repeated%20integrals%20of
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6 matching pages
1: Bibliography
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Repeated integrals and derivatives of Bessel functions.
SIAM J. Math. Anal. 20 (1), pp. 169–175.
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2: 7.23 Tables
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Abramowitz and Stegun (1964, Chapter 7) includes , , , 10D; , , 8S; , , 7D; , , , 6S; , , 10D; , , 9D; , , , 7D; , , , , 15D.
Abramowitz and Stegun (1964, Table 27.6) includes the Goodwin–Staton integral , , 4D; also , , 4D.
Zhang and Jin (1996, pp. 637, 639) includes , , , 8D; , , , 8D.
Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of , , , 7D and 8D, respectively; the real and imaginary parts of , , , 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.
Zhang and Jin (1996, p. 642) includes the first 10 zeros of , 9D; the first 25 distinct zeros of and , 8S.
3: Bibliography G
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Evaluation of the repeated integrals of the coerror function.
ACM Trans. Math. Software 3, pp. 240–252.
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Algorithm 521: Repeated integrals of the coerror function.
ACM Trans. Math. Software 3, pp. 301–302.
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Recursive computation of the repeated integrals of the error function.
Math. Comp. 15 (75), pp. 227–232.
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Algorithm 957: evaluation of the repeated integral of the coerror function by half-range Gauss-Hermite quadrature.
ACM Trans. Math. Softw. 42 (1), pp. 9:1–9:10.
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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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4: 19.36 Methods of Computation
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►Numerical differences between the variables of a symmetric integral can be reduced in magnitude by successive factors of 4 by repeated applications of the duplication theorem, as shown by (19.26.18).
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►Legendre’s integrals can be computed from symmetric integrals by using the relations in §19.25(i).
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►For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20).
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►Numerical quadrature is slower than most methods for the standard integrals but can be useful for elliptic integrals that have complicated representations in terms of standard integrals.
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5: Software Index
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Open Source | With Book | Commercial | |||||||||||||||||||||||
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7.25(ii) , , , | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | NMS | |||
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11.16(iii) Integrals of … | ✓ | ✓ | ✓ | ✓ | a | ||||||||||||||||||||
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11.16(vi) Integrals of … | ✓ | ✓ | ✓ | a | |||||||||||||||||||||
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19 Elliptic Integrals | |||||||||||||||||||||||||
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20 Theta Functions | |||||||||||||||||||||||||
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6: 2.11 Remainder Terms; Stokes Phenomenon
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►As an example consider
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►From §8.19(i) the generalized exponential integral is given by
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►Often the process of re-expansion can be repeated any number of times.
…For integrals, see Berry and Howls (1991), Howls (1992), and Paris and Kaminski (2001, Chapter 6).
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►For example, using double precision is found to agree with (2.11.31) to 13D.
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