Heine integral
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31: 36.9 Integral Identities
§36.9 Integral Identities
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36.9.1
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36.9.8
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►For these results and also integrals over doubly-infinite intervals see Berry and Wright (1980).
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32: 6.4 Analytic Continuation
§6.4 Analytic Continuation
►Analytic continuation of the principal value of yields a multi-valued function with branch points at and . The general value of is given by … ►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.33: 36.10 Differential Equations
§36.10 Differential Equations
►§36.10(i) Equations for
… ► , cusp: … ► , swallowtail: … ►In terms of the normal forms (36.2.2) and (36.2.3), the satisfy the following operator equations …34: 6.11 Relations to Other Functions
35: 6.7 Integral Representations
§6.7 Integral Representations
►§6.7(i) Exponential Integrals
… ► ►§6.7(ii) Sine and Cosine Integrals
… ►§6.7(iii) Auxiliary Functions
…36: Sidebar 7.SB1: Diffraction from a Straightedge
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►The intensity distribution follows , where is the Fresnel integral (See 7.3.4).
Fresnel integrals have many applications in optics.
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37: 12.16 Mathematical Applications
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►PCFs are used as basic approximating functions in the theory of contour integrals with a coalescing saddle point and an algebraic singularity, and in the theory of differential equations with two coalescing turning points; see §§2.4(vi) and 2.8(vi).
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►PCFs are also used in integral transforms with respect to the parameter, and inversion formulas exist for kernels containing PCFs.
…Integral transforms and sampling expansions are considered in Jerri (1982).
38: 36.14 Other Physical Applications
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§36.14(i) Caustics
… ►§36.14(ii) Optics
… ►§36.14(iii) Quantum Mechanics
… ►§36.14(iv) Acoustics
…39: 17.9 Further Transformations of Functions
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