131:
2020:
25:
1758:
1086:
1693:
3667:
3348:
1501:
2015:{\displaystyle {\begin{aligned}H(x)&=\lim _{\varepsilon \to 0^{+}}-{\frac {1}{2\pi i}}\int _{-\infty }^{\infty }{\frac {1}{\tau +i\varepsilon }}e^{-ix\tau }d\tau \\&=\lim _{\varepsilon \to 0^{+}}{\frac {1}{2\pi i}}\int _{-\infty }^{\infty }{\frac {1}{\tau -i\varepsilon }}e^{ix\tau }d\tau .\end{aligned}}}
1705:. In general, however, pointwise convergence need not imply distributional convergence, and vice versa distributional convergence need not imply pointwise convergence. (However, if all members of a pointwise convergent sequence of functions are uniformly bounded by some "nice" function, then
3469:
3156:
1490:
2690:
314:, where it represents a signal that switches on at a specified time and stays switched on indefinitely. Heaviside developed the operational calculus as a tool in the analysis of telegraphic communications and represented the function as
1315:
777:
1688:{\displaystyle {\begin{aligned}H(x)&=\lim _{k\to \infty }\left({\tfrac {1}{2}}+{\tfrac {1}{\pi }}\arctan kx\right)\\H(x)&=\lim _{k\to \infty }\left({\tfrac {1}{2}}+{\tfrac {1}{2}}\operatorname {erf} kx\right)\end{aligned}}}
3448:
1185:
568:
426:
231:
2562:
656:
3065:
2206:
1355:
1025:
2465:
3474:
2407:
2310:
1763:
1506:
3140:
2894:
952:
3662:{\displaystyle {\begin{aligned}{\hat {H}}(s)&=\lim _{N\to \infty }\int _{0}^{N}e^{-sx}H(x)\,dx\\&=\lim _{N\to \infty }\int _{0}^{N}e^{-sx}\,dx\\&={\frac {1}{s}}\end{aligned}}}
2571:
2415:
to preserve the continuity of the limiting functions and ensure the existence of certain solutions. In these cases, the
Heaviside function returns a whole interval of possible solutions,
875:
2946:
1209:
3343:{\displaystyle {\hat {H}}(s)=\lim _{N\to \infty }\int _{-N}^{N}e^{-2\pi ixs}H(x)\,dx={\frac {1}{2}}\left(\delta (s)-{\frac {i}{\pi }}\operatorname {p.v.} {\frac {1}{s}}\right).}
668:
2818:
490:
479:
346:
151:
2476:
2036:
is usually used in integration, and the value of a function at a single point does not affect its integral, it rarely matters what particular value is chosen of
579:
2973:
3388:
1091:
2143:
2024:
where the second representation is easy to deduce from the first, given that the step function is real and thus is its own complex conjugate.
961:
2344:
2247:
3080:
89:
2831:
895:
3153:
of the
Heaviside step function is a distribution. Using one choice of constants for the definition of the Fourier transform we have
1727:
can serve as an approximation, in the limit as the variance approaches zero. For example, all three of the above approximations are
61:
42:
811:
1706:
68:
2434:
2411:
In functional-analysis contexts from optimization and game theory, it is often useful to define the
Heaviside function as a
2903:
75:
2229:
3791:
108:
57:
2225:
1728:
1713:
1039:
2755:
3887:
1485:{\displaystyle H(x)=\lim _{k\to \infty }{\tfrac {1}{2}}(1+\tanh kx)=\lim _{k\to \infty }{\frac {1}{1+e^{-2kx}}}.}
3882:
3706:
46:
3671:
When the bilateral transform is used, the integral can be split in two parts and the result will be the same.
1079:
437:
1075:
299:
are in use. It is an example of the general class of step functions, all of which can be represented as
3877:
3701:
3374:
2048:
1702:
3070:
2685:{\displaystyle H={\begin{cases}0,&n<0,\\{\tfrac {1}{2}},&n=0,\\1,&n>0,\end{cases}}}
1752:
787:
82:
2595:
2500:
370:
175:
1720:
1051:
2951:
2313:
1717:
35:
3382:
3450:. The limit appearing in the integral is also taken in the sense of (tempered) distributions.
2066:) it does not even make sense to talk of a value at zero, since such objects are only defined
3822:
1732:
311:
1495:
1310:{\displaystyle H(x)\approx {\tfrac {1}{2}}+{\tfrac {1}{2}}\tanh kx={\frac {1}{1+e^{-2kx}}},}
3860:
3835:
3761:
3721:
3681:
3463:
3074:
2412:
885:
772:{\displaystyle H(x)=\left(1-{\frac {1}{2\pi i}}\log z,\ -{\frac {1}{2\pi i}}\log z\right).}
307:
8:
2103:
1740:
1736:
3686:
2237:
1082:) may be used to approximate binary cellular switches in response to chemical signals.
484:
340:
300:
3852:
3787:
3758:
3711:
3696:
3459:
3150:
2067:
1204:
1071:
2738:, and cannot authentically be a step function, using the half-maximum convention.
