{4\pi R}}-{\frac {a}{4\pi \rho R'}},\,}
9980:
52:
9713:. 4th ed., Pearson, 2013. Chapter 2: Electrostatics. p. 83-4.
4906:, (a consequence of the divergence theorem) which states that
3571:
in two space dimensions that is independent of time satisfies
2449:
then the Cauchy–Riemann equations will be satisfied if we set
2342:{\displaystyle u_{yy}=(-v_{x})_{y}=-(v_{y})_{x}=-(u_{x})_{x}.}
9734:. 4th ed., Pearson, 2013. Chapter 3: Potentials. p. 119-121.
9464:, a coordinate system under which Laplace's equation becomes
337:, who first studied its properties. This is often written as
9689:
The approach to spherical harmonics taken here is found in (
4548:. Note that, with the opposite sign convention, this is the
4266:
4401:{\displaystyle {\frac {du}{dr}}=-{\frac {1}{4\pi r^{2}}},}
2590:{\displaystyle d\psi =-\varphi _{y}\,dx+\varphi _{x}\,dy.}
7081:
independent solutions of this form, one for each integer
3324:{\displaystyle \nabla \times \mathbf {V} =v_{x}-u_{y}=0.}
2800:
There is an intimate connection between power series and
550:
If the right-hand side is specified as a given function,
5929:
Spherical harmonics § Laplace's spherical harmonics
4155:{\displaystyle \iiint _{V}\nabla \cdot \nabla u\,dV=-1.}
3271:
and the condition that the flow be irrotational is that
666:
444:{\displaystyle \Delta =\nabla \cdot \nabla =\nabla ^{2}}
8683:{\displaystyle \nabla \cdot \mathbf {g} =-4\pi G\rho .}
8344:{\displaystyle \nabla ^{2}V=-\nabla \cdot \mathbf {E} }
8203:{\displaystyle \nabla \times \mathbf {E} =\mathbf {0} }
7044:
Here the solution was assumed to have the special form
8970:
which is Laplace's equation for gravitational fields.
8902:
which is Poisson's equation for gravitational fields.
6319:
Consider the problem of finding solutions of the form
4458:
Note that, with the opposite sign convention (used in
3841:{\displaystyle \varphi _{x}=-u,\quad \varphi _{y}=-v.}
3539:{\displaystyle \varphi _{x}=-u,\quad \varphi _{y}=-v.}
3468:
satisfies the Laplace equation. The harmonic function
1801:
for Laplace's equation consists of finding a solution
9539:
9110:
9004:
8937:
8911:
8860:
8698:
8647:
8627:
8607:
8585:
8554:
8534:
8514:
8490:
8466:
8446:
8417:
8359:
8310:
8240:
8216:
8180:
8143:
8123:
8070:
8039:
8019:
7997:
7957:
7931:
7901:
7877:
7851:
7748:
7576:
7452:
7217:
7116:
6737:
6661:
6370:
6055:
6010:
5969:
5784:
5519:
5387:
5326:
5292:. For the case of the interior of a sphere of radius
5264:
Thus the Green's function describes the influence at
5149:
4914:
4849:
4771:
4668:
4594:
4484:
4418:
4343:
4182:
4113:
3933:
3854:
3848:
The second of Maxwell's equations then implies that
3792:
3731:
3678:
3577:
3490:
3409:
3343:
3277:
3228:
2955:
2896:
2883:{\displaystyle f(z)=\sum _{n=0}^{\infty }c_{n}z^{n},}
2820:
2721:
2686:
2615:
2534:
2455:
2378:
2235:
2143:
2039:
1888:
1526:
1351:
1072:
1034:
838:
683:
597:
556:
512:
488:
461:
412:
380:
343:
6894:
is a linear combination of the complex exponentials
9642:
Differential Equations with Boundary-Value Problems
7549:), and so counting dimensions shows that there are
5906:. A simple consequence of this formula is that if
3899:{\displaystyle \varphi _{xx}+\varphi _{yy}=-\rho ,}
633:. Laplace's equation is also a special case of the
9977:using Laplace's equation from exampleproblems.com.
9574:
9435:of the first and second kind, respectively, while
9403:
9074:
8962:
8923:
8894:
8844:
8682:
8633:
8613:
8593:
8560:
8540:
8520:
8500:
8476:
8452:
8429:
8401:
8343:
8297:
8224:
8202:
8166:
8129:
8107:
8052:
8025:
8005:
7975:
7943:
7913:
7883:
7863:
7835:
7709:
7506:{\displaystyle r^{2}\nabla ^{2}Y=-\ell (\ell +1)Y}
7505:
7346:is called a spherical harmonic function of degree
7327:
7203:
7028:; requiring the solution to be regular throughout
6847:
6724:
6614:
6309:
6031:
5990:
5874:
5770:
5441:
5361:
5254:
5099:
4876:
4823:
4758:
4654:
4528:
4478:. A similar argument shows that in two dimensions
4448:
4400:
4327:
4154:
4056:
3898:
3840:
3772:
3711:
3664:
3538:
3454:
3381:
3323:
3263:
3178:
2941:
2882:
2772:
2707:
2650:
2589:
2514:
2441:
2341:
2202:
2102:
2000:
1701:
1512:
1326:
1058:
1012:
816:
615:
583:
539:
494:
470:
443:
398:
366:
9806:Journal of Mathematical Analysis and Applications
4089:. No function has this property: in fact it is a
1587:
354:
10006:
9971:at EqWorld: The World of Mathematical Equations.
9870:
8411:In the particular case of a source-free region,
7768:
7541:is the expression in spherical coordinates of a
5922:
5510:
5362:{\displaystyle \rho '={\frac {a^{2}}{\rho }}.\,}
4412:that is centered on the source point, and hence
4073:denotes a unit source concentrated at the point
1832:for Laplace's equation specify not the function
1674:
21:Theory of tides § Laplace's tidal equations
9754:
3462:and the irrotationality condition implies that
3455:{\displaystyle \psi _{x}=v,\quad \psi _{y}=-u,}
2442:{\displaystyle f(z)=\varphi (x,y)+i\psi (x,y),}
2203:{\displaystyle u_{x}=v_{y},\quad v_{x}=-u_{y}.}
2019:both satisfy the Laplace equation. That is, if
9844:
9690:
8973:
8232:is also known as the electrostatic condition.
7871:, the solid harmonics with negative powers of
9901:. Providence: American Mathematical Society.
9705:
9703:
9701:
9699:
9672:. 4th ed., Pearson, 2013. Inner front cover.
6915:is regular at the poles of the sphere, where
2603:implies that the integrability condition for
298:
9929:. Boca Raton: Chapman & Hall/CRC Press.
7102:. These angular solutions are a product of
6925:. Imposing this regularity in the solution
5466:denotes the distance to the reflected point
4529:{\displaystyle u=-{\frac {\log(r)}{2\pi }}.}
3331:If we define the differential of a function
2810:in a power series inside a circle of radius
2773:{\displaystyle f(z)=\log z=\log r+i\theta .}
1693:
1677:
9871:Zachmanoglou, E. C.; Thoe, Dale W. (1986).
9662:
9660:
9658:
8978:S. Persides solved the Laplace equation in
4839:is any solution of the Poisson equation in
1861:Solutions of Laplace's equation are called
9861:
9724:
9696:
8800:
8796:
5912:is a harmonic function, then the value of
5299:
3484:. The Cauchy–Riemann equations imply that
2715:then a corresponding analytic function is
2015:The real and imaginary parts of a complex
305:
291:
16:Second-order partial differential equation
9915:
9864:Partial Differential Equations in Physics
9854:Methods of Mathematical Physics, Volume I
9825:
9768:
8895:{\displaystyle \nabla ^{2}V=4\pi G\rho ,}
8460:is specified on the boundary of a region
6039:with respect to the positive order ones.)
5756:
5453:denotes the distance to the source point
5438:
5358:
5251:
5241:
5208:
5129:. In view of the conditions satisfied by
5096:
5086:
5027:
4970:
4952:
4933:
4877:{\displaystyle \nabla \cdot \nabla u=-f,}
4254:
4214:
4136:
3760:
3747:
3712:{\displaystyle \nabla \cdot (u,v)=\rho ,}
3369:
3356:
2577:
2557:
9924:
9799:
9655:
8060:be the permittivity of free space. Then
5932:
2797:, which generally have less regularity.
1739:
10015:Elliptic partial differential equations
9975:Example initial-boundary value problems
9476:, a general case of Laplace's equation.
8167:{\displaystyle \mathbf {E} =-\nabla V,}
7557:linearly independent such polynomials.
7110:, and associated Legendre polynomials:
3918:
3773:{\displaystyle d\varphi =-u\,dx-v\,dy,}
2372:(at least locally). If a trial form is
631:elliptic partial differential equations
506:operator (also symbolized "grad"), and
10007:
9640:Zill, Dennis G, and Michael R Cullen.
9452:is an arbitrary non-negative integer.
6987:, whose solution is a multiple of the
6973:. Furthermore, a change of variables
4449:{\displaystyle u={\frac {1}{4\pi r}}.}
3913:
3206:Laplace equation for irrotational flow
2651:{\displaystyle \psi _{xy}=\psi _{yx},}
1735:
9981:
9896:
9620:Calculus : Early Transcendentals
8484:, then it is uniquely determined. If
2010:
1762:) with Dirichlet boundary conditions
667:Forms in different coordinate systems
9750:
9748:
9575:{\displaystyle \Delta x=x_{1}-x_{2}}
8033:be the electric charge density, and
7739:. Such an expansion is valid in the
2103:{\displaystyle f(z)=u(x,y)+iv(x,y),}
1877:
1724:relative to the new coordinates and
1059:{\displaystyle (r,\theta ,\varphi )}
7986:
7526:. In fact, for any such solution,
6954:for some non-negative integer with
6869:is a complex constant, but because
5937:Real (Laplace) spherical harmonics
5882:is the cosine of the angle between
4571:
3382:{\displaystyle d\psi =v\,dx-u\,dy,}
2942:{\displaystyle c_{n}=a_{n}+ib_{n}.}
2219:is the first partial derivative of
19:For Laplace's tidal equations, see
13:
9890:
9540:
9005:
8939:
8862:
8824:
8802:
8777:
8761:
8749:
8731:
8718:
8648:
8493:
8469:
8361:
8330:
8312:
8283:
8267:
8255:
8241:
8181:
8155:
8071:
7908:
7778:
7620:
7464:
7229:
6983:transforms this equation into the
6813:
6773:
6685:
6667:
6584:
6570:
6514:
6506:
6480:
6476:
6285:
6271:
6215:
6207:
6181:
6177:
6130:
6122:
6095:
6091:
6057:
5791:
5724:
5016:
5004:
4990:
4959:
4953:
4940:
4934:
4856:
4850:
4675:
4669:
4474:force, arising in the solution of
4208:
4202:
4130:
4124:
3934:
3679:
3578:
3278:
3091:
2987:
2852:
2110:then the necessary condition that
1944:
1930:
1907:
1893:
1634:
1626:
1571:
1567:
1528:
1484:
1454:
1446:
1406:
1398:
1374:
1370:
1353:
1302:
1288:
1232:
1224:
1198:
1194:
1147:
1139:
1112:
1108:
1074:
988:
974:
951:
937:
899:
891:
871:
867:
840:
792:
778:
755:
741:
718:
704:
685:
598:
489:
482:operator (also symbolized "div"),
462:
432:
425:
419:
413:
381:
345:
14:
10046:
9944:
9745:
7373:is a normalization constant, and
5998:would be shown rotated about the
4655:{\displaystyle G(x,y,z;x',y',z')}
3550:
2522:This relation does not determine
8986:. Using the canonical variables
8831:
8738:
8704:
8655:
8587:
8337:
8248:
8218:
8196:
8188:
8145:
8078:
8053:{\displaystyle \varepsilon _{0}}
7999:
4560:), which is the solution of the
3927:of Laplace's equation satisfies
3655:
3641:
3285:
2708:{\displaystyle \varphi =\log r,}
367:{\displaystyle \nabla ^{2}\!f=0}
272:
51:
9920:. Philadelphia: W. B. Saunders.
9793:
9757:Journal of Mathematical Physics
9732:Introduction to Electrodynamics
9711:Introduction to Electrodynamics
9669:Introduction to Electrodynamics
8963:{\displaystyle \nabla ^{2}V=0,}
8440:If the electrostatic potential
7951:), to match the terms and find
6445:
4809:
4787:
4781:
4744:
3815:
3513:
3429:
2485:
2170:
2133:be differentiable and that the
1664:
10020:Eponymous equations of physics
9918:Partial Differential Equations
9899:Partial Differential Equations
9683:
9634:
9611:
9585:
9527:
9313:
9307:
9299:
9284:
9263:
9253:
9193:
9184:
9158:
9148:
9136:
9126:
9120:
9114:
9066:
9054:
9041:
9035:
9026:
9008:
8797:
8767:
8755:
8574:
8501:{\displaystyle {\mathcal {R}}}
8477:{\displaystyle {\mathcal {R}}}
8273:
8261:
8174:if the field is irrotational,
7805:
7784:
7775:
7726:are constants and the factors
7701:
7689:
7598:
7580:
7497:
7485:
7388:, or polar angle, ranges from
7365:associated Legendre polynomial
7319:
7307:
7289:
7277:
7265:
7253:
7198:
7184:
7144:
7132:
6989:associated Legendre polynomial
6890:is necessarily an integer and
5991:{\displaystyle Y_{\ell }^{-m}}
5869:
5852:
5728:
5680:
5661:
5639:
5529:
5523:
5509:inside the sphere is given by(
5186:
5153:
4806:
4788:
4741:
4690:
4649:
4598:
4509:
4503:
4172:around the source point, then
4048:
3997:
3786:may be constructed to satisfy
3697:
3685:
3645:
3634:
3608:
3602:
3584:
3264:{\displaystyle u_{x}+v_{y}=0,}
3186:which is a Fourier series for
2965:
2959:
2830:
2824:
2731:
2725:
2433:
2421:
2409:
2397:
2388:
2382:
2327:
2313:
2298:
2284:
2269:
2252:
2094:
2082:
2070:
2058:
2049:
2043:
1696:
1665:
1604:
1596:
1558:
1550:
1053:
1035:
656:, the Laplace equation is the
578:
560:
534:
516:
1:
9604:
8982:on hypersurfaces of constant
7976:{\displaystyle f_{\ell }^{m}}
7416:, may assume all values with
7004:. Finally, the equation for
6032:{\displaystyle 90^{\circ }/m}
5923:Laplace's spherical harmonics
5123:denote normal derivatives on
3393:because it is constant along
3199:
1336:More generally, in arbitrary
471:{\displaystyle \nabla \cdot }
331:partial differential equation
9827:10.1016/0022-247X(73)90277-1
8601:be the gravitational field,
8594:{\displaystyle \mathbf {g} }
8225:{\displaystyle \mathbf {E} }
8006:{\displaystyle \mathbf {E} }
6877:whose period evenly divides
5511:Zachmanoglou & Thoe 1986
5141:, this result simplifies to
4892:assumes the boundary values
2670:conjugate harmonic functions
7:
9957:Encyclopedia of Mathematics
9866:. New York: Academic Press.
