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31: 16.5 Integral Representations and Integrals
§16.5 Integral Representations and Integrals
Then the integral converges when p < q + 1 provided that z 0 , or when p = q + 1 provided that 0 < | z | < 1 , and provides an integral representation of the left-hand side with these conditions. … In the case p = q the left-hand side of (16.5.1) is an entire function, and the right-hand side supplies an integral representation valid when | ph ( z ) | < π / 2 . … For further integral representations and integrals see Apelblat (1983, §16), Erdélyi et al. (1953a, §4.6), Erdélyi et al. (1954a, §§6.9 and 7.5), Luke (1969a, §3.6), and Prudnikov et al. (1990, §§2.22, 4.2.4, and 4.3.1). …
32: 16.15 Integral Representations and Integrals
§16.15 Integral Representations and Integrals
33: 15.6 Integral Representations
§15.6 Integral Representations
The function 𝐅 ( a , b ; c ; z ) (not F ( a , b ; c ; z ) ) has the following integral representations: …
See accompanying text
Figure 15.6.1: t -plane. … Magnify
34: 31.10 Integral Equations and Representations
§31.10 Integral Equations and Representations
Kernel Functions
For suitable choices of the branches of the P -symbols in (31.10.9) and the contour C , we can obtain both integral equations satisfied by Heun functions, as well as the integral representations of a distinct solution of Heun’s equation in terms of a Heun function (polynomial, path-multiplicative solution). …
Kernel Functions
35: 28.28 Integrals, Integral Representations, and Integral Equations
§28.28 Integrals, Integral Representations, and Integral Equations
§28.28(i) Equations with Elementary Kernels
§28.28(iv) Integrals of Products of Mathieu Functions of Integer Order
28.28.49 α ^ n , m ( c ) = 1 2 π 0 2 π cos t ce n ( t , h 2 ) ce m ( t , h 2 ) d t = ( 1 ) p + 1 2 i π ce n ( 0 , h 2 ) ce m ( 0 , h 2 ) h Dc 0 ( n , m , 0 ) .
§28.28(v) Compendia
36: 10.9 Integral Representations
§10.9 Integral Representations
§10.9(ii) Contour Integrals
§10.9(iii) Products
37: 25.12 Polylogarithms
25.12.2 Li 2 ( z ) = 0 z t 1 ln ( 1 t ) d t , z ( 1 , ) .
Integral Representation
25.12.11 Li s ( z ) z Γ ( s ) 0 x s 1 e x z d x ,
25.12.14 F s ( x ) = 1 Γ ( s + 1 ) 0 t s e t x + 1 d t , s > 1 ,
25.12.15 G s ( x ) = 1 Γ ( s + 1 ) 0 t s e t x 1 d t , s > 1 , x < 0 ; or s > 0 , x 0 ,
38: 13.4 Integral Representations
§13.4 Integral Representations
§13.4(i) Integrals Along the Real Line
§13.4(ii) Contour Integrals
§13.4(iii) Mellin–Barnes Integrals
39: 15.19 Methods of Computation
§15.19(iii) Integral Representations
40: 10.74 Methods of Computation
§10.74(iii) Integral Representations
For evaluation of the Hankel functions H ν ( 1 ) ( z ) and H ν ( 2 ) ( z ) for complex values of ν and z based on the integral representations (10.9.18) see Remenets (1973). … The integral representation used is based on (10.32.8). …