4193:
3477:
4188:{\displaystyle {\begin{alignedat}{3}e^{z}&=\,\,\sum _{k=0}^{\infty }{\frac {z^{k}}{k!}}&&=1+z+{\tfrac {1}{2}}z^{2}+{\tfrac {1}{6}}z^{3}+{\tfrac {1}{24}}z^{4}+\dots \\\cosh z&=\sum _{k{\text{ even}}}{\frac {z^{k}}{k!}}&&=1+{\tfrac {1}{2}}z^{2}+{\tfrac {1}{24}}z^{4}+\dots \\\sinh z&=\,\sum _{k{\text{ odd}}}{\frac {z^{k}}{k!}}&&=z+{\tfrac {1}{6}}z^{3}+{\tfrac {1}{120}}z^{5}+\dots \\\cos z&=\sum _{k{\text{ even}}}{\frac {(iz)^{k}}{k!}}&&=1-{\tfrac {1}{2}}z^{2}+{\tfrac {1}{24}}z^{4}-\dots \\i\sin z&=\,\sum _{k{\text{ odd}}}{\frac {(iz)^{k}}{k!}}&&=i\left(z-{\tfrac {1}{6}}z^{3}+{\tfrac {1}{120}}z^{5}-\dots \right)\\\end{alignedat}}}
1097:
1105:
20:
2320:
1971:
1086:
1445:
steadily moving orthogonally to a ray from the origin traces out a hyperbola. In
Euclidean space, the multiple of a given angle traces equal distances around a circle while it traces exponential distances upon the hyperbolic line.
3464:
2872:
cosh and sinh can be presented through the circular functions. But in the
Euclidean plane we might alternately consider circular angle measures to be imaginary and hyperbolic angle measures to be real scalars,
1233:. Given an origin point on one of these ranges, other points correspond to angles. The idea of addition of angles, basic to science, corresponds to addition of points on one of these ranges as follows:
2170:
1821:
3238:
995:
244:
903:
3137:
3061:
821:
393:
167:
sinh, cosh, and tanh, because these functions may be premised on hyperbolic analogies to the corresponding circular (trigonometric) functions by regarding a hyperbolic angle as defining a
1653:
1577:
2116:
2465:
498:
456:
1004:
1813:
4252:
4243:
4233:
2959:
2866:
1718:
2493:. Just as the circular angle is the length of a circular arc using the Euclidean metric, the hyperbolic angle is the length of a hyperbolic arc using the Minkowski metric.
2369:
2020:
1212:
1159:
2912:
2819:
661:
537:
154:
122:
1506:
There is also a curious relation to a hyperbolic angle and the metric defined on
Minkowski space. Just as two dimensional Euclidean geometry defines its line element as
2433:
2401:
2162:
2084:
2052:
2491:
629:
587:
419:
273:
725:
693:
352:
320:
3338:
3278:
754:
3305:
2136:
1781:
1761:
1741:
3344:
2985:
3144:
4372:
3482:
2643:
1001:, the hyperbolic sectors determined by these angles have the same area, which is taken to be the magnitude of the angle. This magnitude is
4250:
4241:
2585:, the logarithm is more familiar than its motivator, the hyperbolic angle. Nevertheless, the hyperbolic angle plays a role when the
4369:
2315:{\displaystyle S=\int _{a}^{b}ds_{m}=\int _{a}^{b}{\sqrt {\left({\frac {dx}{dt}}\right)^{2}-\left({\frac {dy}{dt}}\right)^{2}}}dt,}
1966:{\displaystyle S=\int _{a}^{b}ds_{e}=\int _{a}^{b}{\sqrt {\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}}}dt.}
4230:
912:
190:
826:
1441:
Whereas in
Euclidean geometry moving steadily in an orthogonal direction to a ray from the origin traces out a circle, in a
4419:
160:
3066:
2993:
4464:
4426:
762:
357:
4485:
1588:
1512:
4273:
2089:
501:
1081:{\displaystyle \operatorname {ln} {(bc)}=\operatorname {ln} (c/a)=\operatorname {ln} c-\operatorname {ln} a}
2586:
2438:
477:
435:
2706:
rapidity corresponds a huge velocity, amounting to 3/4 of the velocity of light; more accurately we have
168:
2164:. Now doing the same procedure, except replacing the Euclidean element with the Minkowski line element,
4490:
1786:
2518:
998:
1664:
2917:
2824:
2612:
2502:
2328:
1979:
1171:
1118:
2876:
2783:
2582:
634:
539:, the hyperbolic angle should be negative. This reflects the fact that, as defined, the angle is
510:
135:
103:
2988:
2406:
2374:
2141:
1442:
2057:
2025:
2470:
1454:
1221:
There is also a projective resolution between circular and hyperbolic cases: both curves are
596:
590:
554:
97:
4410:(2004) "Enter, stage center: the early drama of the hyperbolic functions", available in (a)
1469:, and the hyperbolic angle magnitudes stay the same when the plane is squeezed by a mapping
398:
252:
4411:
4347:
2964:
2746:
2662:
2631:
698:
666:
325:
293:
164:
78:
2634:
circulated his essay "The
Imaginary of Algebra", which used hyperbolic angles to generate
730:
8:
4436:
4388:
3310:
3245:
2869:
2742:
2651:
2624:
2616:
1230:
86:
4351:
4384:
4331:
4315:
4231:
Through the
Looking Glass – A glimpse of Euclid's twin geometry, the Minkowski geometry
4210:
4198:
3283:
2658:
2601:
2121:
1766:
1746:
1726:
4460:
4422:
4365:
4298:
2647:
2561:
2538:
2510:
1450:
1434:
to an arbitrary point on the curve as a logarithmic function of the point's value of
1215:
463:
429:
82:
62:
28:
4392:
2970:
2777:
2546:
1237:
1226:
4284:
4432:
4335:
4256:
4247:
4237:
3469:
2635:
2590:
2553:
1466:
1162:
757:
129:
70:
4446:
4480:
4452:
2695:
2620:
2022:
defines a unit circle, a single parameterized solution set to this equation is
1458:
1236:
Circular angles can be characterized geometrically by the property that if two
1166:
74:
27:= 1. A hyperbolic angle has magnitude equal to the area of the corresponding
4474:
4407:
4201:, and the series for sine comes from making sinh into an alternating series.