2221:
281:
3831:
3716:
3691:
3443:{\displaystyle \textstyle \int _{-\infty }^{\infty }{\frac {\varphi (s)}{s}}\,ds}
2823:
2750:
The discrete-time unit impulse is the first difference of the discrete-time step
2330:
2228:
are usually taken to be right continuous, as are functions integrated against in
1200:
1043:
431:
1180:{\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{2}}\tanh(kx)={\frac {1}{1+e^{-2kx}}}}
2967:
2734:. Therefore the "step function" exhibits ramp-like behavior over the domain of
1034:, depending on which formalism one uses to give meaning to integrals involving
805:
563:{\displaystyle H(x):=\mathbf {1} _{x\geq 0}=\mathbf {1} _{\mathbb {R} _{+}}(x)}
1085:
3871:
3736:
3731:
3726:
2963:
2133:
1047:
662:
573:
421:{\displaystyle H(x):={\begin{cases}1,&x\geq 0\\0,&x<0\end{cases}}}
277:
226:{\displaystyle H(x):={\begin{cases}1,&x\geq 0\\0,&x<0\end{cases}}}
2557:{\displaystyle H={\begin{cases}0,&n<0,\\1,&n\geq 0,\end{cases}}}
2129:
1067:
1063:
651:{\displaystyle H(x):={\frac {d}{dx}}\max\{x,0\}\quad {\mbox{for }}x\neq 0}
130:
2074:) then often whatever happens to be the relevant limit at zero is used.
3819:
Heaviside's
Operational Calculus, as applied to Engineering and Physics
3060:{\displaystyle \int _{-\infty }^{x}H(\xi )\,d\xi =xH(x)=\max\{0,x\}\,.}
2241:
889:
289:
285:
3766:
1698:
2431:
An alternative form of the unit step, defined instead as a function
292:
for positive arguments. Different conventions concerning the value
24:
2338:
2058:
1724:
955:
2699:
1723:
that is peaked around zero and has a parameter that controls for
661:
Furthermore, the
Heaviside step function can be represented as a
3756:
3811:
2201:{\displaystyle H(x)={\tfrac {1}{2}}(1+\operatorname {sgn} x).}
2077:
There exist various reasons for choosing a particular value.
1027:
although this expansion may not hold (or even make sense) for
135:
The
Heaviside step function, using the half-maximum convention
1062:
Approximations to the
Heaviside step function are of use in
2678:
2550:
414:
219:
958:
of the Dirac delta function. This is sometimes written as
1755:
representation of the
Heaviside step function is useful:
954:
Hence the
Heaviside function can be considered to be the
1020:{\displaystyle H(x):=\int _{-\infty }^{x}\delta (s)\,ds}
3392:
2623:
2460:{\displaystyle H:\mathbb {Z} \rightarrow \mathbb {R} }
2163:
1653:
1638:
1565:
1550:
1391:
1244:
1229:
1111:
1096:
633:
3472:
3391:
3159:
3083:
2976:
2906:
2834:
2758:
2574:
2479:
2437:
2347:
2250:
2146:
1761:
1504:
1358:
1212:
1094:
964:
898:
814:
671:
582:
493:
440:
349:
154:
1203:
approximation to the step function, one can use the
3786:(3rd ed.). New York: McGraw-Hill. p. 61.
2402:{\displaystyle H(x)=\mathbf {1} _{(0,\infty )}(x).}
2305:{\displaystyle H(x)=\mathbf {1} _{[0,\infty )}(x).}
1707:
convergence holds in the sense of distributions too
1498:to the step function. Among the possibilities are:
49:. Unsourced material may be challenged and removed.
3661:
3466:. Using the unilateral Laplace transform we have:
3442:
3342:
3134:
3059:
2940:
2888:
2812:
2684:
2556:
2459:
2401:
2312:The corresponding probability distribution is the
2304:
2200:
2070:. If using some analytic approximation (as in the
2014:
1687:
1484:
1309:
1179:
1019:
946:
869:
771:
650:
562:
473:
420:
225:
3843:Davies, Brian (2002). "Heaviside step function".
3810:Digital Library of Mathematical Functions, NIST,
3135:{\displaystyle {\frac {dH(x)}{dx}}=\delta (x)\,.}
2822:This function is the cumulative summation of the
1038:. In this context, the Heaviside function is the
3869:
3855:; Naylor, D. (1966). "Heaviside unit function".
3832:"Heaviside, Laplace, and the Inversion Integral"
3582:
3506:
3185:
3035:
2957:
1904:
1786:
1617:
1529:
1430:
1375:
879:
613:
3453:
2889:{\displaystyle H=\sum _{k=-\infty }^{n}\delta }
2132:. In this case the following relation with the
2106:then has rotational symmetry; put another way,
947:{\displaystyle \delta (x)={\frac {d}{dx}}H(x).}
2741:Unlike the continuous case, the definition of
2727:must imply that the function attains unity at
1074:approximations of step functions (such as the
3857:Differential Equations of Applied Mathematics
3817:Berg, Ernst Julius (1936). "Unit function".
3050:
3038:
2071:
628:
616:
335:, the Heaviside function may be defined as:
1746:
870:{\displaystyle H(x)={\frac {x+|x|}{2x}}\,.}
3851:
3845:Integral Transforms and their Applications
3784:The Fourier transform and its applications
1496:many other smooth, analytic approximations
1057:
3781:
3628:
3564:
3432:
3255:
3128:
3053:
3007:
2453:
2445:
1731:of common probability distributions: the
1010:
863:
539:
306:The function was originally developed in
109:Learn how and when to remove this message
2467:(that is, taking in a discrete variable
1084:
788:principal value of the complex logarithm
3829:
1323:corresponds to a sharper transition at
3870:
3842:
3073:of the Heaviside step function is the
2566:or using the half-maximum convention:
16:Indicator function of positive numbers
3847:(3rd ed.). Springer. p. 28.
3757:
2941:{\displaystyle \delta =\delta _{k,0}}
3816:
3462:of the Heaviside step function is a
3144:
47:adding citations to reliable sources
18:
13:
3782:Bracewell, Ronald Newbold (2000).
