Goodwin%E2%80%93Staton%20integral
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11: 19.16 Definitions
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§19.16(i) Symmetric Integrals
… ► … ►All other elliptic cases are integrals of the second kind. …(Note that is not an elliptic integral.) … ►Each of the four complete integrals (19.16.20)–(19.16.23) can be integrated to recover the incomplete integral: …12: 7.1 Special Notation
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►Unless otherwise noted, primes indicate derivatives with respect to the argument.
►The main functions treated in this chapter are the error function ; the complementary error functions and ; Dawson’s integral
; the Fresnel integrals
, , and ; the Goodwin–Staton integral
; the repeated integrals of the complementary error function ; the Voigt functions and .
►Alternative notations are , , , , , , , .
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13: 7.12 Asymptotic Expansions
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§7.12(ii) Fresnel Integrals
►The asymptotic expansions of and are given by (7.5.3), (7.5.4), and … ►They are bounded by times the first neglected terms when . … ►§7.12(iii) Goodwin–Staton Integral
►See Olver (1997b, p. 115) for an expansion of with bounds for the remainder for real and complex values of .14: Bibliography G
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Algorithm 939: computation of the Marcum Q-function.
ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
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Table of
.
Quart. J. Mech. Appl. Math. 1 (1), pp. 319–326.
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Recurrence relations for cross-products of Bessel functions.
Quart. J. Mech. Appl. Math. 2 (1), pp. 72–74.
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The evaluation of integrals of the form
.
Proc. Cambridge Philos. Soc. 45 (2), pp. 241–245.
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Mutual integrability, quadratic algebras, and dynamical symmetry.
Ann. Phys. 217 (1), pp. 1–20.
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15: Software Index
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Open Source | With Book | Commercial | |||||||||||||||||||||||
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7.25(vi) , , , , | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | |||||||||||||
7.25(vii) , , | ✓ | ✓ | ✓ | ||||||||||||||||||||||
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11.16(iii) Integrals of … | ✓ | ✓ | ✓ | ✓ | a | ||||||||||||||||||||
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19 Elliptic Integrals | |||||||||||||||||||||||||
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20 Theta Functions | |||||||||||||||||||||||||
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16: 36.2 Catastrophes and Canonical Integrals
§36.2 Catastrophes and Canonical Integrals
►§36.2(i) Definitions
… ►Canonical Integrals
… ► … ►§36.2(iii) Symmetries
…17: 19.2 Definitions
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§19.2(i) General Elliptic Integrals
… ►is called an elliptic integral. … ►§19.2(ii) Legendre’s Integrals
… ►§19.2(iii) Bulirsch’s Integrals
… ►§19.2(iv) A Related Function:
…18: 8.23 Statistical Applications
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►The functions and are used extensively in statistics as the probability integrals of the gamma distribution; see Johnson et al. (1994, pp. 337–414).
…In queueing theory the Erlang loss function is used, which can be expressed in terms of the reciprocal of ; see Jagerman (1974) and Cooper (1981, pp. 80, 316–319).
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19: Bibliography H
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The Laplace transform for expressions that contain a probability function.
Bul. Akad. Štiince RSS Moldoven. 1973 (2), pp. 78–80, 93 (Russian).
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Integrals containing the Fresnel functions and
.
Bul. Akad. Štiince RSS Moldoven. 1975 (3), pp. 48–60, 93 (Russian).
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Expansions for the probability function in series of Čebyšev polynomials and Bessel functions.
Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 77–80, 96 (Russian).
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Integrals that contain a probability function of complicated arguments.
Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 80–84, 96 (Russian).
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Some properties and applications of the repeated integrals of the error function.
Proc. Manchester Lit. Philos. Soc. 80, pp. 85–102.
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