9455:
9102:spherical harmonic function
8974:In the Schwarzschild metric
7405:at the South Pole, and the
5377:is inside the sphere, then
3397:. The first derivatives of
2528:, but only its increments:
1830:Neumann boundary conditions
616:{\displaystyle \Delta f=h.}
399:{\displaystyle \Delta f=0,}
10:
10051:
9838:
9691:Courant & Hilbert 1962
8528:, and if the total charge
7446:of the eigenvalue problem
7010:has solutions of the form
6933:that forces the parameter
5926:
4174:Gauss's divergence theorem
3780:so the electric potential
3203:
2804:. If we expand a function
2680:are polar coordinates and
1872:principle of superposition
1838:itself on the boundary of
1790:
18:
9916:Petrovsky, I. G. (1967).
8210:. The irrotationality of
7914:{\displaystyle r=\infty }
6965:; this is also explained
4552:generated by a pointlike
2597:The Laplace equation for
1744:Laplace's equation on an
215:Geometric function theory
161:Cauchy's integral formula
151:Cauchy's integral theorem
9925:Polyanin, A. D. (2002).
9521:
7106:, here represented as a
7067:. For a given value of
6971:orbital angular momentum
6900:. The solution function
5473:Poisson integral formula
2135:Cauchy–Riemann equations
584:{\displaystyle h(x,y,z)}
540:{\displaystyle f(x,y,z)}
123:Cauchy–Riemann equations
9862:Sommerfeld, A. (1949).
8980:Schwarzschild spacetime
8924:{\displaystyle \rho =0}
8430:{\displaystyle \rho =0}
8013:be the electric field,
7427:. For a fixed integer
7104:trigonometric functions
6931:Sturm–Liouville problem
6360:separation of variables
3194:multiple angle formulae
1339:curvilinear coordinates
829:cylindrical coordinates
674:rectangular coordinates
495:{\displaystyle \nabla }
108:Complex-valued function
9576:
9405:
9076:
8964:
8925:
8896:
8846:
8684:
8635:
8621:the mass density, and
8615:
8595:
8562:
8542:
8522:
8502:
8478:
8454:
8431:
8403:
8345:
8299:
8226:
8204:
8168:
8131:
8109:
8054:
8027:
8007:
7977:
7945:
7915:
7885:
7865:
7864:{\displaystyle r>R}
7837:
7711:
7648:
7624:
7545:that is harmonic (see
7543:homogeneous polynomial
7507:
7392:at the North Pole, to
7329:
7205:
6849:
6726:
6616:
6311:
6043:Laplace's equation in
6040:
6033:
5992:
5876:
5772:
5443:
5363:
5318:that is at a distance
5256:
5101:
4878:
4825:
4760:
4656:
4530:
4450:
4408:on a sphere of radius
4402:
4329:
4156:
4058:
3900:
3842:
3774:
3713:
3666:
3540:
3456:
3383:
3325:
3265:
3180:
3095:
2991:
2943:
2884:
2856:
2774:
2709:
2652:
2591:
2516:
2443:
2343:
2204:
2104:
2002:
1793:Boundary value problem
1788:
1703:
1514:
1328:
1060:
1014:
818:
617:
585:
541:
496:
472:
445:
400:
368:
279:Mathematics portal
9897:Evans, L. C. (1998).
9800:Persides, S. (1973).
9577:
9406:
9077:
8965:
8926:
8897:
8847:
8685:
8636:
8616:
8614:{\displaystyle \rho }
8596:
8563:
8543:
8523:
8521:{\displaystyle \rho }
8503:
8479:
8455:
8432:
8404:
8346:
8300:
8227:
8205:
8169:
8132:
8110:
8055:
8028:
8026:{\displaystyle \rho }
8008:
7978:
7946:
7916:
7886:
7866:
7838:
7712:
7625:
7604:
7508:
7330:
7206:
6850:
6727:
6617:
6312:
6045:spherical coordinates
6034:
5993:
5936:
5877:
5773:
5492:for the source point
5490:spherical coordinates
5444:
5364:
5257:
5102:
4879:
4826:
4761:
4657:
4531:
4451:
4403:
4330:
4157:
4059:
3901:
3843:
3775:
3714:
3667:
3541:
3474:that is conjugate to
3457:
3384:
3326:
3266:
3181:
3075:
2971:
2944:
2885:
2836:
2775:
2710:
2653:
2592:
2517:
2444:
2344:
2205:
2105:
2003:
1743:
1704:
1515:
1329:
1061:
1025:spherical coordinates
1015:
819:
618:
586:
542:
497:
473:
446:
401:
369:
231:Augustin-Louis Cauchy
33:Mathematical analysis
10035:Pierre-Simon Laplace
9986:"Laplace's Equation"
9856:, Wiley-Interscience
9730:Griffiths, David J.
9709:Griffiths, David J.
9666:Griffiths, David J.
9537:
9516:Fundamental solution
9462:6-sphere coordinates
9446:Schwarzschild radius
9108:
9002:
8935:
8909:
8858:
8696:
8645:
8625:
8605:
8583:
8552:
8532:
8512:
8488:
8464:
8444:
8415:
8357:
8308:
8238:
8214:
8178:
8141:
8121:
8068:
8037:
8017:
7995:
7955:
7929:
7899:
7875:
7849:
7746:
7574:
7450:
7215:
7114:
6735:
6659:
6368:
6053:
6008:
5967:
5953:(top to bottom) and
5782:
5517:
5385:
5324:
5302:): the source point
5147:
4912:
4902:, then we may apply
4847:
4769:
4666:
4592:
4482:
4416:
4341:
4180:
4111:
4066:Dirac delta function
3931:
3925:fundamental solution
3919:Fundamental solution
3852:
3790:
3729:
3676:
3575:
3559:, an electric field
3488:
3407:
3341:
3275:
3226:
2953:
2894:
2818:
2719:
2684:
2613:
2532:
2453:
2376:
2233:
2141:
2121:be analytic is that
2037:
1886:
1524:
1349:
1070:
1032:
836:
681:
595:
554:
510:
486:
459:
410:
378:
341:
335:Pierre-Simon Laplace
241:Carl Friedrich Gauss
176:Isolated singularity
118:Holomorphic function
9875:. New York: Dover.
9818:1973JMAA...43..571P
9346:
9181:
7972:
7944:{\displaystyle r=0}
7802:
7688:
7663:
7399:at the Equator, to
7306:
7252:
7183:
7131:
7108:complex exponential
5987:
5632:
5617:
4566:incompressible flow
4564:in two-dimensional
3914:In three dimensions
3557:Maxwell's equations
3210:Let the quantities
2782:However, the angle
2229:. It follows that
1817:on the boundary of
1736:Boundary conditions
1730:Christoffel symbols
1500:
128:Formal power series
90:Unit complex number
10025:Harmonic functions
9983:Weisstein, Eric W.
9952:"Laplace equation"
9572:
9505:Earnshaw's theorem
9485:Quadrature domains
9480:Spherical harmonic
9474:Helmholtz equation
9433:Legendre functions
9401:
9326:
9167:
9072:
8960:
8921:
8892:
8842:
8840:
8680:
8631:
8611:
8591:
8558:
8538:
8518:
8498:
8474:
8450:
8427:
8399:
8341:
8295:
8222:
8200:
8164:
8127:
8105:
8050:
8023:
8003:
7973:
7958:
7941:
7911:
7881:
7861:
7833:
7788:
7782:
7707:
7674:
7649:
7562:linear combination
7515:linear combination
7503:
7325:
7292:
7238:
7201:
7169:
7117:
6939:to be of the form
6845:
6722:
6612:
6307:
6041:
6029:
5988:
5970:
5872:
5768:
5618:
5600:
5439:
5359:
5252:
5097:
4874:
4821:
4756:
4652:
4526:
4472:inverse-square law
4446:
4398:
4325:
4152:
4054:
3896:
3838:
3770:
3709:
3662:
3536:
3482:velocity potential
3452:
3379:
3321:
3261:
3176:
2939:
2880:
2814:, this means that
2770:
2705:
2648:
2587:
2512:
2439:
2339:
2200:
2100:
2011:Analytic functions
1998:
1863:harmonic functions
1789:
1699:
1510:
1483:
1324:
1056:
1010:
814:
652:. In the study of
646:harmonic functions
635:Helmholtz equation
627:Poisson's equation
613:
581:
537:
492:
468:
441:
396:
364:
329:is a second-order
327:Laplace's equation
206:Laplace's equation
186:Argument principle
9936:978-1-58488-299-2
9908:978-0-8218-0772-9
9779:10.1063/1.3702449
9740:978-1-108-42041-9
9719:978-1-108-42041-9
9678:978-1-108-42041-9
9650:978-1-111-82706-9
9629:978-0-538-49790-9
9500:Bateman transform
9391:
9348:
9243:
9200:
8634:{\displaystyle G}
8561:{\displaystyle V}
8541:{\displaystyle Q}
8453:{\displaystyle V}
8394:
8130:{\displaystyle V}
8100:
7884:{\displaystyle r}
7828:
7767:
7431:, every solution
6985:Legendre equation
6875:periodic function
6825:
6791:
6776:
6704:
6670:
6598:
6564:
6539:
6521:
6487:
6472:
6454:
6429:
6394:
6379:
6299:
6265:
6222:
6188:
6173:
6137:
6102:
6087:
5743:
5739:
5593:
5548:
5433:
5404:
5353:
4813:
4785:
4748:
4588:. For instance,
4546:natural logarithm
4521:
4441:
4393:
4362:
4337:It follows that
4303:
4252:
4099:positive operator
3648:
2017:analytic function
1958:
1921:
1878:In two dimensions
1844:normal derivative
1799:Dirichlet problem
1755:and outer radius
1720:is the Euclidean
1648:
1608:
1585:
1563:
1562:
1468:
1420:
1388:
1316:
1282:
1239:
1205:
1190:
1154:
1119:
1104:
1002:
965:
931:
906:
878:
863:
806:
769:
732:
315:
314:
201:Harmonic function
113:Analytic function
99:Complex functions
85:Complex conjugate
10042:
10030:Fourier analysis
9996:
9995:
9965:
9940:
9921:
9912:
9886:
9867:
9857:
9846:Courant, Richard
9832:
9831:
9829:
9797:
9791:
9790:
9772:
9752:
9743:
9728:
9722:
9707:
9694:
9693:, §V.8, §VII.5).
9687:
9681:
9664:
9653:
9638:
9632:
9617:Stewart, James.
9615:
9599:
9597:
9589:
9583:
9581:
9579:
9578:
9573:
9571:
9570:
9558:
9557:
9531:
9511:Vector Laplacian
9490:Potential theory
9451:
9448:. The parameter
9443:
9430:
9421:
9410:
9408:
9407:
9402:
9397:
9393:
9392:
9390:
9389:
9380:
9372:
9359:
9358:
9349:
9347:
9345:
9334:
9325:
9324:
9305:
9279:
9277:
9276:
9249:
9245:
9244:
9242:
9241:
9232:
9224:
9211:
9210:
9201:
9199:
9182:
9180:
9175:
9166:
9165:
9146:
9144:
9143:
9099:
9081:
9079:
9078:
9073:
9053:
9052:
8998:the solution is
8997:
8993:
8989:
8985:
8969:
8967:
8966:
8961:
8947:
8946:
8930:
8928:
8927:
8922:
8905:In empty space,
8901:
8899:
8898:
8893:
8870:
8869:
8851:
8849:
8848:
8843:
8841:
8834:
8810:
8809:
8785:
8784:
8741:
8707:
8689:
8687:
8686:
8681:
8658:
8640:
8638:
8637:
8632:
8620:
8618:
8617:
8612:
8600:
8598:
8597:
8592:
8590:
8568:is also unique.