1222:
2597:
4197:
The infinite series for cosine is derived from cosh by turning it into an
1113:
1100:
The unit hyperbola has a sector with an area half of the hyperbolic angle
1096:
176:
54:
4338:, Chapter VI: "On the connection of common and hyperbolic trigonometry"
3459:{\displaystyle e^{x+iy}=e^{x}e^{iy}=(\cosh x+\sinh x)(\cos y+i\sin y).}
4437:
Application of hyperbolic functions to electrical engineering problems
1104:
2557:
2530:
2514:
2506:
19:
2666:
2605:
2542:
2513:. It can be shown to be equal to the corresponding area against an
1462:
1461:
on the circle whose magnitude does not vary as the circle turns or
1165:
with an area half of the circular angle in radians. Analogously, a
172:
46:
4259:
exploring
Minkowskian parallels of some standard Euclidean results
1815:). The arclength of this curve in Euclidean space is computed as:
1331:
The same construction can also be applied to the hyperbola. If
4272:, page 1, Translations of Mathematical Monographs volume 170,
3242:
or if the argument is separated into real and imaginary parts
1658:
Consider a curve embedded in two dimensional
Euclidean space,
507:
The formula for the magnitude of the angle suggests that, for
287:
125:
2564:(or "hyperbolic logarithm") is understood as the area under
551:
to that subtended by any interval on the hyperbola. Suppose
2675:
2534:
1449:
Both circular and hyperbolic angle provide instances of an
1427:. It thus makes sense to define the hyperbolic angle from
425:
171:. The parameter thus becomes one of the most useful in the
58:
4240:, ICME-10 Copenhagen 2004; p.14. See also example sheets
4370:
On the introduction of the notion of hyperbolic functions
3280:
the exponential can be split into the product of scaling
3233:{\displaystyle e^{z}=\cosh z+\sinh z=\cos(iz)-i\sin(iz),}
2776:
The hyperbolic angle is often presented as if it were an
1501:
990:{\displaystyle \angle \!\left((1,1),(0,0),(bc,ad)\right)}
1453:. Arcs with an angular magnitude on a circle generate a
239:{\displaystyle \textstyle \{(x,{\frac {1}{x}}):x>0\}}
2737:
which is a little above the velocity of light in water.
898:{\displaystyle \angle \!\left((a,b),(0,0),(c,d)\right)}
81:
as coordinates. In mathematics, hyperbolic angle is an
4149:
4124:
4002:
3977:
3867:
3842:
3742:
3717:
3618:
3593:
3568:
3313:
3286:
3248:
3083:
3069:
3010:
2996:
2973:
2963:
These relationships can be understood in terms of the
2920:
2879:
2827:
2786:
766:
361:
246:, and (by convention) pay particular attention to the
194:
3480:
3347:
3147:
2733:, consequently will represent the velocity .76
2523:
Opus geometricum quadrature circuli et sectionum coni
2473:
2441:
2409:
2377:
2331:
2173:
2144:
2124:
2092:
2060:
2028:
1982:
1824:
1789:
1769:
1749:
1729:
1667:
1591:
1515:
1174:
1121:
1007:
915:
829:
765:
733:
701:
669:
637:
599:
557:
513:
480:
438:
401:
360:
328:
296:
255:
193:
138:
106:
500:
is unbounded); this is related to the fact that the
3132:{\textstyle \sinh z={\tfrac {1}{2}}(e^{z}-e^{-z}),}
2467:(the hyperbolic angle), we arrive at the result of
4187:
3458:
3332:
3299:
3272:
3232:
3131:
3056:{\textstyle \cosh z={\tfrac {1}{2}}(e^{z}+e^{-z})}
3055:
2979:
2953:
2906:
2860:
2813:
2485:
2459:
2427:
2395:
2371:with its corresponding parameterized solution set
2363:
2314:
2156:
2130:
2110:
2078:
2046:
2014:
1965:
1807:
1775:
1755:
1735:
1712:
1647:
1571:
1206:
1153:
1080:
989:
897:
815:
748:
719:
687:
655:
623:
581:
531:
492:
450:
413:
387:
346:
314:
267:
238:
148:
116:
4418:, RE Bradley, LA D'Antonio, CE Sandifer editors,
1091:
919:
833:
4472:
470:Unlike circular angle, the hyperbolic angle is
4459:exercise 9.5.3, p. 298, Springer-Verlag
4373:Bulletin of the American Mathematical Society
2644:Bulletin of the American Mathematical Society
816:{\displaystyle \textstyle f:(x,y)\to (bx,ay)}
462:Note that, because of the role played by the
388:{\displaystyle \textstyle (x,{\frac {1}{x}})}
2661:penned his popular 1914 textbook on the new
1392:, then the parallel condition requires that
232:
195:
2771:
2517:. The quadrature was first accomplished by
1218:with an area half of the hyperbolic angle.
756:and determine an interval on it. Then the
4270:Elliptic Functions and Elliptic Integrals
4268:Viktor Prasolov and Yuri Solovyev (1997)
4052:
3789:
3503:
3502:
1648:{\displaystyle ds_{m}^{2}=dx^{2}-dy^{2}.}
1572:{\displaystyle ds_{e}^{2}=dx^{2}+dy^{2},}
1103:
1095:
73:. The hyperbolic angle parametrizes the
18:
2111:{\displaystyle 0\leqslant t<\theta }
1582:the line element on Minkowski space is
4473:
1502:Relation To The Minkowski Line Element
1465:. For the hyperbola the turning is by
4296:
2460:{\displaystyle 0\leqslant t<\eta }
1279:at the centre of a circle, their sum
493:{\displaystyle \operatorname {ln} x}
451:{\displaystyle \operatorname {ln} x}
16:Argument of the hyperbolic functions
4420:Mathematical Association of America
2560:and thus the geometrically defined
2509:is the evaluation of the area of a
1743:is a real number that runs between
424:The magnitude of this angle is the
187:Consider the rectangular hyperbola
13:
3520:
2702:It seems worth mentioning that to
2669:concept based on hyperbolic angle
1295:is the angle subtended by a chord
916:
830:
547:Finally, extend the definition of
14:
4502:
2600:was extended to the hyperbola by
