relation to logarithmic integral
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11: 7.10 Derivatives
12: 7.11 Relations to Other Functions
§7.11 Relations to Other Functions
►Incomplete Gamma Functions and Generalized Exponential Integral
… ►Confluent Hypergeometric Functions
… ►
7.11.6
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Generalized Hypergeometric Functions
…13: 7.19 Voigt Functions
14: 19.14 Reduction of General Elliptic Integrals
§19.14 Reduction of General Elliptic Integrals
… ►Legendre (1825–1832) showed that every elliptic integral can be expressed in terms of the three integrals in (19.1.2) supplemented by algebraic, logarithmic, and trigonometric functions. The classical method of reducing (19.2.3) to Legendre’s integrals is described in many places, especially Erdélyi et al. (1953b, §13.5), Abramowitz and Stegun (1964, Chapter 17), and Labahn and Mutrie (1997, §3). …A similar remark applies to the transformations given in Erdélyi et al. (1953b, §13.5) and to the choice among explicit reductions in the extensive table of Byrd and Friedman (1971), in which one limit of integration is assumed to be a branch point of the integrand at which the integral converges. …15: 9.13 Generalized Airy Functions
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►Swanson and Headley (1967) define independent solutions and of (9.13.1) by
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►Their relations to the functions and are given by
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►When is a positive integer the relation of these functions to
, is as follows:
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►The are related by
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►Further properties of these functions, and also of similar contour integrals containing an additional factor , , in the integrand, are derived in Reid (1972), Drazin and Reid (1981, Appendix), and Baldwin (1985).
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16: 13.18 Relations to Other Functions
§13.18 Relations to Other Functions
►§13.18(i) Elementary Functions
… ►§13.18(iv) Parabolic Cylinder Functions
… ►§13.18(v) Orthogonal Polynomials
… ►Laguerre Polynomials
…17: 7.13 Zeros
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►
§7.13(iii) Zeros of the Fresnel Integrals
… ►As the and corresponding to the zeros of satisfy … ►As the and corresponding to the zeros of satisfy (7.13.5) with … ►In consequence of (7.5.5) and (7.5.10), zeros of are related to zeros of . … ►18: 1.8 Fourier Series
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►Here is related to
and in (1.8.1), (1.8.2) by , for and .
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►As
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►(1.8.10) continues to apply if either or or both are infinite and/or has finitely many singularities in , provided that the integral converges uniformly (§1.5(iv)) at , and the singularities for all sufficiently large .
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►Then the series (1.8.1) converges to the sum
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►It follows from definition (1.14.1) that the integral in (1.8.14) is equal to
.
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