8567:
8565:
8564:
8559:
8547:
8545:
8544:
8539:
8527:
8525:
8524:
8519:
8507:
8505:
8504:
8499:
8497:
8496:
8483:
8481:
8480:
8475:
8473:
8472:
8459:
8457:
8456:
8451:
8436:
8434:
8433:
8428:
8408:
8406:
8405:
8400:
8395:
8393:
8392:
8380:
8369:
8368:
8350:
8348:
8347:
8342:
8340:
8320:
8319:
8304:
8302:
8301:
8296:
8291:
8290:
8251:
8231:
8229:
8228:
8223:
8221:
8209:
8207:
8206:
8201:
8199:
8191:
8173:
8171:
8170:
8165:
8148:
8136:
8134:
8133:
8128:
8114:
8112:
8111:
8106:
8101:
8099:
8098:
8086:
8081:
8059:
8057:
8056:
8051:
8049:
8048:
8032:
8030:
8029:
8024:
8012:
8010:
8009:
8004:
8002:
7982:
7980:
7979:
7974:
7971:
7966:
7950:
7948:
7947:
7942:
7920:
7918:
7917:
7912:
7890:
7888:
7887:
7882:
7870:
7868:
7867:
7862:
7842:
7840:
7839:
7834:
7829:
7827:
7826:
7825:
7824:
7819:
7814:
7808:
7801:
7796:
7787:
7781:
7762:
7734:
7725:
7716:
7714:
7713:
7708:
7687:
7682:
7673:
7672:
7662:
7657:
7647:
7642:
7623:
7618:
7569:
7556:
7540:
7525:
7512:
7510:
7509:
7504:
7472:
7471:
7462:
7461:
7445:
7430:
7426:
7411:
7404:
7398:
7391:
7387:
7380:
7376:
7372:
7362:
7353:
7349:
7345:
7334:
7332:
7331:
7326:
7305:
7300:
7251:
7246:
7237:
7236:
7227:
7226:
7210:
7208:
7207:
7202:
7197:
7182:
7177:
7168:
7167:
7130:
7125:
7101:
7086:
7080:
7072:
7066:
7040:
7033:
7027:
7009:
7003:
6982:
6969:in terms of the
6964:
6953:
6938:
6928:
6924:
6914:
6899:
6893:
6889:
6883:
6872:
6868:
6862:
6857:for some number
6854:
6852:
6851:
6846:
6844:
6843:
6831:
6827:
6826:
6824:
6816:
6808:
6792:
6790:
6779:
6777:
6772:
6761:
6750:
6749:
6731:
6729:
6728:
6723:
6721:
6720:
6705:
6703:
6702:
6701:
6688:
6684:
6683:
6673:
6671:
6663:
6652:
6629:
6621:
6619:
6618:
6613:
6599:
6597:
6596:
6595:
6582:
6578:
6577:
6567:
6565:
6563:
6556:
6555:
6542:
6540:
6532:
6527:
6523:
6522:
6520:
6512:
6504:
6488:
6486:
6475:
6473:
6471:
6457:
6455:
6447:
6435:
6431:
6430:
6428:
6420:
6412:
6410:
6409:
6395:
6393:
6382:
6380:
6372:
6357:
6316:
6314:
6313:
6308:
6300:
6298:
6297:
6296:
6283:
6279:
6278:
6268:
6266:
6264:
6257:
6256:
6247:
6246:
6233:
6228:
6224:
6223:
6221:
6213:
6205:
6189:
6187:
6176:
6174:
6172:
6162:
6161:
6148:
6143:
6139:
6138:
6136:
6128:
6120:
6118:
6117:
6103:
6101:
6090:
6088:
6086:
6085:
6073:
6065:
6064:
6038:
6036:
6035:
6030:
6025:
6020:
6019:
6003:
5997:
5995:
5994:
5989:
5986:
5978:
5962:
5952:
5945:
5917:
5911:
5905:
5893:
5881:
5879:
5878:
5873:
5868:
5845:
5819:
5777:
5775:
5774:
5769:
5767:
5755:
5744:
5742:
5741:
5740:
5732:
5705:
5704:
5692:
5691:
5678:
5677:
5660:
5649:
5634:
5631:
5626:
5616:
5608:
5599:
5595:
5594:
5592:
5591:
5582:
5581:
5572:
5559:
5558:
5549:
5547:
5536:
5508:
5501:
5497:
5487:
5483:
5479:
5465:
5458:
5452:
5448:
5446:
5445:
5440:
5434:
5432:
5431:
5410:
5405:
5403:
5389:
5376:
5368:
5366:
5365:
5360:
5354:
5349:
5348:
5339:
5334:
5313:
5307:
5297:
5291:
5285:
5279:
5261:
5259:
5258:
5253:
5237:
5236:
5227:
5226:
5201:
5200:
5185:
5174:
5163:
5140:
5134:
5128:
5106:
5104:
5103:
5098:
5085:
5081:
5080:
5079:
5064:
5063:
5046:
5045:
5026:
5022:
4989:
4988:
4969:
4965:
4924:
4923:
4904:Green's identity
4901:
4897:
4891:
4883:
4881:
4880:
4875:
4842:
4838:
4830:
4828:
4827:
4822:
4814:
4811:
4786:
4783:
4765:
4763:
4762:
4757:
4749:
4746:
4740:
4723:
4706:
4661:
4659:
4658:
4653:
4648:
4637:
4626:
4587:
4583:
4578:Green's function
4572:Green's function
4543:
4535:
4533:
4532:
4527:
4522:
4520:
4512:
4495:
4476:Poisson equation
4455:
4453:
4452:
4447:
4442:
4440:
4426:
4411:
4407:
4405:
4404:
4399:
4394:
4392:
4391:
4390:
4371:
4363:
4361:
4353:
4345:
4334:
4332:
4331:
4326:
4321:
4320:
4309:
4305:
4304:
4302:
4294:
4286:
4284:
4283:
4253:
4251:
4243:
4235:
4233:
4232:
4201:
4200:
4171:
4167:
4161:
4159:
4158:
4153:
4123:
4122:
4106:
4088:
4072:
4063:
4061:
4060:
4055:
4047:
4030:
4013:
3987:
3986:
3971:
3970:
3955:
3954:
3908:Poisson equation
3905:
3903:
3902:
3897:
3883:
3882:
3867:
3866:
3847:
3845:
3844:
3839:
3825:
3824:
3802:
3801:
3785:
3779:
3777:
3776:
3771:
3724:
3718:
3716:
3715:
3710:
3671:
3669:
3668:
3663:
3658:
3650:
3649:
3644:
3639:
3633:
3632:
3620:
3619:
3570:
3545:
3543:
3542:
3537:
3523:
3522:
3500:
3499:
3479:
3473:
3467:
3461:
3459:
3458:
3453:
3439:
3438:
3419:
3418:
3402:
3388:
3386:
3385:
3380:
3336:
3330:
3328:
3327:
3322:
3314:
3313:
3301:
3300:
3288:
3270:
3268:
3267:
3262:
3251:
3250:
3238:
3237:
3221:
3215:
3191:
3185:
3183:
3182:
3177:
3172:
3168:
3155:
3154:
3145:
3144:
3120:
3119:
3110:
3109:
3094:
3089:
3068:
3064:
3051:
3050:
3041:
3040:
3016:
3015:
3006:
3005:
2990:
2985:
2948:
2946:
2945:
2940:
2935:
2934:
2919:
2918:
2906:
2905:
2889:
2887:
2886:
2881:
2876:
2875:
2866:
2865:
2855:
2850:
2813:
2809:
2785:
2779:
2777:
2776:
2771:
2714:
2712:
2711:
2706:
2679:
2675:
2663:
2657:
2655:
2654:
2649:
2644:
2643:
2628:
2627:
2608:
2602:
2596:
2594:
2593:
2588:
2576:
2575:
2556:
2555:
2527:
2521:
2519:
2518:
2513:
2508:
2507:
2495:
2494:
2481:
2480:
2465:
2464:
2448:
2446:
2445:
2440:
2371:
2360:
2354:
2348:
2346:
2345:
2340:
2335:
2334:
2325:
2324:
2306:
2305:
2296:
2295:
2277:
2276:
2267:
2266:
2248:
2247:
2228:
2225:with respect to
2224:
2218:
2209:
2207:
2206:
2201:
2196:
2195:
2180:
2179:
2166:
2165:
2153:
2152:
2132:
2126:
2120:
2109:
2107:
2106:
2101:
2032:
2007:
2005:
2004:
1999:
1991:
1990:
1975:
1974:
1959:
1957:
1956:
1955:
1942:
1938:
1937:
1927:
1922:
1920:
1919:
1918:
1905:
1901:
1900:
1890:
1857:
1851:
1841:
1837:
1820:
1816:
1810:
1806:
1787:
1772:
1761:
1754:
1727:
1719:
1708:
1706:
1705:
1700:
1692:
1691:
1654:
1650:
1649:
1647:
1646:
1645:
1632:
1624:
1622:
1621:
1609:
1607:
1599:
1594:
1586:
1584:
1583:
1582:
1566:
1564:
1561:
1553:
1548:
1544:
1536:
1535:
1519:
1517:
1516:
1511:
1499:
1494:
1482:
1481:
1469:
1467:
1466:
1465:
1452:
1444:
1439:
1435:
1434:
1433:
1421:
1419:
1418:
1417:
1404:
1396:
1389:
1387:
1386:
1385:
1369:
1361:
1360:
1344:
1333:
1331:
1330:
1325:
1317:
1315:
1314:
1313:
1300:
1296:
1295:
1285:
1283:
1281:
1274:
1273:
1264:
1263:
1250:
1245:
1241:
1240:
1238:
1230:
1222:
1206:
1204:
1193:
1191:
1189:
1179:
1178:
1165:
1160:
1156:
1155:
1153:
1145:
1137:
1135:
1134:
1120:
1118:
1107:
1105:
1103:
1102:
1090:
1082:
1081:
1065:
1063:
1062:
1057:
1019:
1017:
1016:
1011:
1003:
1001:
1000:
999:
986:
982:
981:
971:
966:
964:
963:
962:
949:
945:
944:
934:
932:
930:
929:
917:
912:
908:
907:
905:
897:
889:
879:
877:
866:
864:
856:
848:
847:
823:
821:
820:
815:
807:
805:
804:
803:
790:
786:
785:
775:
770:
768:
767:
766:
753:
749:
748:
738:
733:
731:
730:
729:
716:
712:
711:
701:
693:
692:
642:potential theory
622:
620:
619:
614:
590:
588:
587:
582:
546:
544:
543:
538:
501:
499:
498:
493:
477:
475:
474:
469:
453:Laplace operator
450:
448:
447:
442:
440:
439:
405:
403:
402:
397:
373:
371:
370:
365:
353:
352:
307:
300:
293:
277:
276:
261:Karl Weierstrass
256:Bernhard Riemann
246:Jacques Hadamard
75:Imaginary number
55:
45:Complex analysis
39:
37:Complex analysis
28:
27:
10050:
10049:
10045:
10044:
10043:
10041:
10040:
10039:
10005:
10004:
9950:
9947:
9937:
9909:
9893:
9891:Further reading
9883:
9841:
9836:
9835:
9798:
9794:
9753:
9746:
9729:
9725:
9708:
9697:
9688:
9684:
9665:
9656:
9639:
9635:
9616:
9612:
9607:
9602:
9592:
9590:
9586:
9566:
9562:
9553:
9549:
9538:
9535:
9534:
9532:
9528:
9524:
9458:
9449:
9441:
9436:
9428:
9423:
9419:
9414:
9385:
9381:
9373:
9371:
9364:
9360:
9354:
9350:
9335:
9330:
9320:
9316:
9306:
9280:
9278:
9266:
9262:
9237:
9233:
9225:
9223:
9216:
9212:
9206:
9202:
9183:
9176:
9171:
9161:
9157:
9147:
9145:
9139:
9135:
9109:
9106:
9105:
9088:
9083:
9048:
9044:
9003:
9000:
8999:
8995:
8991:
8987:
8983:
8976:
8942:
8938:
8936:
8933:
8932:
8910:
8907:
8906:
8865:
8861:
8859:
8856:
8855:
8839:
8838:
8830:
8814:
8805:
8801:
8793:
8792:
8780:
8776:
8742:
8737:
8728:
8727:
8708:
8703:
8699:
8697:
8694:
8693:
8654:
8646:
8643:
8642:
8626:
8623:
8622:
8606:
8603:
8602:
8586:
8584:
8581:
8580:
8577:
8553:
8550:
8549:
8548:is known, then
8533:
8530:
8529:
8513:
8510:
8509:
8492:
8491:
8489:
8486:
8485:
8468:
8467:
8465:
8462:
8461:
8445:
8442:
8441:
8416:
8413:
8412:
8388:
8384:
8379:
8364:
8360:
8358:
8355:
8354:
8336:
8315:
8311:
8309:
8306:
8305:
8286:
8282:
8247:
8239:
8236:
8235:
8217:
8215:
8212:
8211:
8195:
8187:
8179:
8176:
8175:
8144:
8142:
8139:
8138:
8122:
8119:
8118:
8094:
8090:
8085:
8077:
8069:
8066:
8065:
8044:
8040:
8038:
8035:
8034:
8018:
8015:
8014:
7998:
7996:
7993:
7992:
7989:
7967:
7962:
7956:
7953:
7952:
7930:
7927:
7926:
7900:
7897:
7896:
7876:
7873:
7872:
7850:
7847:
7846:
7820:
7815:
7810:
7809:
7804:
7803:
7797:
7792:
7783:
7771:
7766:
7761:
7747:
7744:
7743:
7737:solid harmonics
7732:
7727:
7723:
7718:
7683:
7678:
7668:
7664:
7658:
7653:
7643:
7629:
7619:
7608:
7575:
7572:
7571:
7565:
7550:
7527:
7523:
7518:
7467:
7463:
7457:
7453:
7451:
7448:
7447:
7432:
7428:
7417:
7409:
7400:
7393:
7389:
7385:
7378:
7374:
7368:
7360:
7355:
7351:
7347:
7343:
7338:
7301:
7296:
7247:
7242:
7232:
7228:
7222:
7218:
7216:
7213:
7212:
7193:
7178:
7173:
7157:
7153:
7126:
7121:
7115:
7112:
7111:
7088:
7082:
7074:
7068:
7045:
7035:
7029:
7011:
7005:
6996:
6991:
6974:
6955:
6940:
6934:
6926:
6916:
6901:
6895:
6891:
6885:
6878:
6870:
6864:
6858:
6839:
6835:
6817:
6809:
6807:
6797:
6793:
6783:
6778:
6762:
6760:
6745:
6741:
6736:
6733:
6732:
6716:
6712:
6697:
6693:
6689:
6679:
6675:
6674:
6672:
6662:
6660:
6657:
6656:
6631:
6625:
6591:
6587:
6583:
6573:
6569:
6568:
6566:
6551:
6547:
6546:
6541:
6531:
6513:
6505:
6503:
6493:
6489:
6479:
6474:
6461:
6456:
6446:
6421:
6413:
6411:
6405:
6401:
6400:
6396:
6386:
6381:
6371:
6369:
6366:
6365:
6320:
6292:
6288:
6284:
6274:
6270:
6269:
6267:
6252:
6248:
6242:
6238:
6237:
6232:
6214:
6206:
6204:
6194:
6190:
6180:
6175:
6157:
6153:
6152:
6147:
6129:
6121:
6119:
6113:
6109:
6108:
6104:
6094:
6089:
6081:
6077:
6072:
6060:
6056:
6054:
6051:
6050:
6021:
6015:
6011:
6009:
6006:
6005:
5999:
5979:
5974:
5968:
5965:
5964:
5954:
5947:
5943:
5938:
5931:
5925:
5913:
5907:
5895:
5883:
5861:
5838:
5812:
5783:
5780:
5779:
5760:
5748:
5731:
5727:
5700:
5696:
5687:
5683:
5679:
5670:
5653:
5642:
5635:
5633:
5627:
5622:
5609:
5604:
5587:
5583:
5577:
5573:
5571:
5564:
5560:
5554:
5550:
5540:
5535:
5518:
5515:
5514:
5504:
5499:
5493:
5485:
5481:
5477:
5460:
5454:
5450:
5424:
5414:
5409:
5393:
5388:
5386:
5383:
5382:
5372:
5344:
5340:
5338:
5327:
5325:
5322:
5321:
5309:
5303:
5300:Sommerfeld 1949
5293:
5287:
5281:
5265:
5232:
5228:
5222:
5218:
5196:
5192:
5178:
5167:
5156:
5148:
5145:
5144:
5136:
5130:
5124:
5121:
5114:
5075:
5071:
5059:
5055:
5051:
5047:
5041:
5037:
5000:
4996:
4984:
4980:
4929:
4925:
4919:
4915:
4913:
4910:
4909:
4899:
4893:
4887:
4848:
4845:
4844:
4840:
4834:
4810:
4782:
4770:
4767:
4766:
4745:
4733:
4716:
4699:
4667:
4664:
4663:
4641:
4630:
4619:
4593:
4590:
4589:
4585:
4581:
4574:
4562:Euler equations
4537:
4513:
4496:
4494:
4483:
4480:
4479:
4466:generated by a
4462:), this is the
4430:
4425:
4417:
4414:
4413:
4409:
4386:
4382:
4375:
4370:
4354:
4346:
4344:
4342:
4339:
4338:
4310:
4295:
4287:
4285:
4279:
4275:
4268:
4265:
4264:
4244:
4236:
4234:
4228:
4224:
4196:
4192:
4181:
4178:
4177:
4169:
4165:
4118:
4114:
4112:
4109:
4108:
4102:
4074:
4068:
4040:
4023:
4006:
3979:
3975:
3963:
3959:
3947:
3943:
3932:
3929:
3928:
3921:
3916:
3875:
3871:
3859:
3855:
3853:
3850:
3849:
3820:
3816:
3797:
3793:
3791:
3788:
3787:
3781:
3730:
3727:
3726:
3720:
3677:
3674:
3673:
3654:
3640:
3638:
3637:
3628:
3624:
3615:
3611:
3576:
3573:
3572:
3560:
3553:
3518:
3514:
3495:
3491:
3489:
3486:
3485:
3475:
3469:
3463:
3434:
3430:
3414:
3410:
3408:
3405:
3404:
3398:
3391:stream function
3342:
3339:
3338:
3332:
3309:
3305:
3296:
3292:
3284:
3276:
3273:
3272:
3246:
3242:
3233:
3229:
3227:
3224:
3223:
3217:
3211:
3208:
3202:
3187:
3150:
3146:
3140:
3136:
3115:
3111:
3105:
3101:
3100:
3096:
3090:
3079:
3046:
3042:
3036:
3032:
3011:
3007:
3001:
2997:
2996:
2992:
2986:
2975:
2954:
2951:
2950:
2930:
2926:
2914:
2910:
2901:
2897:
2895:
2892:
2891:
2871:
2867:
2861:
2857:
2851:
2840:
2819:
2816:
2815:
2811:
2805:
2783:
2720:
2717:
2716:
2685:
2682:
2681:
2677:
2673:
2666:Stokes' theorem
2659:
2636:
2632:
2620:
2616:
2614:
2611:
2610:
2604:
2598:
2571:
2567:
2551:
2547:
2533:
2530:
2529:
2523:
2503:
2499:
2490:
2486:
2476:
2472:
2460:
2456:
2454:
2451:
2450:
2377:
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2362:
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2350:
2330:
2326:
2320:
2316:
2301:
2297:
2291:
2287:
2272:
2268:
2262:
2258:
2240:
2236:
2234:
2231:
2230:
2226:
2220:
2216:
2211:
2191:
2187:
2175:
2171:
2161:
2157:
2148:
2144:
2142:
2139:
2138:
2128:
2122:
2111:
2038:
2035:
2034:
2020:
2013:
1983:
1979:
1967:
1963:
1951:
1947:
1943:
1933:
1929:
1928:
1926:
1914:
1910:
1906:
1896:
1892:
1891:
1889:
1887:
1884:
1883:
1880:
1865:; they are all
1853:
1847:
1839:
1833:
1818:
1812:
1808:
1807:on some domain
1802:
1795:
1774:
1763:
1756:
1749:
1738:
1725:
1718:
1710:
1684:
1680:
1641:
1637:
1633:
1625:
1623:
1614:
1610:
1603:
1595:
1593:
1592:
1588:
1578:
1574:
1570:
1565:
1557:
1549:
1543:
1531:
1527:
1525:
1522:
1521:
1495:
1487:
1474:
1470:
1461:
1457:
1453:
1445:
1443:
1426:
1422:
1413:
1409:
1405:
1397:
1395:
1394:
1390:
1381:
1377:
1373:
1368:
1356:
1352:
1350:
1347:
1346:
1342:
1309:
1305:
1301:
1291:
1287:
1286:
1284:
1269:
1265:
1259:
1255:
1254:
1249:
1231:
1223:
1221:
1211:
1207:
1197:
1192:
1174:
1170:
1169:
1164:
1146:
1138:
1136:
1130:
1126:
1125:
1121:
1111:
1106:
1098:
1094:
1089:
1077:
1073:
1071:
1068:
1067:
1033:
1030:
1029:
995:
991:
987:
977:
973:
972:
970:
958:
954:
950:
940:
936:
935:
933:
925:
921:
916:
898:
890:
888:
884:
880:
870:
865:
855:
843:
839:
837:
834:
833:
799:
795:
791:
781:
777:
776:
774:
762:
758:
754:
744:
740:
739:
737:
725:
721:
717:
707:
703:
702:
700:
688:
684:
682:
679:
678:
669:
654:heat conduction
625:This is called
596:
593:
592:
555:
552:
551:
511:
508:
507:
487:
484:
483:
460:
457:
456:
435:
431:
411:
408:
407:
379:
376:
375:
348:
344:
342:
339:
338:
311:
271:
181:Residue theorem
156:Local primitive
146:Zeros and poles
61:Complex numbers
31:
24:
17:
12:
11:
5:
10048:
10038:
10037:
10032:
10027:
10022:
10017:
10003:
10002:
9997:
9978:
9972:
9966:
9946:
9945:External links
9943:
9942:
9941:
9935:
9922:
9913:
9907:
9892:
9889:
9888:
9887:
9881:
9868:
9859:
9850:Hilbert, David
9840:
9837:
9834:
9833:
9812:(3): 571–578.
9792:
9744:
9723:
9695:
9682:
9654:
9633:
9609:
9608:
9606:
9603:
9601:
9600:
9584:
9569:
9565:
9561:
9556:
9552:
9548:
9545:
9542:
9525:
9523:
9520:
9519:
9518:
9513:
9508:
9502:
9497:
9495:Potential flow
9492:
9487:
9482:
9477:
9471:
9457:
9454:
9439:
9426:
9417:
9400:
9396:
9388:
9384:
9379:
9376:
9370:
9367:
9363:
9357:
9353:
9344:
9341:
9338:
9333:
9329:
9323:
9319:
9315:
9312:
9309:
9304:
9301:
9298:
9295:
9292:
9289:
9286:
9283:
9275:
9272:
9269:
9265:
9261:
9258:
9255:
9252:
9248:
9240:
9236:
9231:
9228:
9222:
9219:
9215:
9209:
9205:
9198:
9195:
9192:
9189:
9186:
9179:
9174:
9170:
9164:
9160:
9156:
9153:
9150:
9142:
9138:
9134:
9131:
9128:
9125:
9122:
9119:
9116:
9113:
9086:
9071:
9068:
9065:
9062:
9059:
9056:
9051:
9047:
9043:
9040:
9037:
9034:
9031:
9028:
9025:
9022:
9019:
9016:
9013:
9010:
9007:
8975:
8972:
8959:
8956:
8953:
8950:
8945:
8941:
8920:
8917:
8914:
8891:
8888:
8885:
8882:
8879:
8876:
8873:
8868:
8864:
8837:
8833:
8829:
8826:
8823:
8820:
8817:
8815:
8813:
8808:
8804:
8799:
8795:
8794:
8791:
8788:
8783:
8779:
8775:
8772:
8769:
8766:
8763:
8760:
8757:
8754:
8751:
8748:
8745:
8743:
8740:
8736:
8733:
8730:
8729:
8726:
8723:
8720:
8717:
8714:
8711:
8709:
8706:
8702:
8701:
8679:
8676:
8673:
8670:
8667:
8664:
8661:
8657:
8653:
8650:
8630:
8610:
8589:
8576:
8573:
8557:
8537:
8517:
8495:
8471:
8449:
8426:
8423:
8420:
8398:
8391:
8387:
8383:
8378:
8375:
8372:
8367:
8363:
8339:
8335:
8332:
8329:
8326:
8323:
8318:
8314:
8294:
8289:
8285:
8281:
8278:
8275:
8272:
8269:
8266:
8263:
8260:
8257:
8254:
8250:
8246:
8243:
8220:
8198:
8194:
8190:
8186:
8183:
8163:
8160:
8157:
8154:
8151:
8147:
8126:
8104:
8097:
8093:
8089:
8084:
8080:
8076:
8073:
8047:
8043:
8022:
8001:
7988:
7987:Electrostatics
7985:
7970:
7965:
7961:
7940:
7937:
7934:
7921:), instead of
7910:
7907:
7904:
7893:Laurent series
7880:
7860:
7857:
7854:
7832:
7823:
7818:
7813:
7807:
7800:
7795:
7791:
7786:
7780:
7777:
7774:
7770:
7769:lim sup
7765:
7760:
7757:
7754:
7751:
7730:
7721:
7706:
7703:
7700:
7697:
7694:
7691:
7686:
7681:
7677:
7671:
7667:
7661:
7656:
7652:
7646:
7641:
7638:
7635:
7632:
7628:
7622:
7617:
7614:
7611:
7607:
7603:
7600:
7597:
7594:
7591:
7588:
7585:
7582:
7579:
7521:
7502:
7499:
7496:
7493:
7490:
7487:
7484:
7481:
7478:
7475:
7470:
7466:
7460:
7456:
7358:
7341:
7324:
7321:
7318:
7315:
7312:
7309:
7304:
7299:
7295:
7291:
7288:
7285:
7282:
7279:
7276:
7273:
7270:
7267:
7264:
7261:
7258:
7255:
7250:
7245:
7241:
7235:
7231:
7225:
7221:
7211:which fulfill
7200:
7196:
7192:
7189:
7186:
7181:
7176:
7172:
7166:
7163:
7160:
7156:
7152:
7149:
7146:
7143:
7140:
7137:
7134:
7129:
7124:
7120:
6994:
6842:
6838:
6834:
6830:
6823:
6820:
6815:
6812:
6806:
6803:
6800:
6796:
6789:
6786:
6782:
6775:
6771:
6768:
6765:
6759:
6756:
6753:
6748:
6744:
6740:
6719:
6715:
6711:
6708:
6700:
6696:
6692:
6687:
6682:
6678:
6669:
6666:
6611:
6608:
6605:
6602:
6594:
6590:
6586:
6581:
6576:
6572:
6562:
6559:
6554:
6550:
6545:
6538:
6535:
6530:
6526:
6519:
6516:
6511:
6508:
6502:
6499:
6496:
6492:
6485:
6482:
6478:
6470:
6467:
6464:
6460:
6453:
6450:
6444:
6441:
6438:
6434:
6427:
6424:
6419:
6416:
6408:
6404:
6399:
6392:
6389:
6385:
6378:
6375:
6306:
6303:
6295:
6291:
6287:
6282:
6277:
6273:
6263:
6260:
6255:
6251:
6245:
6241:
6236:
6231:
6227:
6220:
6217:
6212:
6209:
6203:
6200:
6197:
6193:
6186:
6183:
6179:
6171:
6168:
6165:
6160:
6156:
6151:
6146:
6142:
6135:
6132:
6127:
6124:
6116:
6112:
6107:
6100:
6097:
6093:
6084:
6080:
6076:
6071:
6068:
6063:
6059:
6028:
6024:
6018:
6014:
5985:
5982:
5977:
5973:
5941:
5927:Main article:
5924:
5921:
5871:
5867:
5864:
5860:
5857:
5854:
5851:
5848:
5844:
5841:
5837:
5834:
5831:
5828:
5825:
5822:
5818:
5815:
5811:
5808:
5805:
5802:
5799:
5796:
5793:
5790:
5787:
5766:
5763:
5759:
5754:
5751:
5747:
5738:
5735:
5730:
5726:
5723:
5720:
5717:
5714:
5711:
5708:
5703:
5699:
5695:
5690:
5686:
5682:
5676:
5673:
5669:
5666:
5663:
5659:
5656:
5652:
5648:
5645:
5641:
5638:
5630:
5625:
5621:
5615:
5612:
5607:
5603:
5598:
5590:
5586:
5580:
5576:
5570:
5567:
5563:
5557:
5553:
5546:
5543:
5539:
5534:
5531:
5528:
5525:
5522:
5437:
5430:
5427:
5423:
5420:
5417:
5413:
5408:
5402:
5399:
5396:
5392:
5357:
5352:
5347:
5343:
5337:
5333:
5330:
5250:
5247:
5244:
5240:
5235:
5231:
5225:
5221:
5217:
5214:
5211:
5207:
5204:
5199:
5195:
5191:
5188:
5184:
5181:
5177:
5173:
5170:
5166:
5162:
5159:
5155:
5152:
5119:
5112:
5109:The notations
5095:
5092:
5089:
5084:
5078:
5074:
5070:
5067:
5062:
5058:
5054:
5050:
5044:
5040:
5036:
5033:
5030:
5025:
5021:
5018:
5015:
5012:
5009:
5006:
5003:
4999:
4995:
4992:
4987:
4983:
4979:
4976:
4973:
4968:
4964:
4961:
4958:
4955:
4951:
4948:
4945:
4942:
4939:
4936:
4932:
4928:
4922:
4918:
4873:
4870:
4867:
4864:
4861:
4858:
4855:
4852:
4820:
4817:
4808:
4805:
4802:
4799:
4796:
4793:
4790:
4780:
4777:
4774:
4755:
4752:
4743:
4739:
4736:
4732:
4729:
4726:
4722:
4719:
4715:
4712:
4709:
4705:
4702:
4698:
4695:
4692:
4689:
4686:
4683:
4680:
4677:
4674:
4671:
4651:
4647:
4644:
4640:
4636:
4633:
4629:
4625:
4622:
4618:
4615:
4612:
4609:
4606:
4603:
4600:
4597:
4573:
4570:
4558:point particle
4525:
4519:
4516:
4511:
4508:
4505:
4502:
4499:
4493:
4490:
4487:
4468:point particle
4445:
4439:
4436:
4433:
4429:
4424:
4421:
4397:
4389:
4385:
4381:
4378:
4374:
4369:
4366:
4360:
4357:
4352:
4349:
4324:
4319:
4316:
4313:
4308:
4301:
4298:
4293:
4290:
4282:
4278:
4274:
4271:
4267:
4263:
4260:
4257:
4250:
4247:
4242:
4239:
4231:
4227:
4223:
4220:
4217:
4213:
4210:
4207:
4204:
4199:
4195:
4191:
4188:
4185:
4151:
4148:
4145:
4142:
4139:
4135:
4132:
4129:
4126:
4121:
4117:
4053:
4050:
4046:
4043:
4039:
4036:
4033:
4029:
4026:
4022:
4019:
4016:
4012:
4009:
4005:
4002:
3999:
3996:
3993:
3990:
3985:
3982:
3978:
3974:
3969:
3966:
3962:
3958:
3953:
3950:
3946:
3942:
3939:
3936:
3920:
3917:
3915:
3912:
3895:
3892:
3889:
3886:
3881:
3878:
3874:
3870:
3865:
3862:
3858:
3837:
3834:
3831:
3828:
3823:
3819:
3814:
3811:
3808:
3805:
3800:
3796:
3769:
3766:
3763:
3759:
3756:
3753:
3750:
3746:
3743:
3740:
3737:
3734:
3708:
3705:
3702:
3699:
3696:
3693:
3690:
3687:
3684:
3681:
3661:
3657:
3653:
3647:
3643:
3636:
3631:
3627:
3623:
3618:
3614:
3610:
3607:
3604:
3601:
3598:
3595:
3592:
3589:
3586:
3583:
3580:
3552:
3551:Electrostatics
3549:
3535:
3532:
3529:
3526:
3521:
3517:
3512:
3509:
3506:
3503:
3498:
3494:
3480:is called the
3451:
3448:
3445:
3442:
3437:
3433:
3428:
3425:
3422:
3417:
3413:
3378:
3375:
3372:
3368:
3365:
3362:
3359:
3355:
3352:
3349:
3346:
3320:
3317:
3312:
3308:
3304:
3299:
3295:
3291:
3287:
3283:
3280:
3260:
3257:
3254:
3249:
3245:
3241:
3236:
3232:
3204:Main article:
3201:
3198:
3175:
3171:
3167:
3164:
3161:
3158:
3153:
3149:
3143:
3139:
3135:
3132:
3129:
3126:
3123:
3118:
3114:
3108:
3104:
3099:
3093:
3088:
3085:
3082:
3078:
3074:
3071:
3067:
3063:
3060:
3057:
3054:
3049:
3045:
3039:
3035:
3031:
3028:
3025:
3022:
3019:
3014:
3010:
3004:
3000:
2995:
2989:
2984:
2981:
2978:
2974:
2970:
2967:
2964:
2961:
2958:
2938:
2933:
2929:
2925:
2922:
2917:
2913:
2909:
2904:
2900:
2879:
2874:
2870:
2864:
2860:
2854:
2849:
2846:
2843:
2839:
2835:
2832:
2829:
2826:
2823:
2802:Fourier series
2769:
2766:
2763:
2760:
2757:
2754:
2751:
2748:
2745:
2742:
2739:
2736:
2733:
2730:
2727:
2724:
2704:
2701:
2698:
2695:
2692:
2689:
2647:
2642:
2639:
2635:
2631:
2626:
2623:
2619:
2609:is satisfied:
2586:
2583:
2580:
2574:
2570:
2566:
2563:
2560:
2554:
2550:
2546:
2543:
2540:
2537:
2511:
2506:
2502:
2498:
2493:
2489:
2484:
2479:
2475:
2471:
2468:
2463:
2459:
2438:
2435:
2432:
2429:
2426:
2423:
2420:
2417:
2414:
2411:
2408:
2405:
2402:
2399:
2396:
2393:
2390:
2387:
2384:
2381:
2338:
2333:
2329:
2323:
2319:
2315:
2312:
2309:
2304:
2300:
2294:
2290:
2286:
2283:
2280:
2275:
2271:
2265:
2261:
2257:
2254:
2251:
2246:
2243:
2239:
2214:
2199:
2194:
2190:
2186:
2183:
2178:
2174:
2169:
2164:
2160:
2156:
2151:
2147:
2137:be satisfied:
2099:
2096:
2093:
2090:
2087:
2084:
2081:
2078:
2075:
2072:
2069:
2066:
2063:
2060:
2057:
2054:
2051:
2048:
2045:
2042:
2012:
2009:
1997:
1994:
1989:
1986:
1982:
1978:
1973:
1970:
1966:
1962:
1954:
1950:
1946:
1941:
1936:
1932:
1925:
1917:
1913:
1909:
1904:
1899:
1895:
1879:
1876:
1782:=4) = 4 sin(5
1748:(inner radius
1737:
1734:
1714:
1698:
1695:
1690:
1687:
1683:
1679:
1676:
1673:
1670:
1667:
1663:
1660:
1657:
1653:
1644:
1640:
1636:
1631:
1628:
1620:
1617:
1613:
1606:
1602:
1598:
1591:
1581:
1577:
1573:
1569:
1560:
1556:
1552:
1547:
1542:
1539:
1534:
1530:
1509:
1506:
1503:
1498:
1493:
1490:
1486:
1480:
1477:
1473:
1464:
1460:
1456:
1451:
1448:
1442:
1438:
1432:
1429:
1425:
1416:
1412:
1408:
1403:
1400:
1393:
1384:
1380:
1376:
1372:
1367:
1364:
1359:
1355:
1323:
1320:
1312:
1308:
1304:
1299:
1294:
1290:
1280:
1277:
1272:
1268:
1262:
1258:
1253:
1248:
1244:
1237:
1234:
1229:
1226:
1220:
1217:
1214:
1210:
1203:
1200:
1196:
1188:
1185:
1182:
1177:
1173:
1168:
1163:
1159:
1152:
1149:
1144:
1141:
1133:
1129:
1124:
1117:
1114:
1110:
1101:
1097:
1093:
1088:
1085:
1080:
1076:
1055:
1052:
1049:
1046:
1043:
1040:
1037:
1009:
1006:
998:
994:
990:
985:
980:
976:
969:
961:
957:
953:
948:
943:
939:
928:
924:
920:
915:
911:
904:
901:
896:
893:
887:
883:
876:
873:
869:
862:
859:
854:
851:
846:
842:
813:
810:
802:
798:
794:
789:
784:
780:
773:
765:
761:
757:
752:
747:
743:
736:
728:
724:
720:
715:
710:
706:
699:
696:
691:
687:
668:
665:
650:fluid dynamics
612:
609:
606:
603:
600:
580:
577:
574:
571:
568:
565:
562:
559:
536:
533:
530:
527:
524:
521:
518:
515:
491:
467:
464:
438:
434:
430:
427:
424:
421:
418:
415:
395:
392:
389:
386:
383:
363:
360:
357:
351:
347:
313:
312:
310:
309:
302:
295:
287:
284:
283:
282:
281:
266:
265:
264:
263:
258:
253:
248:
243:
238:
236:Leonhard Euler
233:
225:
224:
218:
217:
211:
210:
209:
208:
203:
198:
193:
188:
183:
178:
173:
171:Laurent series
168:
166:Winding number
163:
158:
153:
148:
140:
139:
133:
132:
131:
130:
125:
120:
115:
110:
102:
101:
95:
94:
93:
92:
87:
82:
77:
72:
64:
63:
57:
56:
48:
47:
41:
40:
15:
9:
6:
4:
3:
2:
10047:
10036:
10033:
10031:
10028:
10026:
10023:
10021:
10018:
10016:
10013:
10012:
10010:
10001:
9998:
9993:
9992:
9987:
9984:
9979:
9976:
9973:
9970:
9967:
9963:
9959:
9958:
9953:
9949:
9948:
9938:
9932:
9928:
9923:
9919:
9914:
9910:
9904:
9900:
9895:
9894:
9884:
9882:9780486652511
9878:
9874:
9869:
9865:
9860:
9855:
9851:
9847:
9843:
9842:
9828:
9823:
9819:
9815:
9811:
9807:
9803:
9796:
9788:
9784:
9780:
9776:
9771:
9766:
9763:(4): 042501.