2529:quadrature of a hyperbola to its
2325:and defining a unit hyperbola as
1808:{\displaystyle a\leqslant t<b}
4414:77(1):15–30 or (b) chapter 7 of
2556:interpreted the quadrature as a
159:Hyperbolic angle is used as the
4336:Trigonometry and Double Algebra
2967:, which for a complex argument
2609:Trigonometry and Double Algebra
2525:. As expressed by a historian,
4378:
4359:
4341:
4325:
4309:
4290:
4278:
4262:
4223:
4081:
4071:
3942:
3932:
3450:
3423:
3420:
3396:
3224:
3215:
3200:
3191:
3123:
3094:
3050:
3021:
2954:{\textstyle \sinh ix=i\sin x.}
2861:{\textstyle \sin ix=i\sinh x,}
1713:{\displaystyle x=f(t),y=g(t).}
1704:
1698:
1683:
1677:
1315:is required to be parallel to
1092:Comparison with circular angle
1051:
1037:
1024:
1015:
979:
961:
955:
943:
937:
925:
887:
875:
869:
857:
851:
839:
809:
791:
788:
785:
773:
714:
702:
682:
670:
381:
362:
341:
329:
309:
297:
217:
198:
1:
4401:
4274:American Mathematical Society
2615:used the hyperbolic angle to
2364:{\displaystyle y^{2}-x^{2}=1}
2015:{\displaystyle x^{2}+y^{2}=1}
1207:{\displaystyle x^{2}-y^{2}=1}
1154:{\displaystyle x^{2}+y^{2}=1}
1108:Circular vs. hyperbolic angle
182:
2907:{\textstyle \cosh ix=\cos x}
2814:{\textstyle \cos ix=\cosh x}
727:are points on the hyperbola
124:, analogous to the circular
7:
4204:
1225:, and hence are treated as
656:{\displaystyle c>a>1}
532:{\displaystyle 0<x<1}
149:{\displaystyle {\sqrt {2}}}
117:{\displaystyle {\sqrt {2}}}
10:
4507:
2623:, describing it as "quasi-
2496:
2118:, computing the arclength
2533:, and showed that as the
2519:Gregoire de Saint-Vincent
2428:{\displaystyle x=\sinh t}
2396:{\displaystyle y=\cosh t}
2157:{\displaystyle S=\theta }
1338:is taken to be the point
999:Gregoire de Saint-Vincent
85:as it is preserved under
69:= 1 in Quadrant I of the
4389:The Theory of Relativity
4356:, B. Westerman, New York
4353:Papers on Space Analysis
4216:
2772:Imaginary circular angle
2690:, the ratio of velocity
2640:Papers on Space Analysis
2587:theorem of Saint-Vincent
2079:{\displaystyle y=\sin t}
2047:{\displaystyle x=\cos t}
432:, which turns out to be
282:The hyperbolic angle in
132:in a circle with radius
4447:Hyperbolic Trigonometry
2583:transcendental function
2486:{\displaystyle S=\eta }
624:{\displaystyle ab=cd=1}
582:{\displaystyle a,b,c,d}
128:equaling the area of a
4320:History of Mathematics
4189:
3524:
3460:
3334:
3301:
3274:
3234:
3133:
3057:
2981:
2955:
2908:
2862:
2815:
2741:Silberstein also uses
2581:. As an example of a
2487:
2461:
2429:
2397:
2365:
2316:
2158:
2132:
2112:
2080:
2048:
2016:
1967:
1809:
1777:
1757:
1737:
1714:
1649:
1573:
1443:pseudo-Euclidean plane
1208:
1155:
1109:
1101:
1082:
991:
899:
817:
750:
721:
689:
657:
625:
583:
533:
494:
452:
415:
414:{\displaystyle x>1}
389:
348:
316:
269:
268:{\displaystyle x>1}
240:
150:
118:
42:
4486:Differential calculus
4443:Exploring Precalculus
4391:, pp. 180–1 via
4303:mathworld.wolfram.com
4190:
3504:
3461:
3335:
3302:
3275:
3235:
3134:
3058:
2982:
2956:
2909:
2863:
2816:
2642:. The following year
2488:
2462:
2430:
2398:
2366:
2317:
2159:
2133:
2113:
2081:
2049:
2017:
1968:
1810:
1778:
1758:
1738:
1715:
1650:
1574:
1209:
1156:
1107:
1099:
1083:
992:
900:
818:
751:
722:
720:{\displaystyle (c,d)}
690:
688:{\displaystyle (a,b)}
658:
626:
591:positive real numbers
584:
534:
495:
453:
428:of the corresponding
416:
390:
349:
347:{\displaystyle (1,1)}
317:
315:{\displaystyle (0,0)}
270:
241:
151:
119:
100:with semi-major axis
61:of the corresponding
23:The curve represents
22:
4457:Numbers and Geometry
4412:Mathematics Magazine
4348:Alexander Macfarlane
3478:
3345:
3333:{\textstyle e^{iy},}
3311:
3284:
3273:{\textstyle z=x+iy,}
3246:
3145:
3067:
2994:
2971:
2965:exponential function
2918:
2877:
2870:hyperbolic functions
2825:
2784:
2747:angle of parallelism
2663:theory of relativity
2652:hyperbolic functions
2632:Alexander Macfarlane
2471:
2439:
2407:
2375:
2329:
2171:
2142:
2122:
2090:
2058:
2026:
1980:
1822:
1787:
1767:
1747:
1727:
1723:Where the parameter
1665:
1589:
1513:
1172:
1119:
1005:
913:
827:
763:
749:{\displaystyle xy=1}
731:
699:
667:
635:
597:
555:
511:
478:
436:
399:
358:
326:
294:
253:
191:
165:hyperbolic functions
161:independent variable
136:
104:
79:hyperbolic functions
4408:Janet Heine Barnett
4297:Weisstein, Eric W.
4285:Hyperbolic Geometry
3139:respectively. Then
2987:can be broken into
2225:
2194:
1876:
1845:
1609:
1533:
1231:projective geometry
997:. By the result of
322:between the ray to
169:hyperbolic triangle
87:hyperbolic rotation
4445:, § The Number e,
4385:Ludwik Silberstein
4332:Augustus De Morgan
4316:David Eugene Smith
4299:"Minkowski Metric"
4255:2008-11-21 at the
4246:2009-01-06 at the
4236:2011-07-16 at the
4211:Transcendent angle
4199:alternating series
4185:
4183:
4158:
4133:
4067:
4011:
3986:
3928:
3876:
3851:
3804:
3751:
3726:
3679:
3627:
3602:
3577:
3456:
3330:
3300:{\textstyle e^{x}}
3297:
3270:
3230:
3129:
3092:
3053:
3019:
2989:even and odd parts
2977:
2951:
2904:
2858:
2811:
2659:Ludwik Silberstein
2650:'s outline of the
2636:hyperbolic versors
2602:Augustus De Morgan
2483:
2457:
2425:
2393:
2361:
2312:
2211:
2180:
2154:
2128:
2108:
2076:
2044:
2012:
1963:
1862:
1831:
1805:
1773:
1753:
1733:
1710:
1645:
1595:
1569:
1519:
1204:
1151:
1110:
1102:
