generalized sine and cosine integrals
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1: 8.21 Generalized Sine and Cosine Integrals
§8.21 Generalized Sine and Cosine Integrals
… ►From here on it is assumed that unless indicated otherwise the functions , , , and have their principal values. … ►§8.21(iv) Interrelations
… ►For the corresponding expansions for and apply (8.21.20) and (8.21.21).2: 8.1 Special Notation
…
►Unless otherwise indicated, primes denote derivatives with respect to the argument.
►The functions treated in this chapter are the incomplete gamma functions , , , , and ; the incomplete beta functions and ; the generalized exponential integral
; the generalized sine and cosine integrals
, , , and .
►Alternative notations include: Prym’s functions
, , Nielsen (1906a, pp. 25–26), Batchelder (1967, p. 63); , , Dingle (1973); , , Magnus et al. (1966); , , Luke (1975).
3: 6.4 Analytic Continuation
§6.4 Analytic Continuation
… ►The general value of is given by … ►The general values of the other functions are defined in a similar manner, and … ►Unless indicated otherwise, in the rest of this chapter and elsewhere in the DLMF the functions , , , , and assume their principal values, that is, the branches that are real on the positive real axis and two-valued on the negative real axis.4: 6.2 Definitions and Interrelations
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§6.2(i) Exponential and Logarithmic Integrals
… ►§6.2(ii) Sine and Cosine Integrals
… ►Values at Infinity
… ►Hyperbolic Analogs of the Sine and Cosine Integrals
… ►§6.2(iii) Auxiliary Functions
…5: 6 Exponential, Logarithmic, Sine, and
Cosine Integrals
Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals
…6: 10.15 Derivatives with Respect to Order
7: 7.14 Integrals
§7.14 Integrals
… ►Fourier Transform
… ►§7.14(ii) Fresnel Integrals
►Laplace Transforms
… ►In a series of ten papers Hadži (1968, 1969, 1970, 1972, 1973, 1975a, 1975b, 1976a, 1976b, 1978) gives many integrals containing error functions and Fresnel integrals, also in combination with the hypergeometric function, confluent hypergeometric functions, and generalized hypergeometric functions.8: Software Index
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6 Exponential, Logarithmic, Sine, and Cosine Integrals | |||||||||||||||||||||||||
6.21(ii) , , , , , , | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | |||
6.21(iii) , , , , , | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | |||||||||||||||
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7.25(iv) , , , , | ✓ | ✓ | ✓ | a | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ||||||||||
7.25(v) , , | ✓ | a | ✓ | ✓ | ✓ | ✓ | |||||||||||||||||||
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9: 7.11 Relations to Other Functions
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Incomplete Gamma Functions and Generalized Exponential Integral
… ►Confluent Hypergeometric Functions
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7.11.6
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Generalized Hypergeometric Functions
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7.11.7
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10: 6.10 Other Series Expansions
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§6.10(i) Inverse Factorial Series
… ►For a more general result (incomplete gamma function), and also for a result for the logarithmic integral, see Nielsen (1906a, p. 283: Formula (3) is incorrect). ►§6.10(ii) Expansions in Series of Spherical Bessel Functions
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6.10.4
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►An expansion for can be obtained by combining (6.2.4) and (6.10.8).