9762:
9758:
9751:
9749:
9741:
9737:
9733:
9727:
9720:
9716:
9712:
9706:
9704:
9702:
9700:
9692:
9686:
9679:
9675:
9671:
9670:
9663:
9661:
9659:
9651:
9647:
9643:
9637:
9630:
9626:
9622:
9621:
9614:
9610:
9595:
9588:
9567:
9563:
9559:
9554:
9550:
9546:
9543:
9530:
9526:
9517:
9514:
9512:
9509:
9506:
9503:
9501:
9498:
9496:
9493:
9491:
9488:
9486:
9483:
9481:
9478:
9475:
9472:
9470:
9468:
9463:
9460:
9459:
9453:
9447:
9442:
9434:
9429:
9420:
9411:
9398:
9394:
9386:
9382:
9377:
9374:
9368:
9365:
9361:
9355:
9351:
9342:
9339:
9336:
9331:
9327:
9321:
9317:
9310:
9302:
9296:
9293:
9290:
9287:
9281:
9273:
9270:
9267:
9259:
9256:
9250:
9246:
9238:
9234:
9229:
9226:
9220:
9217:
9213:
9207:
9203:
9196:
9190:
9187:
9177:
9172:
9168:
9162:
9154:
9151:
9140:
9132:
9129:
9123:
9117:
9111:
9103:
9097:
9093:
9089:
9069:
9063:
9060:
9057:
9049:
9045:
9038:
9032:
9029:
9023:
9020:
9017:
9014:
9011:
8981:
8971:
8957:
8954:
8951:
8948:
8943:
8918:
8915:
8912:
8903:
8889:
8886:
8883:
8880:
8877:
8874:
8871:
8866:
8852:
8835:
8827:
8821:
8818:
8816:
8811:
8806:
8789:
8786:
8781:
8773:
8770:
8764:
8758:
8752:
8746:
8744:
8734:
8724:
8721:
8715:
8712:
8710:
8690:
8677:
8674:
8671:
8668:
8665:
8662:
8659:
8651:
8628:
8608:
8572:
8569:
8555:
8535:
8515:
8447:
8438:
8424:
8421:
8418:
8409:
8396:
8389:
8385:
8381:
8376:
8373:
8370:
8365:
8351:
8333:
8327:
8324:
8321:
8316:
8292:
8287:
8279:
8276:
8270:
8264:
8258:
8252:
8244:
8233:
8192:
8184:
8161:
8158:
8152:
8149:
8124:
8115:
8102:
8095:
8091:
8087:
8082:
8074:
8063:
8045:
8041:
8020:
7984:
7968:
7963:
7959:
7938:
7935:
7932:
7924:
7923:Taylor series
7905:
7902:
7894:
7878:
7858:
7855:
7852:
7843:
7830:
7821:
7816:
7811:
7798:
7793:
7789:
7772:
7763:
7758:
7755:
7752:
7749:
7742:
7738:
7735:are known as
7733:
7724:
7704:
7698:
7695:
7692:
7684:
7679:
7675:
7669:
7665:
7659:
7654:
7650:
7644:
7639:
7636:
7633:
7630:
7626:
7615:
7612:
7609:
7605:
7601:
7595:
7592:
7589:
7586:
7583:
7577:
7568:
7563:
7558:
7554:
7548:
7544:
7538:
7534:
7530:
7524:
7516:
7500:
7494:
7491:
7488:
7482:
7479:
7476:
7473:
7468:
7458:
7454:
7443:
7439:
7435:
7425:
7421:
7415:
7408:
7403:
7396:
7384:
7371:
7366:
7361:
7344:
7335:
7322:
7316:
7313:
7310:
7302:
7297:
7293:
7286:
7283:
7280:
7274:
7271:
7268:
7262:
7259:
7256:
7248:
7243:
7239:
7233:
7223:
7219:
7194:
7190:
7187:
7179:
7174:
7170:
7164:
7161:
7158:
7154:
7150:
7147:
7141:
7138:
7135:
7127:
7122:
7118:
7109:
7105:
7100:
7096:
7092:
7085:
7078:
7071:
7064:
7060:
7056:
7052:
7048:
7042:
7038:
7032:
7026:
7022:
7018:
7014:
7008:
7001:
6997:
6990:
6986:
6981:
6977:
6972:
6968:
6962:
6958:
6951:
6947:
6943:
6937:
6932:
6923:
6919:
6912:
6908:
6904:
6898:
6888:
6882:
6876:
6867:
6861:
6855:
6840:
6836:
6832:
6828:
6821:
6818:
6810:
6804:
6801:
6798:
6794:
6787:
6784:
6780:
6769:
6766:
6763:
6757:
6754:
6751:
6746:
6742:
6738:
6717:
6713:
6709:
6706:
6698:
6694:
6690:
6680:
6676:
6664:
6654:
6650:
6646:
6642:
6638:
6634:
6630:has the form
6628:
6622:
6609:
6606:
6603:
6600:
6592:
6588:
6579:
6574:
6560:
6557:
6552:
6548:
6543:
6536:
6533:
6528:
6524:
6517:
6509:
6500:
6497:
6494:
6490:
6483:
6468:
6465:
6462:
6458:
6451:
6448:
6442:
6439:
6436:
6432:
6425:
6422:
6417:
6414:
6406:
6402:
6397:
6390:
6387:
6383:
6376:
6373:
6363:
6361:
6355:
6351:
6347:
6343:
6339:
6335:
6331:
6327:
6323:
6317:
6304:
6301:
6293:
6289:
6280:
6275:
6261:
6258:
6253:
6249:
6243:
6239:
6234:
6229:
6225:
6218:
6210:
6201:
6198:
6195:
6191:
6184:
6169:
6166:
6163:
6158:
6154:
6149:
6144:
6140:
6133:
6125:
6114:
6110:
6105:
6098:
6082:
6078:
6074:
6069:
6066:
6061:
6048:
6046:
6026:
6022:
6016:
6012:
6002:
5983:
5980:
5975:
5971:
5961:
5957:
5950:
5944:
5935:
5930:
5920:
5916:
5910:
5903:
5899:
5891:
5887:
5865:
5862:
5858:
5855:
5849:
5846:
5842:
5839:
5835:
5832:
5829:
5826:
5823:
5820:
5816:
5813:
5809:
5806:
5803:
5800:
5797:
5794:
5788:
5785:
5764:
5761:
5757:
5752:
5749:
5745:
5736:
5733:
5721:
5718:
5715:
5712:
5709:
5706:
5701:
5697:
5693:
5688:
5684:
5674:
5671:
5667:
5664:
5657:
5654:
5650:
5646:
5643:
5636:
5628:
5623:
5619:
5613:
5610:
5605:
5601:
5596:
5588:
5584:
5578:
5574:
5568:
5565:
5561:
5555:
5551:
5544:
5541:
5537:
5532:
5526:
5520:
5512:
5507:
5496:
5491:
5475:
5474:
5469:
5463:
5457:
5435:
5428:
5425:
5421:
5418:
5415:
5411:
5406:
5400:
5397:
5394:
5390:
5380:
5375:
5371:Note that if
5369:
5355:
5350:
5345:
5341:
5335:
5331:
5328:
5319:
5317:
5312:
5306:
5301:
5296:
5290:
5284:
5277:
5273:
5269:
5262:
5248:
5245:
5242:
5238:
5233:
5229:
5223:
5219:
5215:
5212:
5209:
5205:
5202:
5197:
5193:
5189:
5182:
5179:
5175:
5171:
5168:
5164:
5160:
5157:
5150:
5142:
5139:
5133:
5127:
5122:
5115:
5107:
5093:
5090:
5087:
5082:
5076:
5072:
5068:
5065:
5060:
5056:
5052:
5048:
5042:
5038:
5034:
5031:
5028:
5023:
5019:
5013:
5010:
5007:
5001:
4997:
4993:
4985:
4981:
4977:
4974:
4971:
4966:
4962:
4956:
4949:
4946:
4943:
4937:
4930:
4926:
4920:
4916:
4907:
4905:
4896:
4890:
4884:
4871:
4868:
4865:
4862:
4859:
4853:
4837:
4831:
4818:
4815:
4803:
4800:
4797:
4794:
4791:
4778:
4775:
4772:
4753:
4750:
4737:
4734:
4730:
4727:
4724:
4720:
4717:
4713:
4710:
4707:
4703:
4700:
4696:
4693:
4687:
4684:
4681:
4678:
4672:
4645:
4642:
4638:
4634:
4631:
4627:
4623:
4620:
4616:
4613:
4610:
4607:
4604:
4601:
4595:
4579:
4569:
4567:
4563:
4559:
4555:
4551:
4547:
4541:
4523:
4517:
4514:
4506:
4500:
4497:
4491:
4488:
4485:
4477:
4473:
4469:
4465:
4461:
4456:
4443:
4437:
4434:
4431:
4427:
4422:
4419:
4395:
4387:
4383:
4379:
4376:
4372:
4367:
4364:
4358:
4355:
4350:
4347:
4335:
4322:
4317:
4314:
4311:
4306:
4299:
4296:
4291:
4288:
4280:
4276:
4272:
4269:
4261:
4258:
4255:
4248:
4245:
4240:
4237:
4229:
4225:
4221:
4218:
4215:
4211:
4205:
4197:
4193:
4189:
4186:
4183:
4176:implies that
4175:
4162:
4149:
4146:
4143:
4140:
4137:
4133:
4127:
4119:
4115:
4105:
4100:
4096:
4095:weak solution
4092:
4086:
4082:
4078:
4071:
4067:
4051:
4044:
4041:
4037:
4034:
4031:
4027:
4024:
4020:
4017:
4014:
4010:
4007:
4003:
4000:
3994:
3991:
3988:
3983:
3980:
3976:
3972:
3967:
3964:
3960:
3956:
3951:
3948:
3944:
3940:
3937:
3926:
3911:
3909:
3906:which is the
3893:
3890:
3887:
3884:
3879:
3876:
3872:
3868:
3863:
3860:
3856:
3835:
3832:
3829:
3826:
3821:
3817:
3812:
3809:
3806:
3803:
3798:
3794:
3784:
3767:
3764:
3761:
3757:
3754:
3751:
3748:
3744:
3741:
3738:
3735:
3732:
3723:
3706:
3703:
3700:
3694:
3691:
3688:
3682:
3659:
3651:
3629:
3625:
3621:
3616:
3612:
3605:
3599:
3596:
3593:
3590:
3587:
3581:
3568:
3564:
3558:
3555:According to
3548:
3533:
3530:
3527:
3524:
3519:
3515:
3510:
3507:
3504:
3501:
3496:
3492:
3483:
3478:
3472:
3466:
3449:
3446:
3443:
3440:
3435:
3431:
3426:
3423:
3420:
3415:
3411:
3403:are given by
3401:
3396:
3392:
3376:
3373:
3370:
3366:
3363:
3360:
3357:
3353:
3350:
3347:
3344:
3335:
3318:
3315:
3310:
3306:
3302:
3297:
3293:
3289:
3281:
3258:
3255:
3252:
3247:
3243:
3239:
3234:
3230:
3220:
3214:
3207:
3197:
3195:
3190:
3173:
3169:
3165:
3162:
3159:
3156:
3151:
3147:
3141:
3137:
3133:
3130:
3127:
3124:
3121:
3116:
3112:
3106:
3102:
3097:
3086:
3083:
3080:
3076:
3072:
3069:
3065:
3061:
3058:
3055:
3052:
3047:
3043:
3037:
3033:
3029:
3026:
3023:
3020:
3017:
3012:
3008:
3002:
2998:
2993:
2982:
2979:
2976:
2972:
2968:
2962:
2956:
2936:
2931:
2927:
2923:
2920:
2915:
2911:
2907:
2902:
2898:
2877:
2872:
2868:
2862:
2858:
2847:
2844:
2841:
2837:
2833:
2827:
2821:
2808:
2803:
2798:
2796:
2795:wave equation
2792:
2787:
2780:
2767:
2764:
2761:
2758:
2755:
2752:
2749:
2746:
2743:
2740:
2737:
2734:
2728:
2722:
2702:
2699:
2696:
2693:
2690:
2687:
2671:
2667:
2662:
2645:
2640:
2637:
2633:
2629:
2624:
2621:
2617:
2607:
2601:
2584:
2581:
2578:
2572:
2568:
2564:
2561:
2558:
2552:
2548:
2544:
2541:
2538:
2535:
2526:
2509:
2504:
2500:
2496:
2491:
2487:
2482:
2477:
2473:
2469:
2466:
2461:
2457:
2436:
2430:
2427:
2424:
2418:
2415:
2412:
2406:
2403:
2400:
2394:
2391:
2385:
2379:
2369:
2365:
2359:
2353:
2336:
2331:
2321:
2317:
2310:
2307:
2302:
2292:
2288:
2281:
2278:
2273:
2263:
2259:
2255:
2249:
2244:
2241:
2237:
2223:
2217:
2197:
2192:
2188:
2184:
2181:
2176:
2172:
2167:
2162:
2158:
2154:
2149:
2145:
2136:
2131:
2125:
2118:
2114:
2097:
2091:
2088:
2085:
2079:
2076:
2073:
2067:
2064:
2061:
2055:
2052:
2046:
2040:
2031:
2027:
2023:
2018:
2008:
1995:
1992:
1987:
1984:
1980:
1976:
1971:
1968:
1964:
1960:
1952:
1948:
1939:
1934:
1923:
1915:
1911:
1902:
1897:
1875:
1873:
1868:
1864:
1859:
1856:
1850:
1845:
1836:
1831:
1826:
1824:
1823:heat equation
1815:
1805:
1800:
1794:
1785:
1781:
1777:
1770:
1766:
1759:
1752:
1747:
1742:
1733:
1731:
1723:
1722:metric tensor
1717:
1713:
1688:
1685:
1681:
1671:
1668:
1661:
1658:
1655:
1651:
1642:
1638:
1629:
1618:
1615:
1611:
1600:
1589:
1579:
1575:
1554:
1545:
1540:
1537:
1532:
1507:
1504:
1501:
1496:
1491:
1488:
1478:
1475:
1471:
1462:
1458:
1449:
1440:
1436:
1430:
1427:
1423:
1414:
1410:
1401:
1391:
1382:
1378:
1365:
1362:
1357:
1341:
1340:
1334:
1321:
1318:
1310:
1306:
1297:
1292:
1278:
1275:
1270:
1266:
1260:
1256:
1251:
1246:
1242:
1235:
1227:
1218:
1215:
1212:
1208:
1201:
1186:
1183:
1180:
1175:
1171:
1166:
1161:
1157:
1150:
1142:
1131:
1127:
1122:
1115:
1099:
1095:
1091:
1086:
1083:
1078:
1050:
1047:
1044:
1041:
1038:
1027:
1026:
1020:
1007:
1004:
996:
992:
983:
978:
967:
959:
955:
946:
941:
926:
922:
918:
913:
909:
902:
894:
885:
881:
874:
860:
857:
852:
849:
844:
831:
830:
824:
811:
808:
800:
796:
787:
782:
771:
763:
759:
750:
745:
734:
726:
722:
713:
708:
697:
694:
689:
677:
675:
664:
662:
661:heat equation
659:
655:
651:
647:
643:
638:
636:
632:
628:
623:
610:
607:
604:
601:
575:
572:
569:
566:
563:
557:
548:
531:
528:
525:
522:
519:
513:
505:
481:
465:
454:
436:
428:
422:
416:
393:
390:
387:
384:
361:
358:
355:
349:
336:
332:
328:
324:
320:
308:
303:
301:
296:
294:
289:
288:
286:
285:
280:
275:
270:
269:
268:
267:
262:
259:
257:
254:
252:
249:
247:
244:
242:
239:
237:
234:
232:
229:
228:
227:
226:
223:
220:
219:
216:
213:
212:
207:
204:
202:
199:
197:
196:Schwarz lemma
194:
192:
191:Conformal map
189:
187:
184:
182:
179:
177:
174:
172:
169:
167:
164:
162:
159:
157:
154:
152:
149:
147:
144:
143:
142:
141:
138:
135:
134:
129:
126:
124:
121:
119:
116:
114:
111:
109:
106:
105:
104:
103:
100:
97:
96:
91:
88:
86:
83:
81:
80:Complex plane
78:
76:
73:
71:
68:
67:
66:
65:
62:
59:
58:
54:
50:
49:
46:
43:
42:
38:
34:
30:
29:
26:
22:
9989:
9955:
9926:
9917:
9898:
9872:
9863:
9853:
9809:
9805:
9795:
9760:
9756:
9731:
9726:
9710:
9685:
9667:
9641:
9636:
9618:
9613:
9593:
9587:
9529:
9466:
9437:
9424:
9415:
9412:
9095:
9091:
9084:
8977:
8931:and we have
8904:
8853:
8691:
8578:
8570:
8439:
8410:
8352:
8234:
8116:
7990:
7844:
7728:
7719:
7566:
7559:
7552:
7536:
7532:
7528:
7519:
7441:
7437:
7433:
7423:
7419:
7401:
7394:
7369:
7356:
7339:
7336:
7098:
7094:
7090:
7083:
7076:
7073:, there are
7069:
7062:
7058:
7054:
7050:
7046:
7043:
7036:
7030:
7024:
7020:
7016:
7012:
7006:
6999:
6992:
6979:
6975:
6960:
6956:
6949:
6945:
6941:
6935:
6921:
6917:
6910:
6906:
6902:
6896:
6886:
6880:
6865:
6863:. A priori,
6859:
6856:
6655:
6648:
6644:
6640:
6636:
6632:
6626:
6623:
6364:
6353:
6349:
6345:
6341:
6337:
6333:
6329:
6325:
6321:
6318:
6049:
6042:
6000:
5959:
5955:
5948:
5939:
5914:
5908:
5901:
5897:
5889:
5885:
5505:
5494:
5471:
5467:
5461:
5455:
5378:
5373:
5370:
5320:
5315:
5310:
5308:at distance
5304:
5294:
5288:
5282:
5280:of the data
5275:
5271:
5267:
5263:
5143:
5137:
5131:
5125:
5117:
5110:
5108:
4908:
4894:
4888:
4885:
4835:
4832:
4662:may satisfy
4584:of a volume
4575:
4544:denotes the
4539:
4457:
4336:
4163:
4103:
4091:distribution
4084:
4080:
4076:
4069:
3922:
3782:
3721:
3566:
3562:
3554:
3476:
3470:
3464:
3399:
3333:
3218:
3212:
3209:
3188:
2806:
2799:
2791:power series
2788:
2781:
2669:
2660:
2605:
2599:
2524:
2367:
2363:
2357:
2351:
2221:
2212:
2129:
2123:
2116:
2112:
2029:
2025:
2021:
2014:
1881:
1860:
1854:
1848:
1834:
1827:
1813:
1803:
1796:
1783:
1779:
1775:
1768:
1764:
1757:
1750:
1728:denotes its
1715:
1711:
1337:
1335:
1066:convention,
1028:, using the
1023:
1021:
827:
825:
672:
670:
658:steady-state
639:
624:
549:
333:named after
326:
316:
205:
137:Basic theory
36:
25:
8575:Gravitation
8062:Gauss's law
5951:= 0, ..., 4
319:mathematics
251:Kiyoshi Oka
70:Real number
10009:Categories
9605:References
9469:-separable
7717:where the
7383:colatitude
7350:and order
6873:must be a
5958:= 0, ...,
5513:, p. 228)
4064:where the
3395:flow lines
3200:Fluid flow
2949:Therefore
2349:Therefore
2033:, and if
1811:such that
1791:See also:
591:, we have
480:divergence
9991:MathWorld
9962:EMS Press
9787:118707082
9770:1111.4702
9560:−
9541:Δ
9369:−
9257:−
9221:−
9130:−
9064:φ
9058:θ
9024:φ
9018:θ
9006:Ψ
8940:∇
8913:ρ
8887:ρ
8881:π
8863:∇
8828:⋅
8825:∇
8822:−
8803:∇
8798:⟹
8778:∇
8774:−
8762:∇
8759:−
8753:⋅
8750:∇
8735:⋅
8732:∇
8719:∇
8716:−
8675:ρ
8669:π
8663:−
8652:⋅
8649:∇
8609:ρ
8516:ρ
8419:ρ
8386:ε
8382:ρ
8377:−
8362:∇
8334:⋅
8331:∇
8328:−
8313:∇
8284:∇
8280:−
8268:∇
8265:−
8259:⋅
8256:∇
8245:⋅
8242:∇
8185:×
8182:∇
8156:∇
8153:−
8092:ε
8088:ρ
8075:⋅
8072:∇
8042:ε
8021:ρ
7964:ℓ
7909:∞
7822:ℓ
7794:ℓ
7779:∞
7776:→
7773:ℓ
7699:φ
7693:θ
7680:ℓ
7670:ℓ
7655:ℓ
7645:ℓ
7640:ℓ
7637:−
7627:∑
7621:∞
7610:ℓ
7606:∑
7596:φ
7590:θ
7489:ℓ
7483:ℓ
7480:−
7465:∇
7407:longitude
7317:φ
7311:θ
7298:ℓ
7281:ℓ
7275:ℓ
7272:−
7263:φ
7257:θ
7244:ℓ
7230:∇
7195:θ
7191:
7175:ℓ
7165:φ
7142:φ
7136:θ
7123:ℓ
6822:θ
6814:Θ
6805:θ
6802:
6788:θ
6774:Θ
6770:θ
6767:
6755:θ
6752:
6739:λ
6710:−
6695:φ
6686:Φ
6668:Φ
6607:λ
6604:−
6589:φ
6585:∂
6571:∂
6561:θ
6558:
6518:θ
6515:∂
6507:∂
6501:θ
6498:
6484:θ
6481:∂
6477:∂
6469:θ
6466:
6440:λ
6290:φ
6286:∂
6272:∂
6262:θ
6259:
6219:θ
6216:∂
6208:∂
6202:θ
6199:
6185:θ
6182:∂
6178:∂
6170:θ
6167:
6131:∂
6123:∂
6096:∂
6092:∂
6058:∇
6017:∘
5981:−
5976:ℓ
5863:φ
5859:−
5856:φ
5850:
5840:θ
5836:
5830:θ
5827:
5814:θ
5810:
5804:θ
5801:
5792:Θ
5789:
5762:φ
5750:θ
5725:Θ
5722:
5716:ρ
5707:−
5698:ρ
5672:θ
5668:
5655:φ
5644:θ
5629:π
5620:∫
5614:π
5602:∫
5575:ρ
5569:−
5545:π
5422:ρ
5419:π
5407:−
5398:π
5351:ρ
5329:ρ
5220:∬
5194:∭
5066:−
5039:∬
5017:∇
5011:−
5005:∇
4994:⋅
4991:∇
4982:∭
4960:∇
4957:⋅
4954:∇
4947:−
4941:∇
4938:⋅
4935:∇
4917:∭
4866:−
4857:∇
4854:⋅
4851:∇
4731:−
4714:−
4697:−
4688:δ
4685:−
4676:∇
4673:⋅
4670:∇
4550:potential
4518:π
4501:
4492:−
4470:, for an
4464:potential
4435:π
4380:π
4368:−
4273:π
4226:∬
4209:∇
4206:⋅
4203:∇
4194:∭
4184:−
4147:−
4131:∇
4128:⋅
4125:∇
4116:∭
4038:−
4021:−
4004:−
3995:δ
3992:−
3935:Δ
3891:ρ
3888:−
3873:φ
3857:φ
3830:−
3818:φ
3807:−
3795:φ
3755:−
3742:−
3736:φ
3704:ρ
3683:⋅
3680:∇
3646:^
3622:−
3582:×
3579:∇
3528:−
3516:φ
3505:−
3493:φ
3444:−
3432:ψ
3412:ψ
3364:−
3348:ψ
3303:−
3282:×
3279:∇
3166:θ
3160:
3131:θ
3125:
3092:∞
3077:∑
3062:θ
3056:
3030:−
3027:θ
3021:
2988:∞
2973:∑
2853:∞
2838:∑
2765:θ
2753:
2741:
2697:
2688:φ
2658:and thus
2634:ψ
2618:ψ
2569:φ
2549:φ
2545:−
2539:ψ
2501:φ
2488:ψ
2474:φ
2470:−
2458:ψ
2419:ψ
2395:φ
2311:−
2282:−
2256:−
2185:−
1981:ψ
1965:ψ
1961:≡
1945:∂
1940:ψ
1931:∂
1908:∂
1903:ψ
1894:∂
1858:is zero.
1639:ξ
1635:∂
1627:∂
1576:ξ
1572:∂
1568:∂
1529:∇
1485:Γ
1459:ξ
1455:∂
1447:∂
1411:ξ
1407:∂
1399:∂
1379:ξ
1375:∂
1371:∂
1354:∇
1307:φ
1303:∂
1289:∂
1279:θ
1276:
1236:θ
1233:∂
1225:∂
1219:θ
1216:
1202:θ
1199:∂
1195:∂
1187:θ
1184:
1148:∂
1140:∂
1113:∂
1109:∂
1075:∇
1051:φ
1045:θ
989:∂
975:∂
956:ϕ
952:∂
938:∂
900:∂
892:∂
872:∂
868:∂
841:∇
793:∂
779:∂
756:∂
742:∂
719:∂
705:∂
686:∇
599:Δ
490:∇
466:⋅
463:∇
433:∇
426:∇
423:⋅
420:∇
414:Δ
382:Δ
346:∇
9852:(1962),
9456:See also
6004:axis by
5866:′
5843:′
5817:′
5765:′
5753:′
5675:′
5658:′
5647:′
5498:. Here
5429:′
5379:P′
5332:′
5183:′
5172:′
5161:′
4812:on
4747:in
4738:′
4721:′
4704:′
4646:′
4635:′
4624:′
4045:′
4028:′
4011:′
1867:analytic
1842:but its
1343:(ξ)
504:gradient
9964:, 2001
9839:Sources
9814:Bibcode
9444:is the
7925:(about
7895:(about
7414:azimuth
7034:forces
4833:Now if
4460:physics
1771:=2) = 0
1746:annulus
502:is the
478:is the
451:is the
323:physics
9933:
9905:
9879:
9785:
9738:
9717:
9676:
9648:
9627:
9104:, and
9082:where
7422:< 2
7363:is an
7057:) = Θ(
6978:= cos
6643:) = Θ(
6358:. By
5778:where
5484:, and
5476:. Let
5449:where
4536:where
3719:where
2210:where
1726:Γ
1709:where
406:where
222:People
9783:S2CID
9765:arXiv
9522:Notes
9413:Here
9100:is a
7547:below
7513:is a
7412:, or
7337:Here
7087:with
6998:(cos
6967:below
6920:= 0,
4556:(see
9931:ISBN
9903:ISBN
9877:ISBN
9736:ISBN
9715:ISBN
9674:ISBN
9646:ISBN
9625:ISBN
9431:are
9422:and
8579:Let
7991:Let
7856:>
7845:For
7753:<
7741:ball
7418:0 ≤
7377:and
7061:) Φ(
7019:) =
6952:+ 1)
6647:) Φ(
6336:) =
6047:is:
5946:for
5894:and
5459:and
5286:and
5135:and
5116:and
4886:and
4554:sink
4538:log(
3672:and
3216:and
2676:and
2127:and
1828:The
1797:The
1773:and
1520:or
321:and
9822:doi
9775:doi
9596:= 0
7729:r Y
7555:+ 1
7529:r Y
7517:of
7188:cos
7079:+ 1
7039:= 0
7025:B r
7021:A r
6959:≥ |
6799:sin
6764:sin
6743:sin
6549:sin
6495:sin
6463:sin
6250:sin
6196:sin
6164:sin
5900:′,
5847:cos
5833:sin
5824:sin
5807:cos
5798:cos
5786:cos
5719:cos
5665:sin
5488:be
5274:′,
5270:′,
4898:on
4498:log
4083:′,
4079:′,
3337:by
3157:cos
3122:sin
3053:sin
3018:cos
2750:log
2738:log
2694:log
1760:= 4
1753:= 2
1675:det
1267:sin
1213:sin
1181:sin
1022:In
826:In
671:In
374:or
317:In
10011::
9988:.
9960:,
9954:,
9848:;
9820:.
9810:43
9808:.
9804:.
9781:.
9773:.
9761:53
9759:.
9747:^
9698:^
9657:^
9094:,
8994:,
8990:,
8137:,
7983:.
7570:,
7535:,
7440:,
7397:/2
7367:,
7354:,
7097:≤
7093:≤
7053:,
7041:.
7023:+
6944:=
6909:,
6884:,
6639:,
6352:,
6344:)
6332:,
6328:,
6305:0.
6013:90
5904:′)
5888:,
5480:,
5316:P'
5278:′)
4843::
4784:if
4576:A
4568:.
4150:1.
4087:′)
3923:A
3565:,
3319:0.
3196:.
2030:iy
2028:+
2024:=
1996:0.
1732:.
1716:ij
1345:,
1322:0.
1008:0.
832:,
812:0.
637:.
455:,
325:,
35:→
9994:.
9939:.
9911:.
9885:.
9858:.
9830:.
9824::
9816::
9789:.
9777::
9767::
9742:.
9721:.
9680:.
9652:.
9631:.
9594:A
9568:2
9564:x
9555:1
9551:x
9547:=
9544:x
9467:R
9450:l
9440:s
9438:r
9427:l
9425:Q
9418:l
9416:P
9399:.
9395:)
9387:s
9383:r
9378:r
9375:2
9366:1
9362:(
9356:l
9352:Q
9343:1
9340:+
9337:l
9332:s
9328:r
9322:2
9318:!
9314:)
9311:l
9308:(
9303:!
9300:)
9297:1
9294:+
9291:l
9288:2
9285:(
9282:2
9274:1
9271:+
9268:l
9264:)
9260:1
9254:(
9251:+
9247:)
9239:s
9235:r
9230:r
9227:2
9218:1
9214:(
9208:l
9204:P
9197:!
9194:)
9191:l
9188:2
9185:(
9178:l
9173:s
9169:r
9163:2
9159:)
9155:!
9152:l
9149:(
9141:l
9137:)
9133:1
9127:(
9124:=
9121:)
9118:r
9115:(
9112:R
9098:)
9096:φ
9092:θ
9090:(
9087:l
9085:Y
9070:,
9067:)
9061:,
9055:(
9050:l
9046:Y
9042:)
9039:r
9036:(
9033:R
9030:=
9027:)
9021:,
9015:,
9012:r
9009:(
8996:φ
8992:θ
8988:r
8984:t
8958:,
8955:0
8952:=
8949:V
8944:2
8919:0
8916:=
8890:,
8884:G
8878:4
8875:=
8872:V
8867:2
8836:.