1078:
987:
895:
813:
812:
746:
717:
685:
653:
621:
579:
529:
490:
448:
411:
385:
384:
344:
312:
265:
236:
235:
146:
114:
57:determined by the
43:
4491:Integral calculus
4441:William Mueller,
4366:Mellen W. Haskell
4322:, pp. 424,5 v. 1
4157:
4132:
4099:
4064:
4053:
4010:
3985:
3960:
3925:
3914:
3875:
3850:
3825:
3801:
3790:
3750:
3725:
3700:
3676:
3665:
3626:
3601:
3576:
3545:
3091:
3018:
2648:Mellen W. Haskell
2589:is advanced with
2562:natural logarithm
2539:arithmetic series
2511:hyperbolic sector
2435:, and by letting
2301:
2289:
2251:
2131:{\displaystyle S}
1952:
1940:
1902:
1776:{\displaystyle b}
1756:{\displaystyle a}
1736:{\displaystyle t}
1451:invariant measure
1227:projective ranges
1216:hyperbolic sector
907:standard position
464:natural logarithm
430:hyperbolic sector
379:
284:standard position
215:
144:
112:
83:invariant measure
63:hyperbolic sector
33:standard position
29:hyperbolic sector
4498:
4395:
4393:Internet Archive
4382:
4376:
4363:
4357:
4345:
4339:
4329:
4323:
4313:
4307:
4306:
4294:
4288:
4287:pp 5–6, Fig 15.1
4282:
4276:
4266:
4260:
4229:Bjørn Felsager,
4227:
4194:
4192:
4191:
4186:
4184:
4180:
4176:
4169:
4168:
4159:
4150:
4144:
4143:
4134:
4125:
4102:
4100:
4098:
4090:
4089:
4088:
4069:
4066:
4065:
4062:
4022:
4021:
4012:
4003:
3997:
3996:
3987:
3978:
3963:
3961:
3959:
3951:
3950:
3949:
3930:
3927:
3926:
3923:
3887:
3886:
3877:
3868:
3862:
3861:
3852:
3843:
3828:
3826:
3824:
3816:
3815:
3806:
3803:
3802:
3799:
3762:
3761:
3752:
3743:
3737:
3736:
3727:
3718:
3703:
3701:
3699:
3691:
3690:
3681:
3678:
3677:
3674:
3638:
3637:
3628:
3619:
3613:
3612:
3603:
3594:
3588:
3587:
3578:
3569:
3548:
3546:
3544:
3536:
3535:
3526:
3523:
3518:
3494:
3493:
3465:
3463:
3462:
3457:
3392:
3391:
3379:
3378:
3366:
3365:
3339:
3337:
3336:
3331:
3326:
3325:
3306:
3304:
3303:
3298:
3296:
3295:
3279:
3277:
3276:
3271:
3239:
3237:
3236:
3231:
3157:
3156:
3138:
3136:
3135:
3130:
3122:
3121:
3106:
3105:
3093:
3084:
3062:
3060:
3059:
3054:
3049:
3048:
3033:
3032:
3020:
3011:
2986:
2984:
2983:
2978:
2960:
2958:
2957:
2952:
2913:
2911:
2910:
2905:
2867:
2865:
2864:
2859:
2820:
2818:
2817:
2812:
2778:imaginary number
2767:
2732:
2722:
2715:
2689:
2580:
2574:to the right of
2573:
2547:geometric series
2492:
2490:
2489:
2484:
2466:
2464:
2463:
2458:
2434:
2432:
2431:
2426:
2402:
2400:
2399:
2394:
2370:
2368:
2367:
2362:
2354:
2353:
2341:
2340:
2321:
2319:
2318:
2313:
2302:
2300:
2299:
2294:
2290:
2288:
2280:
2272:
2262:
2261:
2256:
2252:
2250:
2242:
2234:
2227:
2224:
2219:
2207:
2206:
2193:
2188:
2163:
2161:
2160:
2155:
2137:
2135:
2134:
2129:
2117:
2115:
2114:
2109:
2085:
2083:
2082:
2077:
2053:
2051:
2050:
2045:
2021:
2019:
2018:
2013:
2005:
2004:
1992:
1991:
1972:
1970:
1969:
1964:
1953:
1951:
1950:
1945:
1941:
1939:
1931:
1923:
1913:
1912:
1907:
1903:
1901:
1893:
1885:
1878:
1875:
1870:
1858:
1857:
1844:
1839:
1814:
1812:
1811:
1806:
1782:
1780:
1779:
1774:
1762:
1760:
1759:
1754:
1742:
1740:
1739:
1734:
1719:
1717:
1716:
1711:
1654:
1652:
1651:
1646:
1641:
1640:
1625:
1624:
1608:
1603:
1578:
1576:
1575:
1570:
1565:
1564:
1549:
1548:
1532:
1527:
1426:
1391:
1366:
1341:
1294:
1213:
1211:
1210:
1205:
1197:
1196:
1184:
1183:
1160:
1158:
1157:
1152:
1144:
1143:
1131:
1130:
1087:
1085:
1084:
1079:
1047:
1027:
996:
994:
993:
988:
986:
982:
904:
902:
901:
896:
894:
890:
822:
820:
819:
814:
755:
753:
752:
747:
726:
724:
723:
718:
694:
692:
691:
686:
662:
660:
659:
654:
630:
628:
627:
622:
588:
586:
585:
580:
549:hyperbolic angle
538:
536:
535:
530:
499:
497:
496:
491:
457:
455:
454:
449:
420:
418:
417:
412:
394:
392:
391:
386:
380:
372:
353:
351:
350:
345:
321:
319:
318:
313:
274:
272:
271:
266:
245:
243:
242:
237:
216:
208:
155:
153:
152:
147:
145:
140:
123:
121:
120:
115:
113:
108:
51:hyperbolic angle
41:
4506:
4505:
4501:
4500:
4499:
4497:
4496:
4495:
4471:
4470:
4433:Arthur Kennelly
4404:
4399:
4398:
4383:
4379:
4364:
4360:
4346:
4342:
4330:
4326:
4314:
4310:
4295:
4291:
4283:
4279:
4267:
4263:
4257:Wayback Machine
4248:Wayback Machine
4238:Wayback Machine
4228:
4224:
4219:
4207:
4182:
4181:
4164:
4160:
4148:
4139:
4135:
4123:
4116:
4112:
4101:
4091:
4084:
4080:
4070:
4068:
4061:
4057:
4045:
4030:
4029:
4017:
4013:
4001:
3992:
3988:
3976:
3962:
3952:
3945:
3941:
3931:
3929:
3922:
3918:
3907:
3895:
3894:
3882:
3878:
3866:
3857:
3853:
3841:
3827:
3817:
3811:
3807:
3805:
3798:
3794:
3782:
3770:
3769:
3757:
3753:
3741:
3732:
3728:
3716:
3702:
3692:
3686:
3682:
3680:
3673:
3669:
3658:
3646:
3645:
3633:
3629:
3617:
3608:
3604:
3592:
3583:
3579:
3567:
3547:
3537:
3531:
3527:
3525:
3519:
3508:
3495:
3489:
3485:
3481:
3479:
3476:
3475:
3470:infinite series
3384:
3380:
3374:
3370:
3352:
3348:
3346:
3343:
3342:
3318:
3314:
3312:
3309:
3308:
3291:
3287:
3285:
3282:
3281:
3247:
3244:
3243:
3152:
3148:
3146:
3143:
3142:
3114:
3110:
3101:
3097:
3082:
3068:
3065:
3064:
3041:
3037:
3028:
3024:
3009:
2995:
2992:
2991:
2972:
2969:
2968:
2919:
2916:
2915:
2878:
2875:
2874:
2826:
2823:
2822:
2785:
2782:
2781:
2774:
2754:
2727:
2717:
2707:
2674:
2625:harmonic motion
2591:squeeze mapping
2575:
2565:
2554:A. A. de Sarasa
2499:
2472:
2469:
2468:
2440:
2437:
2436:
2408:
2405:
2404:
2376:
2373:
2372:
2349:
2345:
2336:
2332:
2330:
2327:
2326:
2295:
2281:
2273:
2271:
2267:
2266:
2257:
2243:
2235:
2233:
2229:
2228:
2226:
2220:
2215:
2202:
2198:
2189:
2184:
2172:
2169:
2168:
2143:
2140:
2139:
2123:
2120:
2119:
2091:
2088:
2087:
2059:
2056:
2055:
2027:
2024:
2023:
2000:
1996:
1987:
1983:
1981:
1978:
1977:
1946:
1932:
1924:
1922:
1918:
1917:
1908:
1894:
1886:
1884:
1880:
1879:
1877:
1871:
1866:
1853:
1849:
1840:
1835:
1823:
1820:
1819:
1788:
1785:
1784:
1768:
1765:
1764:
1748:
1745:
1744:
1728:
1725:
1724:
1666:
1663:
1662:
1636:
1632:
1620:
1616:
1604:
1599:
1590:
1587:
1586:
1560:
1556:
1544:
1540:
1528:
1523:
1514:
1511:
1510:
1504:
1467:squeeze mapping
1459:measurable sets
1433:
1424:
1417:
1410:
1404:
1397:
1389:
1382:
1375:
1373:
1364:
1357:
1350:
1348:
1339:
1337:
1327:
1321:
1311:
1301:
1293:
1286:
1280:
1278:
1271:
1265:subtend angles
1264:
1258:
1251:
1245:
1192:
1188:
1179:
1175:
1173:
1170:
1169:
1163:circular sector
1139:
1135:
1126:
1122:
1120:
1117:
1116:
1094:
1043:
1014:
1006:
1003:
1002:
924:
920:
914:
911:
910:
838:
834:
828:
825:
824:
823:maps the angle
764:
761:
760:
758:squeeze mapping
732:
729:
728:
700:
697:
696:
668:
665:
664:
636:
633:
632:
598:
595:
594:
556:
553:
552:
512:
509:
508:
502:harmonic series
479:
476:
475:
437:
434:
433:
400:
397:
396:
371:
359:
356:
355:
354:and the ray to
327:
324:
323:
295:
292:
291:
254:
251:
250:
207:
192:
189:
188:
185:
139:
137:
134:
133:
130:circular sector
107:
105:
102:
101:
71:Cartesian plane
36:
17:
12:
11:
5:
4504:
4494:
4493:
4488:
4483:
4469:
4468:
4453:John Stillwell
4450:
4439:
4430:
4403:
4400:
4397:
4396:
4377:
4358:
4340:
4324:
4308:
4289:
4277:
4261:
4221:
4220:
4218:
4215:
4214:
4213:
4206:
4203:
4179:
4175:
4172:
4167:
4163:
4156:
4153:
4147:
4142:
4138:
4131:
4128:
4122:
4119:
4115:
4111:
4108:
4105:
4103:
4097:
4094:
4087:
4083:
4079:
4076:
4073:
4060:
4056:
4051:
4048:
4046:
4044:
4041:
4038:
4035:
4032:
4031:
4028:
4025:
4020:
4016:
4009:
4006:
4000:
3995:
3991:
3984:
3981:
3975:
3972:
3969:
3966:
3964:
3958:
3955:
3948:
3944:
3940:
3937:
3934:
3921:
3917:
3913:
3910:
3908:
3906:
3903:
3900:
3897:
3896:
3893:
3890:
3885:
3881:
3874:
3871:
3865:
3860:
3856:
3849:
3846:
3840:
3837:
3834:
3831:
3829:
3823:
3820:
3814:
3810:
3797:
3793:
3788:
3785:
3783:
3781:
3778:
3775:
3772:
3771:
3768:
3765:
3760:
3756:
3749:
3746:
3740:
3735:
3731:
3724:
3721:
3715:
3712:
3709:
3706:
3704:
3698:
3695:
3689:
3685:
3672:
3668:
3664:
3661:
3659:
3657:
3654:
3651:
3648:
3647:
3644:
3641:
3636:
3632:
3625:
3622:
3616:
3611:
3607:
3600:
3597:
3591:
3586:
3582:
3575:
3572:
3566:
3563:
3560:
3557:
3554:
3551:
3549:
3543:
3540:
3534:
3530:
3522:
3517:
3514:
3511:
3507:
3501:
3498:
3496:
3492:
3488:
3484:
3483:
3455:
3452:
3449:
3446:
3443:
3440:
3437:
3434:
3431:
3428:
3425:
3422:
3419:
3416:
3413:
3410:
3407:
3404:
3401:
3398:
3395:
3390:
3387:
3383:
3377:
3373:
3369:
3364:
3361:
3358:
3355:
3351:
3329:
3324:
3321:
3317:
3294:
3290:
3269:
3266:
3263:
3260:
3257:
3254:
3251:
3229:
3226:
3223:
3220:
3217:
3214:
3211:
3208:
3205:
3202:
3199:
3196:
3193:
3190:
3187:
3184:
3181:
3178:
3175:
3172:
3169:
3166:
3163:
3160:
3155:
3151:
3128:
3125:
3120:
3117:
3113:
3109:
3104:
3100:
3096:
3090:
3087:
3081:
3078:
3075:
3072:
3052:
3047:
3044:
3040:
3036:
3031:
3027:
3023:
3017:
3014:
3008:
3005:
3002:
2999:
2980:{\textstyle z}
2976:
2950:
2947:
2944:
2941:
2938:
2935:
2932:
2929:
2926:
2923:
2903:
2900:
2897:
2894:
2891:
2888:
2885:
2882:
2857:
2854:
2851:
2848:
2845:
2842:
2839:
2836:
2833:
2830:
2810:
2807:
2804:
2801:
2798:
2795:
2792:
2789:
2773:
2770:
2745:'s concept of
2739:
2738:
2724:
2696:speed of light
2665:, he used the
2638:, in his book
2621:unit hyperbola
2551:
2550:
2498:
2495:
2482:
2479:
2476:
2456:
2453:
2450:
2447:
2444:
2424:
2421:
2418:
2415:
2412:
2392:
2389:
2386:
2383:
2380:
2360:
2357:
2352:
2348:
2344:
2339:
2335:
2323:
2322:
2311:
2308:
2305:
2298:
2293:
2287:
2284:
2279:
2276:
2270:
2265:
2260:
2255:
2249:
2246:
2241:
2238:
2232:
2223:
2218:
2214:
2210:
2205:
2201:
2197:
2192:
2187:
2183:
2179:
2176:
2153:
2150:
2147:
2127:
2107:
2104:
2101:
2098:
2095:
2075:
2072:
2069:
2066:
2063:
2043:
2040:
2037:
2034:
2031:
2011:
2008:
2003:
1999:
1995:
1990:
1986:
1974:
1973:
1962:
1959:
1956:
1949:
1944:
1938:
1935:
1930:
1927:
1921:
1916:
1911:
1906:
1900:
1897:
1892:
1889:
1883:
1874:
1869:
1865:
1861:
1856:
1852:
1848:
1843:
1838:
1834:
1830:
1827:
1804:
1801:
1798:
1795:
1792:
1772:
1752:
1732:
1721:
1720:
1709:
1706:
1703:
1700:
1697:
1694:
1691:
1688:
1685:
1682:
1679:
1676:
1673:
1670:
1656:
1655:
1644:
1639:
1635:
1631:
1628:
1623:
1619:
1615:
1612:
1607:
1602:
1598:
1594:
1580:
1579:
1568:
1563:
1559:
1555:
1552:
1547:
1543:
1539:
1536:
1531:
1526:
1522:
1518:
1503:
1500:
1499:
1498:
1431:
1422:
1415:
1408:
1402:
1387:
1380:
1371:
1362:
1355:
1346:
1335:
1325:
1319:
1309:
1299:
1291:
1284:
1276:
1269:
1262:
1256:
1249:
1243:
1223:conic sections
1203:
1200:
1195:
1191:
1187:
1182:
1178:
1167:unit hyperbola
1150:
1147:
1142:
1138:
1134:
1129:
1125:
1093:
1090:
1077:
1074:
1071:
1068:
1065:
1062:
1059:
1056:
1053:
1050:
1046:
1042:
1039:
1036:
1033:
1030:
1026:
1023:
1020:
1017:
1013:
1010:
985:
981:
978:
975:
972:
969:
966:
963:
960:
957:
954:
951:
948:
945:
942:
939:
936:
933:
930:
927:
923:
918:
893:
889:
886:
883:
880:
877:
874:
871:
868:
865:
862:
859:
856:
853:
850:
847:
844:
841:
837:
832:
811:
808:
805:
802:
799:
796:
793:
790:
787:
784:
781:
778:
775:
772:
769:
745:
742:
739:
736:
716:
713:
710:
707:
704:
684:
681:
678:
675:
672:
652:
649:
646:
643:
640:
620:
617:
614:
611:
608:
605:
602:
578:
575:
572:
569:
566:
563:
560:
545:
544:
528:
525:
522:
519:
516:
505:
489:
486:
483:
460:
459:
447:
444:
441:
422:
410:
407:
404:
383:
378:
375:
370:
367:
364:
343:
340:
337:
334:
331:
311:
308:
305:
302:
299:
278:First define:
264:
261:
258:
234:
231:
228:
225:
222:
219:
214:
211:
206:
203:
200:
197:
184:
181:
143:
111:
92:The hyperbola
75:unit hyperbola
31:, which is in
15:
9:
6:
4:
3:
2:
4503:
4492:
4489:
4487:
4484:
4482:
4479:
4478:
4476:
4466:
4465:0-387-98289-2
4462:
4458:
4454:
4451:
4448:
4444:
4440:
4438:
4434:
4431:
4428:
4427:0-88385-565-8
4424:
4421:
4417:
4413:
4409:
4406:
4405:
4394:
4390:
4386:
4381:
4374:
4371:
4367:
4362:
4355:
4354:
4349:
4344:
4337:
4333:
4328:
4321:
4317:
4312:
4304:
4300:
4293:
4286:
4281:
4275:
4271:
4265:
4258:
4254:
4251:
4249:
4245:
4242:
4239:
4235:
4232:
4226:
4222:
4212:
4209:
4208:
4202:
4200:
4195:
4177:
4173:
4170:
4165:
4161:
4154:
4151:
4145:
4140:
4136:
4129:
4126:
4120:
4117:
4113:
4109:
4106:
4104:
4095:
4092:
4085:
4077:
4074:
4058:
4054:
4049:
4047:
4042:
4039:
4036:
4033:
4026:
4023:
4018:
4014:
4007:
4004:
3998:
3993:
3989:
3982:
3979:
3973:
3970:
3967:
3965:
3956:
3953:
3946:
3938:
3935:
3919:
3915:
3911:
3909:
3904:
3901:
3898:
3891:
3888:
3883:
3879:
3872:
3869:
3863:
3858:
3854:
3847:
3844:
3838:
3835:
3832:
3830:
3821:
3818:
3812:
3808:
3795:
3791:
3786:
3784:
3779:
3776:
3773:
3766:
3763:
3758:
3754:
3747:
3744:
3738:
3733:
3729:
3722:
3719:
3713:
3710:
3707:
3705:
3696:
3693:
3687:
3683:
3670:
3666:
3662:
3660:
3655:
3652:
3649:
3642:
3639:
3634:
3630:
3623:
3620:
3614:
3609:
3605:
3598:
3595:
3589:
3584:
3580:
3573:
3570:
3564:
3561:
3558:
3555:
3552:
3550:
3541:
3538:
3532:
3528:
3515:
3512:
3509:
3505:
3499:
3497:
3490:
3486:
3473:
3471:
3466:
3453:
3447:
3444:
3441:
3438:
3435:
3432:
3429:
3426:
3417:
3414:
3411:
3408:
3405:
3402:
3399:
3393:
3388:
3385:
3381:
3375:
3371:
3367:
3362:
3359:
3356:
3353:
3349:
3340:
3327:
3322:
3319:
3315:
3307:and rotation
3292:
3288:
3267:
3264:
3261:
3258:
3255:
3252:
3249:
3240:
3227:
3221:
3218:
3212:
3209:
3206:
3203:
3197:
3194:
3188:
3185:
3182:
3179:
3176:
3173:
3170:
3167:
3164:
3161:
3158:
3153:
3149:
3140:
3126:
3118:
3115:
3111:
3107:
3102:
3098:
3088:
3085:
3079:
3076:
3073:
3070:
3045:
3042:
3038:
3034:
3029:
3025:
3015:
3012:
3006:
3003:
3000:
2997:
2990:
2974:
2966:
2961:
2948:
2945:
2942:
2939:
2936:
2933:
2930:
2927:
2924:
2921:
2901:
2898:
2895:
2892:
2889:
2886:
2883:
2880:
2871:
2855:
2852:
2849:
2846:
2843:
2840:
2837:
2834:
2831:
2828:
2808:
2805:
2802:
2799:
2796:
2793:
2790:
2787:
2779:
2769:
2766:
2762:
2758:
2752:
2748:
2744:
2736:
2730:
2726:the rapidity
2725:
2720:
2714:
2710:
2705:
2701:
2700:
2699:
2697:
2693:
2688:
2684:
2680:
2677:
2672:
2668:
2664:
2660:
2655:
2653:
2649:
2645:
2641:
2637:
2633:
2628:
2626:
2622:
2618:
2614:
2613:W.K. Clifford
2610:
2607:
2603:
2599:
2594:
2592:
2588:
2584:
2578:
2572:
2568:
2563:
2559:
2555:
2548:
2545:increased in
2544:
2540:
2537:increased in
2536:
2532:
2528:
2527:
2526:
2524:
2520:
2516:
2512:
2508:
2504:
2494:
2480:
2477:
2474:
2454:
2451:
2448:
2445:
2442:
2422:
2419:
2416:
2413:
2410:
2390:
2387:
2384:
2381:
2378:
2358:
2355:
2350:
2346:
2342:
2337:
2333:
2309:
2306:
2303:
2296:
2291:
2285:
2282:
2277:
2274:
2268:
2263:
2258:
2253:
2247:
2244:
2239:
2236:
2230:
2221:
2216:
2212:
2208:
2203:
2199:
2195:
2190:
2185:
2181:
2177:
2174:
2167:
2166:
2165:
2151:
2148:
2145:
2125:
2105:
2102:
2099:
2096:
2093:
2073:
2070:
2067:
2064:
2061:
2041:
2038:
2035:
2032:
2029:
2009:
2006:
2001:
1997:
1993:
1988:
1984:
1960:
1957:
1954:
1947:
1942:
1936:
1933:
1928:
1925:
1919:
1914:
1909:
1904:
1898:
1895:
1890:
1887:
1881:
1872:
1867:
1863:
1859:
1854:
1850:
1846:
1841:
1836:
1832:
1828:
1825:
1818:
1817:
1816:
1802:
1799:
1796:
1793:
1790:
1770:
1750:
1730:
1707:
1701:
1695:
1692:
1689:
1686:
1680:
1674:
1671:
1668:
1661:
1660:
1659:
1642:
1637:
1633:
1629:
1626:
1621:
1617:
1613:
1610:
1605:
1600:
1596:
1592:
1585:
1584:
1583:
1566:
1561:
1557:
1553:
1550:
1545:
1541:
1537:
1534:
1529:
1524:
1520:
1516:
1509:
1508:
1507:
1496:
1492:
1488:
1484:
1480:
1476:
1472:
1471:
1470:
1468:
1464:
1460:
1456:
1452:
1447:
1444:
1439:
1437:
1430:
1421:
1414:
1407:
1401:
1396:be the point
1395:
1386:
1379:
1370:
1361:
1354:
1345:
1334:
1329:
1324:
1318:
1314:
1308:
1304:
1298:
1290:
1283:
1275:
1268:
1261:
1255:
1248:
1242:
1239:
1234:
1232:
1228:
1224:
1219:
1217:
1201:
1198:
1193:
1189:
1185:
1180:
1176:
1168:
1164:
1148:
1145:
1140:
1136:
1132:
1127:
1123:
1115:
1106:
1098:
1089:
1075:
1072:
1069:
1066:
1063:
1060:
1057:
1054:
1048:
1044:
1040:
1034:
1031:
1028:
1021:
1018:
1011:
1008:
1000:
983:
976:
973:
970:
967:
964:
958:
952:
949:
946:
940:
934:
931:
928:
921:
908:
891:
884:
881:
878:
872:
866:
863:
860:
854:
848:
845:
842:
835:
806:
803:
800:
797:
794:
782:
779:
776:
770:
767:
759:
743:
740:
737:
734:
711:
708:
705:
679:
676:
673:
650:
647:
644:
641:
638:
618:
615:
612:
609:
606:
603:
600:
592:
576:
573:
570:
567:
564:
561:
558:
550:
542:
526:
523:
520:
517:
514:
506:
504:is unbounded.