3592:
3516:
3406:
3401:
3312:
3306:
3195:
2985:
2864:
2379:
2282:
1957:
1952:
1842:
1837:
1627:
1539:
1440:
1385:
988:
14:
3899:
3804:
2970:of the Heaviside step function:
2226:cumulative distribution functions
1729:cumulative distribution functions
2426:
2365:
2268:
2027:
1714:cumulative distribution function
1188:approaches the step function as
1040:cumulative distribution function
532:
511:
129:
23:
2337:is an indicator function of an
1352:, equality holds in the limit:
631:
34:needs additional citations for
3775:
3750:
3707:List of mathematical functions
3589:
3561:
3555:
3513:
3495:
3489:
3483:
3423:
3417:
3289:
3283:
3252:
3246:
3192:
3178:
3172:
3166:
3125:
3119:
3099:
3093:
3029:
3023:
3004:
2998:
2952:discrete unit impulse function
2916:
2910:
2883:
2877:
2844:
2838:
2804:
2792:
2783:
2777:
2768:
2762:
2584:
2578:
2489:
2483:
2449:
2393:
2387:
2382:
2370:
2357:
2351:
2296:
2290:
2285:
2273:
2260:
2254:
2230:LebesgueāStieltjes integration
2192:
2174:
2156:
2150:
1911:
1793:
1775:
1769:
1624:
1606:
1600:
1536:
1518:
1512:
1437:
1423:
1402:
1382:
1368:
1362:
1222:
1216:
1137:
1128:
1007:
1001:
974:
968:
938:
932:
908:
902:
848:
840:
824:
818:
681:
675:
592:
586:
557:
551:
503:
497:
468:
456:
450:
444:
359:
353:
323:
164:
158:
1:
3743:
2958:Antiderivative and derivative
1743:distributions, respectively.
880:Relationship with Dirac delta
797:It can also be expressed for
303:of translations of this one.
3454:Unilateral Laplace transform
2813:{\displaystyle \delta =H-H.}
7:
3674:
3377:that takes a test function
892:of the Heaviside function:
328:Taking the convention that
288:for negative arguments and
10:
3906:
3830:Calvert, James B. (2002).
3702:Laplacian of the indicator
1080:MichaelisāMenten equations
3762:"Heaviside Step Function"
3071:distributional derivative
236:
145:
140:
128:
123:
58:"Heaviside step function"
2341:semi-infinite interval:
2244:semi-infinite interval:
2102:is often used since the
1747:Integral representations
1721:probability distribution
1052:Constant random variable
284:, the value of which is
2314:degenerate distribution
1058:Analytic approximations
248:Heaviside step function
3888:Schwartz distributions
3663:
3444:
3383:Cauchy principal value
3344:
3136:
3061:
2942:
2890:
2873:
2814:
2686:
2558:
2461:
2403:
2306:
2202:
2016:
1689:
1486:
1311:
1196:
1181:
1021:
948:
871:
773:
652:
572:the derivative of the
564:
475:
474:{\displaystyle H(x):=}
422:
312:differential equations
227:
3883:Generalized functions
3861:John Wiley & Sons
3823:McGraw-Hill Education
3664:
3445:
3345:
3137:
3062:
2943:
2891:
2850:
2815:
2736:[−1, 1]
2687:
2559:
2462:
2404:
2307:
2203:
2017:
1690:
1487:
1312:
1182:
1088:
1022:
949:
872:
774:
653:
565:
476:
423:
254:, usually denoted by
237:Fields of application
228:
3836:University of Denver
3722:Rectangular function
3682:Dirac delta function
3470:
3464:meromorphic function
3389:
3157:
3081:
3075:Dirac delta function
2974:
2904:
2832:
2756:
2706:is an integer, then
2572:
2477:
2435:
2345:
2248:
2144:
1759:
1701:and in the sense of
1502:
1356:
1210:
1092:
962:
896:
886:Dirac delta function
812:
669:
580:
491:
438:
347:
310:for the solution of
308:operational calculus
240:Operational calculus
152:
43:improve this article
3611:
3535:
3410:
3217:
2994:
2413:set-valued function
2047:is considered as a
1961:
1846:
997:
301:linear combinations
141:General information
3853:Duff, George F. D.
3759:Weisstein, Eric W.
3687:Indicator function
3659:
3657:
3597:
3596:
3521:
3520:
3440:
3439:
3393:
3340:
3200:
3199:
3132:
3057:
2977:
2938:
2886:
2810:
2682:
2677:
2632:
2554:
2549:
2457:
2399:
2302:
2238:indicator function
2198:
2172:
2012:
2010:
1944:
1925:
1829:
1807:
1697:These limits hold
1685:
1683:
1662:
1647:
1631:
1574:
1559:
1543:
1482:
1444:
1400:
1389:
1307:
1253:
1238:
1197:
1177:
1120:
1105:
1017:
980:
944:
867:
769:
648:
637:
560:
485:indicator function
471:
418:
413:
341:piecewise function
252:unit step function
223:
218:
146:General definition
3878:Special functions
3712:Macaulay brackets
3697:Laplace transform
3653:
3581:
3505:
3486:
3460:Laplace transform
3430:
3330:
3303:
3273:
3184:
3169:
3151:Fourier transform
3145:Fourier transform
3111:
2631:
2171:
2068:almost everywhere
2051:or an element of
1981:
1942:
1903:
1866:
1827:
1785:
1661:
1646:
1616:
1573:
1558:
1528:
1477:
1429:
1399:
1374:
1302:
1252:
1237:
1205:logistic function
1175:
1119:
1104:
927:
861:
750:
730:
714:
636:
611:
244:
243:
119:
118:
111:
93:
3895:
3864:
3848:
3839:
3826:
3798:
3797:
3779:
3773:
3772:
3771:
3754:
3668:
3666:
3665:
3660:
3658:
3654:
3646:
3638:
3627:
3626:
3610:
3605:
3595:
3574:
3551:
3550:
3534:
3529:
3519:
3488:
3487:
3479:
3449:
3447:
3446:
3441:
3431:
3426:
3412:
3409:
3404:
3380:
3372:
3371:
3369:
3368:
3363:
3360:
3349:
3347:
3346:
3341:
3336:
3332:
3331:
3323:
3318:
3304:
3296:
3274:
3266:
3242:
3241:
3216:
3211:
3198:
3171:
3170:
3162:
3141:
3139:
3138:
3133:
3112:
3110:
3102:
3085:
3066:
3064:
3063:
3058:
2993:
2988:
2947:
2945:
2944:
2939:
2937:
2936:
2895:
2893:
2892:
2887:
2872:
2867:
2819:
2817:
2816:
2811:
2747:is significant.