8832:g
8819:=
8812:V
8807:2
8790:,
8787:V
8782:2
8771:=
8768:)
8765:V
8756:(
8747:=
8739:g
8725:,
8722:V
8713:=
8705:g
8678:.
8672:G
8666:4
8660:=
8656:g
8629:G
8588:g
8556:V
8536:Q
8494:R
8470:R
8448:V
8425:0
8422:=
8397:.
8390:0
8374:=
8371:V
8366:2
8338:E
8325:=
8322:V
8317:2
8293:V
8288:2
8277:=
8274:)
8271:V
8262:(
8253:=
8249:E
8219:E
8197:0
8193:=
8189:E
8162:,
8159:V
8150:=
8146:E
8125:V
8103:.
8096:0
8083:=
8079:E
8046:0
8000:E
7969:m
7960:f
7939:0
7936:=
7933:r
7906:=
7903:r
7879:r
7859:R
7853:r
7831:.
7817:/
7812:1
7806:|
7799:m
7790:f
7785:|
7764:1
7759:=
7756:R
7750:r
7731:ℓ
7722:ℓ
7720:f
7705:,
7702:)
7696:,
7690:(
7685:m
7676:Y
7666:r
7660:m
7651:f
7634:=
7631:m
7616:0
7613:=
7602:=
7599:)
7593:,
7587:,
7584:r
7581:(
7578:f
7567:r
7553:ℓ
7551:2
7539:)
7537:φ
7533:θ
7531:(
7522:ℓ
7520:Y
7501:Y
7498:)
7495:1
7492:+
7486:(
7477:=
7474:Y
7469:2
7459:2
7455:r
7444:)
7442:φ
7438:θ
7436:(
7434:Y
7429:ℓ
7424:π
7420:φ
7410:φ
7402:π
7395:π
7390:0
7386:θ
7379:φ
7375:θ
7370:N
7359:ℓ
7357:P
7352:m
7348:ℓ
7342:ℓ
7340:Y
7323:.
7320:)
7314:,
7308:(
7303:m
7294:Y
7290:)
7287:1
7284:+
7278:(
7269:=
7266:)
7260:,
7254:(
7249:m
7240:Y
7234:2
7224:2
7220:r
7199:)
7185:(
7180:m
7171:P
7162:m
7159:i
7155:e
7151:N
7148:=
7145:)
7139:,
7133:(
7128:m
7119:Y
7099:ℓ
7095:m
7091:ℓ
7089:−
7084:m
7077:ℓ
7075:2
7070:ℓ
7065:)
7063:φ
7059:θ
7055:φ
7051:θ
7049:(
7047:Y
7037:B
7031:R
7017:r
7015:(
7013:R
7007:R
7002:)
7000:θ
6995:ℓ
6993:P
6980:θ
6976:t
6963:|
6961:m
6957:ℓ
6950:ℓ
6948:(
6946:ℓ
6942:λ
6936:λ
6927:Θ
6922:π
6918:θ
6913:)
6911:φ
6907:θ
6905:(
6903:Y
6897:e
6892:Φ
6887:m
6881:π
6879:2
6871:Φ
6866:m
6860:m
6841:2
6837:m
6833:=
6829:)
6819:d
6811:d
6795:(
6785:d
6781:d
6758:+
6747:2
6718:2
6714:m
6707:=
6699:2
6691:d
6681:2
6677:d
6665:1
6651:)
6649:φ
6645:θ
6641:φ
6637:θ
6635:(
6633:Y
6627:Y
6610:.
6601:=
6593:2
6580:Y
6575:2
6553:2
6544:1
6537:Y
6534:1
6529:+
6525:)
6510:Y
6491:(
6459:1
6452:Y
6449:1
6443:,
6437:=
6433:)
6426:r
6423:d
6418:R
6415:d
6407:2
6403:r
6398:(
6391:r
6388:d
6384:d
6377:R
6374:1
6356:)
6354:φ
6350:θ
6348:(
6346:Y
6342:r
6340:(
6338:R
6334:φ
6330:θ
6326:r
6324:(
6322:f
6302:=
6294:2
6281:f
6276:2
6254:2
6244:2
6240:r
6235:1
6230:+
6226:)
6211:f
6192:(
6159:2
6155:r
6150:1
6145:+
6141:)
6134:r
6126:f
6115:2
6111:r
6106:(
6099:r
6083:2
6079:r
6075:1
6070:=
6067:f
6062:2
6027:m
6023:/
6001:z
5984:m
5972:Y
5960:ℓ
5956:m
5949:ℓ
5942:ℓ
5940:Y
5915:u
5909:u
5902:φ
5898:θ
5896:(
5892:)
5890:φ
5886:θ
5884:(
5870:)
5853:(
5821:+
5795:=
5758:d
5746:d
5737:2
5734:3
5729:)
5713:a
5710:2
5702:2
5694:+
5689:2
5685:a
5681:(
5662:)
5651:,
5640:(
5637:g
5624:0
5611:2
5606:0
5597:)
5589:2
5585:a
5579:2
5566:1
5562:(
5556:3
5552:a
5542:4
5538:1
5533:=
5530:)
5527:P
5524:(
5521:u
5506:g
5500:θ
5495:P
5486:φ
5482:θ
5478:ρ
5468:P
5464:′
5462:R
5456:P
5451:R
5436:,
5426:R
5416:4
5412:a
5401:R
5395:4
5391:1
5374:P
5356:.
5346:2
5342:a
5336:=
5311:ρ
5305:P
5295:a
5289:g
5283:f
5276:z
5272:y
5268:x
5266:(
5249:.
5246:S
5243:d
5239:g
5234:n
5230:G
5224:S
5216:+
5213:V
5210:d
5206:f
5203:G
5198:V
5190:=
5187:)
5180:z
5176:,
5169:y
5165:,
5158:x
5154:(
5151:u
5138:G
5132:u
5126:S
5120:n
5118:G
5113:n
5111:u
5094:.
5091:S
5088:d
5083:]
5077:n
5073:G
5069:u
5061:n
5057:u
5053:G
5049:[
5043:S
5035:=
5032:V
5029:d
5024:]
5020:G
5014:u
5008:u
5002:G
4998:[
4986:V
4978:=
4975:V
4972:d
4967:]
4963:G
4950:u
4944:u
4931:G
4927:[
4921:V
4900:S
4895:g
4889:u
4872:,
4869:f
4863:=
4860:u
4841:V
4836:u
4819:.
4816:S
4807:)
4804:z
4801:,
4798:y
4795:,
4792:x
4789:(
4779:0
4776:=
4773:G
4754:,
4751:V
4742:)
4735:z
4728:z
4725:,
4718:y
4711:y
4708:,
4701:x
4694:x
4691:(
4682:=
4679:G
4650:)
4643:z
4639:,
4632:y
4628:,
4621:x
4617:;
4614:z
4611:,
4608:y
4605:,
4602:x
4599:(
4596:G
4586:V
4582:S
4542:)
4540:r
4524:.
4515:2
4510:)
4507:r
4504:(
4489:=
4486:u
4444:.
4438:r
4432:4
4428:1
4423:=
4420:u
4410:r
4396:,
4388:2
4384:r
4377:4
4373:1
4365:=
4359:r
4356:d
4351:u
4348:d
4323:.
4318:a
4315:=
4312:r
4307:|
4300:r
4297:d
4292:u
4289:d
4281:2
4277:a
4270:4
4262:=
4259:S
4256:d
4249:r
4246:d
4241:u
4238:d
4230:S
4222:=
4219:V
4216:d
4212:u
4198:V
4190:=
4187:1
4170:a
4166:r
4144:=
4141:V
4138:d
4134:u
4120:V
4104:u
4085:z
4081:y
4077:x
4075:(
4070:δ
4052:,
4049:)
4042:z
4035:z
4032:,
4025:y
4018:y
4015:,
4008:x
4001:x
3998:(
3989:=
3984:z
3981:z
3977:u
3973:+
3968:y
3965:y
3961:u
3957:+
3952:x
3949:x
3945:u
3941:=
3938:u
3894:,
3885:=
3880:y
3877:y
3869:+
3864:x
3861:x
3836:.
3833:v
3827:=
3822:y
3813:,
3810:u
3804:=
3799:x
3783:φ
3768:,
3765:y
3762:d
3758:v
3752:x
3749:d
3745:u
3739:=
3733:d
3722:ρ
3707:,
3701:=
3698:)
3695:v
3692:,
3689:u
3686:(
3660:,
3656:0
3652:=
3642:k
3635:)
3630:y
3626:u
3617:x
3613:v
3609:(
3606:=
3603:)
3600:0
3597:,
3594:v
3591:,
3588:u
3585:(
3569:)
3567:v
3563:u
3561:(
3534:.
3531:v
3525:=
3520:y
3511:,
3508:u
3502:=
3497:x
3477:ψ
3471:φ
3465:ψ
3450:,
3447:u
3441:=
3436:y
3427:,
3424:v
3421:=
3416:x
3400:ψ
3377:,
3374:y
3371:d
3367:u
3361:x
3358:d
3354:v
3351:=
3345:d
3334:ψ
3316:=
3311:y
3307:u
3298:x
3294:v
3290:=
3286:V
3259:,
3256:0
3253:=
3248:y
3244:v
3240:+
3235:x
3231:u
3219:v
3213:u
3189:f
3174:,
3170:]
3163:n
3152:n
3148:r
3142:n
3138:b
3134:+
3128:n
3117:n
3113:r
3107:n
3103:a
3098:[
3087:1
3084:=
3081:n
3073:i
3070:+
3066:]
3059:n
3048:n
3044:r
3038:n
3034:b
3024:n
3013:n
3009:r
3003:n
2999:a
2994:[
2983:0
2980:=
2977:n
2969:=
2966:)
2963:z
2960:(
2957:f
2937:.
2932:n
2928:b
2924:i
2921:+
2916:n
2912:a
2908:=
2903:n
2899:c
2878:,
2873:n
2869:z
2863:n
2859:c
2848:0
2845:=
2842:n
2834:=
2831:)
2828:z
2825:(
2822:f
2812:R
2807:f
2784:θ
2768:.
2762:i
2759:+
2756:r
2747:=
2744:z
2735:=
2732:)
2729:z
2726:(
2723:f
2703:,
2700:r
2691:=
2678:θ
2674:r
2661:ψ
2646:,
2641:x
2638:y
2630:=
2625:y
2622:x
2606:ψ
2600:φ
2585:.
2582:y
2579:d
2573:x
2565:+
2562:x
2559:d
2553:y
2542:=
2536:d
2525:ψ
2510:.
2505:x
2497:=
2492:y
2483:,
2478:y
2467:=
2462:x
2437:,
2434:)
2431:y
2428:,
2425:x
2422:(
2416:i
2413:+
2410:)
2407:y
2404:,
2401:x
2398:(
2392:=
2389:)
2386:z
2383:(
2380:f
2370:)
2368:z
2366:(
2364:f
2358:v
2352:u
2337:.
2332:x
2328:)
2322:x
2318:u
2314:(
2308:=
2303:x
2299:)
2293:y
2289:v
2285:(
2279:=
2274:y
2270:)
2264:x
2260:v
2253:(
2250:=
2245:y
2242:y
2238:u
2227:x
2222:u
2215:x
2213:u
2198:.
2193:y
2189:u
2182:=
2177:x
2173:v
2168:,
2163:y
2159:v
2155:=
2150:x
2146:u
2130:v
2124:u
2119:)
2117:z
2115:(
2113:f
2098:,
2095:)
2092:y
2089:,
2086:x
2083:(
2080:v
2077:i
2074:+
2071:)
2068:y
2065:,
2062:x
2059:(
2056:u
2053:=
2050:)
2047:z
2044:(
2041:f
2026:x
2022:z
1993:=
1988:y
1985:y
1977:+
1972:x
1969:x
1953:2
1949:y
1935:2
1924:+
1916:2
1912:x
1898:2
1855:φ
1849:D
1840:D
1835:φ
1819:D
1814:φ
1809:D
1804:φ
1786:)
1784:θ
1780:R
1778:(
1776:u
1769:r
1767:(
1765:u
1758:R
1751:r
1712:g
1697:)
1694:}
1689:j
1686:i
1682:g
1678:{
1672:=
1669:g
1666:(
1662:,
1659:0
1656:=
1652:)
1643:j
1630:f
1619:j
1616:i
1612:g
1605:|
1601:g
1597:|
1590:(
1580:i
1559:|
1555:g
1551:|
1546:1
1541:=
1538:f
1533:2
1508:,
1505:0
1502:=
1497:n
1492:n
1489:m
1479:m
1476:j
1472:g
1463:j
1450:f
1441:+
1437:)
1431:j
1428:k
1424:g
1415:k
1402:f
1392:(
1383:j
1366:=
1363:f
1358:2
1319:=
1311:2
1298:f
1293:2
1271:2
1261:2
1257:r
1252:1
1247:+
1243:)
1228:f
1209:(
1176:2
1172:r
1167:1
1162:+
1158:)
1151:r
1143:f
1132:2
1128:r
1123:(
1116:r
1100:2
1096:r
1092:1
1087:=
1084:f
1079:2
1054:)
1048:,
1042:,
1039:r
1036:(
1005:=
997:2
993:z
984:f
979:2
968:+
960:2
947:f
942:2
927:2
923:r
919:1
914:+
910:)
903:r
895:f
886:r
882:(
875:r
861:r
858:1
853:=
850:f
845:2
809:=
801:2
797:z
788:f
783:2
772:+
764:2
760:y
751:f
746:2
735:+
727:2
723:x
714:f
709:2
698:=
695:f
690:2
676:,
611:.
608:h
605:=
602:f
579:)
576:z
573:,
570:y
567:,
564:x
561:(
558:h
535:)
532:z
529:,
526:y
523:,
520:x
517:(
514:f
437:2
429:=
417:=
394:,
391:0
388:=
385:f
362:0
359:=
356:f
350:2
306:e
299:t
292:v
23:.
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