503:
487:
484:
481:
473:
469:
468:
467:
465:
445:
442:
439:
431:
427:
423:
408:
405:
402:
376:
373:
368:
365:
338:
335:
332:
306:
303:
300:
289:
285:
281:
280:
279:
276:
262:
259:
256:
249:
229:
226:
223:
220:
212:
209:
204:
201:
180:
178:
174:
170:
166:
162:
157:
141:
131:
127:
109:
99:
95:
90:
88:
84:
80:
76:
72:
68:
64:
60:
56:
52:
48:
39:
34:
30:
26:
21:
4456:
4442:
4416:Euler at 300
4415:
4380:
4361:
4352:
4343:
4327:
4319:
4311:
4302:
4292:
4280:
4269:
4264:
4225:
4196:
3474:
3467:
3341:
3241:
3141:
2962:
2868:so that the
2775:
2764:
2760:
2756:
2753:) to obtain
2750:
2740:
2734:
2728:
2718:
2712:
2708:
2703:
2698:. He wrote:
2691:
2686:
2682:
2678:
2670:
2656:
2639:
2629:
2608:
2598:trigonometry
2595:
2576:
2570:
2566:
2552:
2522:
2500:
2324:
1975:
1722:
1657:
1581:
1505:
1494:
1490:
1486:
1482:
1478:
1474:
1448:
1440:
1435:
1428:
1419:
1412:
1405:
1399:
1393:
1384:
1377:
1368:
1359:
1352:
1343:
1332:
1330:
1322:
1316:
1312:
1306:
1302:
1296:
1288:
1281:
1273:
1266:
1259:
1253:
1246:
1240:
1235:
1220:
1111:
906:
548:
546:
540:
471:
461:
283:
277:
247:
186:
158:
93:
91:
77:, which has
66:
50:
44:
37:
32:
24:
2743:Lobachevsky
2617:parametrize
2521:in 1647 in
1457:on certain
1114:unit circle
179:variables.
98:rectangular
55:real number
4475:Categories
4402:References
4375:1(6):155–9
3924: even
3675: even
2646:published
2611:. In 1878
2531:asymptotes
2503:quadrature
2086:. Letting
1374:the point
1349:the point
663:, so that
593:such that
183:Definition
4174:…
4171:−
4121:−
4063: odd
4055:∑
4040:
4027:…
4024:−
3974:−
3916:∑
3902:
3892:…
3800: odd
3792:∑
3777:
3767:…
3667:∑
3653:
3643:…
3521:∞
3506:∑
3445:
3430:
3415:
3403:
3213:
3204:−
3189:
3177:
3165:
3116:−
3108:−
3074:
3043:−
3001:
2943:
2925:
2899:
2884:
2850:
2832:
2806:
2791:
2711:= (.7616)
2596:Circular
2558:logarithm
2543:abscissas
2515:asymptote
2507:hyperbola
2481:η
2455:η
2446:⩽
2420:
2388:
2343:−
2264:−
2213:∫
2182:∫
2152:θ
2106:θ
2097:⩽
2071:
2039:
1864:∫
1833:∫
1794:⩽
1627:−
1186:−
1073:
1067:−
1061:
1035:
1012:
917:∠
831:∠
789:→
485:
474:(because
472:unbounded
443:
4253:Archived
4244:Archived
4234:Archived
4205:See also
2673:, where
2667:rapidity
2630:In 1894
2606:textbook
1497:> 0 .
1493:), with
1305:, where
541:directed
395:, where
173:calculus
163:for the
47:geometry
4455:(1998)
4435:(1912)
4387:(1914)
4368:(1895)
4350:(1894)
4334:(1849)
4318:(1925)
2694:to the
2604:in his
2505:of the
2497:History
1463:rotates
1455:measure
905:to the
286:is the
96:= 1 is
4463:
4425:
2755:cos Π(
2138:gives
1367:, and
1340:(1, 1)
1238:chords
1214:has a
1161:has a
909:angle
248:branch
4481:Angle
4217:Notes
2657:When
1481:) ↦ (
288:angle
126:angle
53:is a
4461:ISBN
4423:ISBN
3774:sinh
3650:cosh
3412:sinh
3400:cosh
3174:sinh
3162:cosh
3071:sinh
3063:and
2998:cosh
2922:sinh
2914:and
2881:cosh
2847:sinh
2821:and
2803:cosh
2759:) =
2716:for
2704:unit
2676:tanh
2569:= 1/
2541:the
2535:area
2501:The
2452:<
2417:sinh
2403:and
2385:cosh
2103:<
2054:and
1800:<
1763:and
1411:, 1/
1383:, 1/
1358:, 1/
1272:and
1252:and
695:and
648:>
642:>
631:and
589:are
524:<
518:<
426:area
406:>
260:>
227:>
177:real
59:area
4155:120
4037:sin
3899:cos
3873:120
3468:As
3442:sin
3427:cos
3210:sin
3186:cos
2940:sin
2896:cos
2829:sin
2788:cos
2731:= 1
2721:= 1
2627:".
2579:= 1
2068:sin
2036:cos
1976:If
1229:in
290:at
175:of
65:of
45:In
40:= 1
35:if
4477::
4301:.
4008:24
3748:24
3624:24
3472:,
2780:,
2768:.
2749:Π(
2681:=
2654:.
2619:a
2593:.
1489:/
1485:,
1483:rx
1477:,
1438:.
1418:1/
1342:,
1328:.