2746:
2737:
2733:
2726:
2719:
2713:must imply that
2712:
2705:
2697:
2691:
2689:
2688:
2683:
2681:
2680:
2633:
2624:
2563:
2561:
2560:
2555:
2553:
2552:
2470:
2466:
2464:
2463:
2458:
2456:
2448:
2421:
2408:
2406:
2405:
2400:
2386:
2385:
2368:
2336:
2328:
2324:
2311:
2309:
2308:
2303:
2289:
2288:
2271:
2235:
2222:right-continuous
2219:
2215:
2207:
2205:
2204:
2199:
2173:
2164:
2139:
2127:
2126:
2124:
2123:
2120:
2117:
2101:
2100:
2098:
2097:
2094:
2091:
2063:
2056:
2046:
2042:
2035:
2021:
2019:
2018:
2013:
2011:
1998:
1997:
1982:
1980:
1963:
1960:
1955:
1943:
1941:
1927:
1924:
1923:
1922:
1896:
1886:
1885:
1867:
1865:
1848:
1845:
1840:
1828:
1826:
1812:
1806:
1805:
1804:
1712:In general, any
1694:
1692:
1691:
1686:
1684:
1680:
1676:
1663:
1654:
1648:
1639:
1630:
1592:
1588:
1575:
1566:
1560:
1551:
1542:
1491:
1489:
1488:
1483:
1478:
1476:
1475:
1474:
1446:
1443:
1401:
1392:
1388:
1351:
1350:
1348:
1347:
1344:
1341:
1329:
1322:
1316:
1314:
1313:
1308:
1303:
1301:
1300:
1299:
1271:
1254:
1245:
1239:
1230:
1194:
1186:
1184:
1183:
1178:
1176:
1174:
1173:
1172:
1144:
1121:
1112:
1106:
1097:
1037:
1033:
1026:
1024:
1023:
1018:
996:
991:
953:
951:
950:
945:
928:
926:
915:
876:
874:
873:
868:
862:
860:
852:
851:
843:
831:
804:in terms of the
803:
793:
785:
778:
776:
775:
770:
765:
761:
751:
749:
735:
728:
715:
713:
699:
657:
655:
654:
649:
638:
634:
612:
610:
599:
569:
567:
566:
561:
550:
549:
548:
547:
542:
535:
526:
525:
514:
480:
478:
477:
472:
427:
425:
424:
419:
417:
416:
334:
319:
298:
282:Oliver Heaviside
275:
271:
265:
261:
257:
232:
230:
229:
224:
222:
221:
133:
121:
120:
114:
107:
103:
100:
94:
92:
51:
27:
19:
3905:
3904:
3898:
3897:
3896:
3894:
3893:
3892:
3868:
3867:
3807:
3802:
3801:
3794:
3780:
3776:
3755:
3751:
3746:
3741:
3717:Negative number
3692:Iverson bracket
3677:
3656:
3655:
3645:
3636:
3635:
3616:
3612:
3606:
3601:
3585:
3572:
3571:
3540:
3536:
3530:
3525:
3509:
3498:
3478:
3477:
3473:
3471:
3468:
3467:
3456:
3413:
3411:
3405:
3397:
3390:
3387:
3386:
3378:
3364:
3361:
3358:
3357:
3355:
3353:
3322:
3305:
3295:
3279:
3275:
3265:
3222:
3218:
3212:
3204:
3188:
3161:
3160:
3158:
3155:
3154:
3147:
3103:
3086:
3084:
3082:
3079:
3078:
2989:
2981:
2975:
2972:
2971:
2960:
2926:
2922:
2905:
2902:
2901:
2868:
2854:
2833:
2830:
2829:
2824:Kronecker delta
2757:
2754:
2753:
2742:
2735:
2728:
2721:
2714:
2707:
2703:
2695:
2676:
2675:
2661:
2652:
2651:
2637:
2622:
2619:
2618:
2604:
2591:
2590:
2573:
2570:
2569:
2548:
2547:
2533:
2524:
2523:
2509:
2496:
2495:
2478:
2475:
2474:
2468:
2452:
2444:
2436:
2433:
2432:
2429:
2416:
2369:
2364:
2363:
2346:
2343:
2342:
2334:
2333:. In this case
2331:left-continuous
2326:
2319:
2272:
2267:
2266:
2249:
2246:
2245:
2233:
2232:. In this case
2224:. For instance
2217:
2210:
2162:
2145:
2142:
2141:
2137:
2121:
2118:
2115:
2114:
2112:
2107:
2095:
2092:
2089:
2088:
2086:
2081:
2059:
2052:
2044:
2037:
2033:
2030:
2009:
2008:
1987:
1983:
1967:
1962:
1956:
1948:
1931:
1926:
1918:
1914:
1907:
1894:
1893:
1872:
1868:
1852:
1847:
1841:
1833:
1816:
1811:
1800:
1796:
1789:
1778:
1762:
1760:
1757:
1756:
1749:
1682:
1681:
1652:
1637:
1636:
1632:
1620:
1609:
1594:
1593:
1564:
1549:
1548:
1544:
1532:
1521:
1505:
1503:
1500:
1499:
1461:
1457:
1450:
1445:
1433:
1390:
1378:
1357:
1354:
1353:
1345:
1342:
1339:
1338:
1336:
1331:
1324:
1320:
1319:where a larger
1286:
1282:
1275:
1270:
1243:
1228:
1211:
1208:
1207:
1189:
1187:
1159:
1155:
1148:
1143:
1110:
1095:
1093:
1090:
1089:
1060:
1044:random variable
1035:
1028:
992:
984:
963:
960:
959:
919:
914:
897:
894:
893:
882:
853:
847:
839:
832:
830:
813:
810:
809:
798:
791:
780:
739:
734:
703:
698:
691:
687:
670:
667:
666:
632:
603:
598:
581:
578:
577:
543:
538:
537:
536:
531:
530:
515:
510:
509:
492:
489:
488:
439:
436:
435:
432:Iverson bracket
412:
411:
400:
391:
390:
379:
366:
365:
348:
345:
344:
329:
326:
315:
293:
273:
267:
263:
262:(but sometimes
259:
255:
217:
216:
205:
196:
195:
184:
171:
170:
153:
150:
149:
136:
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
3903:
3902:
3891:
3890:
3885:
3880:
3866:
3865:
3849:
3840:
3827:
3814:
3806:
3805:External links
3803:
3800:
3799:
3792:
3774:
3748:
3747:
3745:
3742:
3740:
3739:
3734:
3729:
3724:
3719:
3714:
3709:
3704:
3699:
3694:
3689:
3684:
3678:
3676:
3673:
3652:
3649:
3644:
3641:
3639:
3637:
3634:
3631:
3625:
3622:
3619:
3615:
3609:
3604:
3600:
3594:
3591:
3588:
3584:
3580:
3577:
3575:
3573:
3570:
3567:
3563:
3560:
3557:
3554:
3549:
3546:
3543:
3539:
3533:
3528:
3524:
3518:
3515:
3512:
3508:
3504:
3501:
3499:
3497:
3494:
3491:
3485:
3482:
3476:
3475:
3455:
3452:
3438:
3435:
3429:
3425:
3422:
3419:
3416:
3408:
3403:
3400:
3396:
3339:
3335:
3329:
3326:
3321:
3317:
3314:
3311:
3308:
3302:
3299:
3294:
3291:
3288:
3285:
3282:
3278:
3272:
3269:
3264:
3261:
3258:
3254:
3251:
3248:
3245:
3240:
3237:
3234:
3231:
3228:
3225:
3221:
3215:
3210:
3207:
3203:
3197:
3194:
3191:
3187:
3183:
3180:
3177:
3174:
3168:
3165:
3146:
3143:
3131:
3127:
3124:
3121:
3118:
3115:
3109:
3106:
3101:
3098:
3095:
3092:
3089:
3056:
3052:
3049:
3046:
3043:
3040:
3037:
3034:
3031:
3028:
3025:
3022:
3019:
3016:
3013:
3010:
3006:
3003:
3000:
2997:
2992:
2987:
2984:
2980:
2968:antiderivative
2959:
2956:
2935:
2932:
2929:
2925:
2921:
2918:
2915:
2912:
2909:
2885:
2882:
2879:
2876:
2871:
2866:
2863:
2860:
2857:
2853:
2849:
2846:
2843:
2840:
2837:
2809:
2806:
2803:
2800:
2797:
2794:
2791:
2788:
2785:
2782:
2779:
2776:
2773:
2770:
2767:
2764:
2761:
2679:
2674:
2671:
2668:
2665:
2662:
2660:
2657:
2654:
2653:
2650:
2647:
2644:
2641:
2638:
2636:
2630:
2627:
2621:
2620:
2617:
2614:
2611:
2608:
2605:
2603:
2600:
2597:
2596:
2594:
2589:
2586:
2583:
2580:
2577:
2551:
2546:
2543:
2540:
2537:
2534:
2532:
2529:
2526:
2525:
2522:
2519:
2516:
2513:
2510:
2508:
2505:
2502:
2501:
2499:
2494:
2491:
2488:
2485:
2482:
2455:
2451:
2447:
2443:
2440:
2428:
2425:
2424:
2423:
2409:
2398:
2395:
2392:
2389:
2384:
2381:
2378:
2375:
2372:
2367:
2362:
2359:
2356:
2353:
2350:
2317:
2301:
2298:
2295:
2292:
2287:
2284:
2281:
2278:
2275:
2270:
2265:
2262:
2259:
2256:
2253:
2208:
2197:
2194:
2191:
2188:
2185:
2182:
2179:
2176:
2170:
2167:
2161:
2158:
2155:
2152:
2149:
2136:holds for all
2072:examples above
2043:. Indeed when
2029:
2026:
2007:
2004:
2001:
1996:
1993:
1990:
1986:
1979:
1976:
1973:
1970:
1966:
1959:
1954:
1951:
1947:
1940:
1937:
1934:
1930:
1921:
1917:
1913:
1910:
1906:
1902:
1899:
1897:
1895:
1892:
1889:
1884:
1881:
1878:
1875:
1871:
1864:
1861:
1858:
1855:
1851:
1844:
1839:
1836:
1832:
1825:
1822:
1819:
1815:
1810:
1803:
1799:
1795:
1792:
1788:
1784:
1781:
1779:
1777:
1774:
1771:
1768:
1765:
1764:
1748:
1745:
1679:
1675:
1672:
1669:
1666:
1660:
1657:
1651:
1645:
1642:
1635:
1629:
1626:
1623:
1619:
1615:
1612:
1610:
1608:
1605:
1602:
1599:
1596:
1595:
1591:
1587:
1584:
1581:
1578:
1572:
1569:
1563:
1557:
1554:
1547:
1541:
1538:
1535:
1531:
1527:
1524:
1522:
1520:
1517:
1514:
1511:
1508:
1507:
1481:
1473:
1470:
1467:
1464:
1460:
1456:
1453:
1449:
1442:
1439:
1436:
1432:
1428:
1425:
1422:
1419:
1416:
1413:
1410:
1407:
1404:
1398:
1395:
1387:
1384:
1381:
1377:
1373:
1370:
1367:
1364:
1361:
1306:
1298:
1295:
1292:
1289:
1285:
1281:
1278:
1274:
1269:
1266:
1263:
1260:
1257:
1251:
1248:
1242:
1236:
1233:
1227:
1224:
1221:
1218:
1215:
1171:
1168:
1165:
1162:
1158:
1154:
1151:
1147:
1142:
1139:
1136:
1133:
1130:
1127:
1124:
1118:
1115:
1109:
1103:
1100:
1059:
1056:
1016:
1013:
1009:
1006:
1003:
1000:
995:
990:
987:
983:
979:
976:
973:
970:
967:
943:
940:
937:
934:
931:
925:
922:
918:
913:
910:
907:
904:
901:
881:
878:
866:
859:
856:
850:
846:
842:
838:
835:
829:
826:
823:
820:
817:
806:absolute value
768:
764:
760:
757:
754:
748:
745:
742:
738:
733:
727:
724:
721:
718:
712:
709:
706:
702:
697:
694:
690:
686:
683:
680:
677:
674:
659:
658:
647:
644:
641:
630:
627:
624:
621:
618:
615:
609:
606:
602:
597:
594:
591:
588:
585:
570:
559:
556:
553:
546:
541:
534:
529:
524:
521:
518:
513:
508:
505:
502:
499:
496:
481:
470:
467:
464:
461:
458:
455:
452:
449:
446:
443:
428:
415:
410:
407:
404:
401:
399:
396:
393:
392:
389:
386:
383:
380:
378:
375:
372:
371:
369:
364:
361:
358:
355:
352:
325:
322:
242:
241:
238:
234:
233:
220:
215:
212:
209:
206:
204:
201:
198:
197:
194:
191:
188:
185:
183:
180:
177:
176:
174:
169:
166:
163:
160:
157:
147:
143:
142:
138:
137:
134:
126:
125:
124:Heaviside step
117:
116:
31:
29:
22:
15:
9:
6:
4:
3:
2:
3901:
3900:
3889:
3886:
3884:
3881:
3879:
3876:
3875:
3873:
3863:. p. 42.
3862:
3858:
3854:
3850:
3846:
3841:
3837:
3833:
3828:
3824:
3820:
3815:
3812:
3809:
3808:
3795:
3793:0-07-303938-1
3789:
3785:
3778:
3769:
3768:
3763:
3760:
3753:
3749:
3738:
3737:Step response
3735:
3733:
3732:Sine integral
3730:
3728:
3727:Sign function
3725:
3723:
3720:
3718:
3715:
3713:
3710:
3708:
3705:
3703:
3700:
3698:
3695:
3693:
3690:
3688:
3685:
3683:
3680:
3679:
3672:
3669:
3650:
3647:
3642:
3640:
3632:
3629:
3623:
3620:
3617:
3613:
3607:
3602:
3598:
3586:
3578:
3576:
3568:
3565:
3558:
3552:
3547:
3544:
3541:
3537:
3531:
3526:
3522:
3510:
3502:
3500:
3492:
3480:
3465:
3461:
3451:
3436:
3433:
3427:
3420:
3414:
3398:
3394:
3384:
3376:
3367:
3350:
3337:
3333:
3327:
3324:
3319:
3315:
3309:
3300:
3297:
3292:
3286:
3280:
3276:
3270:
3267:
3262:
3259:
3256:
3249:
3243:
3238:
3235:
3232:
3229:
3226:
3223:
3219:
3213:
3208:
3205:
3201:
3189:
3181:
3175:
3163:
3152:
3142:
3129:
3122:
3116:
3113:
3107:
3104:
3096:
3090:
3087:
3076:
3072:
3067:
3054:
3047:
3044:
3041:
3032:
3026:
3020:
3017:
3014:
3011:
3008:
3001:
2995:
2990:
2982:
2978:
2969:
2965:
2964:ramp function
2955:
2953:
2948:
2933:
2930:
2927:
2923:
2919:
2913:
2907:
2899:
2896:
2880:
2874:
2869:
2861:
2858:
2855:
2851:
2847:
2841:
2835:
2827:
2825:
2820:
2807:
2801:
2798:
2795:
2789:
2786:
2780:
2774:
2771:
2765:
2759:
2751:
2748:
2745:
2739:
2731:
2724:
2717:
2710:
2701:
2692:
2672:
2669:
2666:
2663:
2658:
2655:
2648:
2645:
2642:
2639:
2634:
2628:
2625:
2615:
2612:
2609:
2606:
2601:
2598:
2592:
2587:
2581:
2575:
2567:
2564:
2544:
2541:
2538:
2535:
2530:
2527:
2520:
2517:
2514:
2511:
2506:
2503:
2497:
2492:
2486:
2480:
2472:
2441:
2438:
2427:Discrete form
2419:
2414:
2410:
2396:
2390:
2376:
2373:
2360:
2354:
2348:
2340:
2332:
2325:is used when
2322:
2318:
2315:
2299:
2293:
2279:
2276:
2263:
2257:
2251:
2243:
2239:
2231:
2227:
2223:
2216:is used when
2213:
2209:
2195:
2189:
2186:
2183:
2180:
2177:
2168:
2165:
2159:
2153:
2147:
2135:
2134:sign function
2131:
2110:
2105:
2084:
2080:
2079:
2078:
2075:
2073:
2069:
2065:
2062:
2055:
2050:
2040:
2028:Zero argument
2025:
2022:
2005:
2002:
1999:
1994:
1991:
1988:
1984:
1977:
1974:
1971:
1968:
1964:
1949:
1945:
1938:
1935:
1932:
1928:
1919:
1915:
1908:
1900:
1898:
1890:
1887:
1882:
1879:
1876:
1873:
1869:
1862:
1859:
1856:
1853:
1849:
1834:
1830:
1823:
1820:
1817:
1813:
1808:
1801:
1797:
1790:
1782:
1780:
1772:
1766:
1754:
1744:
1742:
1738:
1734:
1730:
1726:
1722:
1719:
1715:
1710:
1708:
1704:
1703:distributions
1700:
1695:
1677:
1673:
1670:
1667:
1664:
1658:
1655:
1649:
1643:
1640:
1633:
1621:
1613:
1611:
1603:
1597:
1589:
1585:
1582:
1579:
1576:
1570:
1567:
1561:
1555:
1552:
1545:
1533:
1525:
1523:
1515:
1509:
1497:
1492:
1479:
1471:
1468:
1465:
1462:
1458:
1454:
1451:
1447:
1434:
1426:
1420:
1417:
1414:
1411:
1408:
1405:
1396:
1393:
1379:
1371:
1365:
1359:
1334:
1330:. If we take
1327:
1317:
1304:
1296:
1293:
1290:
1287:
1283:
1279:
1276:
1272:
1267:
1264:
1261:
1258:
1255:
1249:
1246:
1240:
1234:
1231:
1225:
1219:
1213:
1206:
1202:
1192:
1169:
1166:
1163:
1160:
1156:
1152:
1149:
1145:
1140:
1134:
1131:
1125:
1122:
1116:
1113:
1107:
1101:
1098:
1087:
1083:
1081:
1077:
1073:
1069:
1065:
1055:
1053:
1049:
1048:almost surely
1045:
1041:
1031:
1014:
1011:
1004:
998:
993:
985:
981:
977:
971:
965:
957:
941:
935:
929:
923:
920:
916:
911:
905:
899:
891:
887:
877:
864:
857:
854:
844:
836:
833:
827:
821:
815:
807:
801:
795:
789:
784:
766:
762:
758:
755:
752:
746:
743:
740:
736:
731:
725:
722:
719:
716:
710:
707:
704:
700:
695:
692:
688:
684:
678:
672:
664:
663:hyperfunction
645:
642:
639:
625:
622:
619:
607:
604:
600:
595:
589:
583:
575:
574:ramp function
571:
554:
544:
527:
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278:step function
270:
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99:December 2012
91:
88:
84:
81:
77:
74:
70:
67:
63:
60: ā
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
3856:
3844:
3825:. p. 5.
3818:
3783:
3777:
3765:
3752:
3670:
3457:
3375:distribution
3365:
3351:
3148:
3068:
2961:
2949:
2900:
2897:
2828:
2821:
2752:
2749:
2743:
2740:
2729:
2722:
2715:
2708:
2693:
2568:
2565:
2473:
2430:
2417:
2329:needs to be
2320:
2220:needs to be
2211:
2130:odd function
2108:
2082:
2076:
2060:
2053:
2049:distribution
2038:
2031:
2023:
1750:
1711:
1696:
1493:
1332:
1325:
1318:
1198:
1190:
1068:neuroscience
1064:biochemistry
1061:
1029:
883:
808:function as
799:
796:
782:
660:
330:
327:
316:
305:
294:
280:named after
268:
251:
247:
245:
105:
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
2128:is then an
324:Formulation
3872:Categories
3744:References
2718:ā¤ −1
1718:continuous
1494:There are
890:derivative
434:notation:
430:using the
69:newspapers
3767:MathWorld
3618:−
3599:∫
3593:∞
3590:→
3542:−
3523:∫
3517:∞
3514:→
3484:^
3415:φ
3407:∞
3402:∞
3399:−
3395:∫
3320:
3301:π
3293:−
3281:δ
3230:π
3224:−
3206:−
3202:∫
3196:∞
3193:→
3167:^
3117:δ
3012:ξ
3002:ξ
2986:∞
2983:−
2979:∫
2924:δ
2908:δ
2875:δ
2865:∞
2862:−
2852:∑
2799:−
2787:−
2760:δ
2539:≥
2450:→
2380:∞
2283:∞
2187:
2003:τ
1995:τ
1978:ε
1972:−
1969:τ
1958:∞
1953:∞
1950:−
1946:∫
1936:π
1912:→
1909:ε
1891:τ
1883:τ
1874:−
1863:ε
1854:τ
1843:∞
1838:∞
1835:−
1831:∫
1821:π
1809:−
1794:→
1791:ε
1751:Often an
1699:pointwise
1668:
1628:∞
1625:→
1580:
1571:π
1540:∞
1537:→
1463:−
1441:∞
1438:→
1415:
1386:∞
1383:→
1288:−
1259:
1226:≈
1161:−
1126:
1046:which is
999:δ
989:∞
986:−
982:∫
900:δ
756:
744:π
732:−
720:
708:π
696:−
643:≠
635:for
520:≥
463:≥
385:≥
250:, or the
190:≥
3675:See also
2720:, while
1753:integral
1733:logistic
1725:variance
1078:and the
1072:logistic
1070:, where
1050:0. (See
956:integral
276:), is a
3381:to the
3373:is the
3370:
3356:
2950:is the
2700:integer
2471:), is:
2323:(0) = 0
2236:is the
2214:(0) = 1
2125:
2113:
2099:
2087:
1349:
1337:
888:is the
786:is the
333:(0) = 1
83:scholar
3790:
2966:is an
2898:where
2725:> 0
2711:< 0
2698:is an
2694:where
2420:(0) =
2242:closed
2085:(0) =
2032:Since
1741:normal
1737:Cauchy
1577:arctan
1335:(0) =
1201:smooth
1199:For a
779:where
729:
85:
78:
71:
64:
56:
3352:Here
2702:. If
2240:of a
2104:graph
2064:space
2057:(see
1716:of a
1042:of a
90:JSTOR
76:books
3788:ISBN
3458:The
3354:p.v.