1287:+
1112:A
1088:.
1070:ln
1058:ln
1032:ln
1009:ln
482:ln
466::
440:ln
275:.
156:.
94:xy
89:.
67:xy
49:,
25:xy
4467:.
4449:.
4429:.
4305:.
4178:)
4166:5
4162:z
4152:1
4146:+
4141:3
4137:z
4130:6
4127:1
4118:z
4114:(
4110:i
4107:=
4096:!
4093:k
4086:k
4082:)
4078:z
4075:i
4072:(
4059:k
4050:=
4043:z
4034:i
4019:4
4015:z
4005:1
3999:+
3994:2
3990:z
3983:2
3980:1
3971:1
3968:=
3957:!
3954:k
3947:k
3943:)
3939:z
3936:i
3933:(
3920:k
3912:=
3905:z
3889:+
3884:5
3880:z
3870:1
3864:+
3859:3
3855:z
3848:6
3845:1
3839:+
3836:z
3833:=
3822:!
3819:k
3813:k
3809:z
3796:k
3787:=
3780:z
3764:+
3759:4
3755:z
3745:1
3739:+
3734:2
3730:z
3723:2
3720:1
3714:+
3711:1
3708:=
3697:!
3694:k
3688:k
3684:z
3671:k
3663:=
3656:z
3640:+
3635:4
3631:z
3621:1
3615:+
3610:3
3606:z
3599:6
3596:1
3590:+
3585:2
3581:z
3574:2
3571:1
3565:+
3562:z
3559:+
3556:1
3553:=
3542:!
3539:k
3533:k
3529:z
3516:0
3513:=
3510:k
3500:=
3491:z
3487:e
3454:.
3451:)
3448:y
3439:i
3436:+
3433:y
3424:(
3421:)
3418:x
3409:+
3406:x
3397:(
3394:=
3389:y
3386:i
3382:e
3376:x
3372:e
3368:=
3363:y
3360:i
3357:+
3354:x
3350:e
3328:,
3323:y
3320:i
3316:e
3293:x
3289:e
3268:,
3265:y
3262:i
3259:+
3256:x
3253:=
3250:z
3228:,
3225:)
3222:z
3219:i
3216:(
3207:i
3201:)
3198:z
3195:i
3192:(
3183:=
3180:z
3171:+
3168:z
3159:=
3154:z
3150:e
3127:,
3124:)
3119:z
3112:e
3103:z
3099:e
3095:(
3089:2
3086:1
3080:=
3077:z
3051:)
3046:z
3039:e
3035:+
3030:z
3026:e
3022:(
3016:2
3013:1
3007:=
3004:z
2975:z
2949:.
2946:x
2937:i
2934:=
2931:x
2928:i
2902:x
2893:=
2890:x
2887:i
2856:,
2853:x
2844:i
2841:=
2838:x
2835:i
2809:x
2800:=
2797:x
2794:i
2765:c
2763:/
2761:v
2757:a
2751:a
2735:c
2729:a
2723:.
2719:a
2713:c
2709:v
2692:v
2687:c
2685:/
2683:v
2679:a
2671:a
2577:x
2571:x
2567:y
2549:.
2478:=
2475:S
2449:t
2443:0
2423:t
2414:=
2411:x
2391:t
2382:=
2379:y
2359:1
2356:=
2351:2
2347:x
2338:2
2334:y
2310:,
2307:t
2304:d
2297:2
2292:)
2286:t
2283:d
2278:y
2275:d
2269:(
2259:2
2254:)
2248:t
2245:d
2240:x
2237:d
2231:(
2222:b
2217:a
2209:=
2204:m
2200:s
2196:d
2191:b
2186:a
2178:=
2175:S
2149:=
2146:S
2126:S
2100:t
2094:0
2074:t
2065:=
2062:y
2042:t
2033:=
2030:x
2010:1
2007:=
2002:2
1998:y
1994:+
1989:2
1985:x
1961:.
1958:t
1955:d
1948:2
1943:)
1937:t
1934:d
1929:y
1926:d
1920:(
1915:+
1910:2
1905:)
1899:t
1896:d
1891:x
1888:d
1882:(
1873:b
1868:a
1860:=
1855:e
1851:s
1847:d
1842:b
1837:a
1829:=
1826:S
1803:b
1797:t
1791:a
1783:(
1771:b
1751:a
1731:t
1708:.
1705:)
1702:t
1699:(
1696:g
1693:=
1690:y
1687:,
1684:)
1681:t
1678:(
1675:f
1672:=
1669:x
1643:.
1638:2
1634:y
1630:d
1622:2
1618:x
1614:d
1611:=
1606:2
1601:m
1597:s
1593:d
1567:,
1562:2
1558:y
1554:d
1551:+
1546:2
1542:x
1538:d
1535:=
1530:2
1525:e
1521:s
1517:d
1495:r
1491:r
1487:y
1479:y
1475:x
1473:(
1436:x
1432:0
1429:P
1425:)
1423:2
1420:x
1416:1
1413:x
1409:2
1406:x
1403:1
1400:x
1398:(
1394:Q
1390:)
1388:2
1385:x
1381:2
1378:x
1376:(
1372:2
1369:P
1365:)
1363:1
1360:x
1356:1
1353:x
1351:(
1347:1
1344:P
1336:0
1333:P
1326:2
1323:P
1320:1
1317:P
1313:Q
1310:0
1307:P
1303:Q
1300:0
1297:P
1292:2
1289:L
1285:1
1282:L
1277:2
1274:L
1270:1
1267:L
1263:2
1260:P
1257:0
1254:P
1250:1
1247:P
1244:0
1241:P
1202:1
1199:=
1194:2
1190:y
1181:2
1177:x
1149:1
1146:=
1141:2
1137:y
1133:+
1128:2
1124:x
1076:a
1064:c
1055:=
1052:)
1049:a
1045:/
1041:c
1038:(
1029:=
1025:)
1022:c
1019:b
1016:(
984:)
980:)
977:d
974:a
971:,
968:c
965:b
962:(
959:,
956:)
953:0
950:,
947:0
944:(
941:,
938:)
935:1
932:,
929:1
926:(
922:(
892:)
888:)
885:d
882:,
879:c
876:(
873:,
870:)
867:0
864:,
861:0
858:(
855:,
852:)
849:b
846:,
843:a
840:(
836:(
810:)
807:y
804:a
801:,
798:x
795:b
792:(
786:)
783:y
780:,
777:x
774:(
771::
768:f
744:1
741:=
738:y
735:x
715:)
712:d
709:,
706:c
703:(
683:)
680:b
677:,
674:a
671:(
651:1
645:a
639:c
619:1
616:=
613:d
610:c
607:=
604:b
601:a
577:d
574:,
571:c
568:,
565:b
562:,
559:a
543:.
527:1
521:x
515:0
488:x
458:.
446:x
421:.
409:1
403:x
382:)
377:x
374:1
369:,
366:x
363:(
342:)
339:1
336:,
333:1
330:(
310:)
307:0
304:,
301:0
298:(
263:1
257:x
233:}
230:0
224:x
221::
218:)
213:x
210:1
205:,
202:x
199:(
196:{
142:2
110:2
38:a
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