3149:The
3069:The
2962:The
2667:>
2610:<
2515:<
2339:open
1739:and
1412:tanh
1256:tanh
1123:tanh
1076:Hill
1066:and
884:The
781:log
406:<
286:zero
246:The
211:<
62:news
3583:lim
3507:lim
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3186:lim
3036:max
2732:= 1
2184:sgn
2041:(0)
1905:lim
1787:lim
1709:.)
1665:erf
1618:lim
1530:lim
1431:lim
1376:lim
1328:= 0
1193:ā ā
1054:.)
1032:= 0
802:ā 0
790:of
753:log
717:log
665:as
614:max
483:an
297:(0)
290:one
272:or
258:or
45:by
3874::
3859:.
3834:.
3821:.
3764:.
3077::
2954:.
2826::
2140::
2111:ā
1735:,
978::=
794:.
596::=
576::
507::=
487::
454::=
363::=
343::
339:a
320:.
274:š
266:,
168::=
3838:.
3813:.
3796:.
3770:.
3651:s
3648:1
3643:=
3633:x
3630:d
3624:x
3621:s
3614:e
3608:N
3603:0
3587:N
3579:=
3569:x
3566:d
3562:)
3559:x
3556:(
3553:H
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3503:=
3496:)
3493:s
3490:(
3481:H
3437:s
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3428:s
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3421:s
3418:(
3379:Ļ
3366:s
3362:/
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3338:.
3334:)
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3316:.
3313:v
3310:.
3307:p
3298:i
3290:)
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3284:(
3277:(
3271:2
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3263:=
3260:x
3257:d
3253:)
3250:x
3247:(
3244:H
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3220:e
3214:N
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3190:N
3182:=
3179:)
3176:s
3173:(
3164:H
3130:.
3126:)
3123:x
3120:(
3114:=
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3105:d
3100:)
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3055:.
3051:}
3048:x
3045:,
3042:0
3039:{
3033:=
3030:)
3027:x
3024:(
3021:H
3018:x
3015:=
3009:d
3005:)
2999:(
2996:H
2991:x
2934:0
2931:,
2928:k
2920:=
2917:]
2914:k
2911:[
2884:]
2881:k
2878:[
2870:n
2859:=
2856:k
2848:=
2845:]
2842:n
2839:[
2836:H
2808:.
2805:]
2802:1
2796:n
2793:[
2790:H
2784:]
2781:n
2778:[
2775:H
2772:=
2769:]
2766:n
2763:[
2744:H
2730:n
2723:n
2716:n
2709:n
2704:n
2696:n
2673:,
2670:0
2664:n
2659:,
2656:1
2649:,
2646:0
2643:=
2640:n
2635:,
2629:2
2626:1
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2613:0
2607:n
2602:,
2599:0
2593:{
2588:=
2585:]
2582:n
2579:[
2576:H
2545:,
2542:0
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2531:,
2528:1
2521:,
2518:0
2512:n
2507:,
2504:0
2498:{
2493:=
2490:]
2487:n
2484:[
2481:H
2469:n
2454:R
2446:Z
2442::
2439:H
2422:.
2418:H
2397:.
2394:)
2391:x
2388:(
2383:)
2377:,
2374:0
2371:(
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2361:=
2358:)
2355:x
2352:(
2349:H
2335:H
2327:H
2321:H
2316:.
2300:.
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2291:(
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2277:0
2274:[
2269:1
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2255:(
2252:H
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2212:H
2196:.
2193:)
2190:x
2181:+
2178:1
2175:(
2169:2
2166:1
2160:=
2157:)
2154:x
2151:(
2148:H
2138:x
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2119:/
2116:1
2109:H
2096:2
2093:/
2090:1
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2061:L
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2045:H
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2034:H
2006:.
2000:d
1992:x
1989:i
1985:e
1975:i
1965:1
1939:i
1933:2
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1920:+
1916:0
1901:=
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1880:x
1877:i
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1850:1
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1798:0
1783:=
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1773:x
1770:(
1767:H
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1614:=
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1568:1
1562:+
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1480:.
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1421:x
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1005:s
1002:(
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972:x
969:(
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912:=
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906:x
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501:x
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