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1: 14.26 Uniform Asymptotic Expansions
See also Frenzen (1990), Gil et al. (2000), Shivakumar and Wong (1988), Ursell (1984), and Wong (1989) for uniform asymptotic approximations obtained from integral representations.
2: 9.13 Generalized Airy Functions
§9.13(ii) Generalizations from Integral Representations
Each of the functions A k ( z , p ) and B k ( z , p ) satisfies the differential equation …and the difference equation … Connection formulas for the solutions of (9.13.31) include …
3: 22.10 Maclaurin Series
Further terms may be derived from the differential equations (22.13.13), (22.13.14), (22.13.15), or from the integral representations of the inverse functions in §22.15(ii). …
4: 19.16 Definitions
19.16.2_5 R G ( x , y , z ) = 1 4 0 1 s ( t ) ( x t + x + y t + y + z t + z ) t d t .
5: 18.10 Integral Representations
§18.10 Integral Representations
Ultraspherical
Legendre
Jacobi
Ultraspherical
6: 25.11 Hurwitz Zeta Function
25.11.27 ζ ( s , a ) = 1 2 a s + a 1 s s 1 + 1 Γ ( s ) 0 ( 1 e x 1 1 x + 1 2 ) x s 1 e a x d x , s > 1 , s 1 , a > 0 .
25.11.28 ζ ( s , a ) = 1 2 a s + a 1 s s 1 + k = 1 n B 2 k ( 2 k ) ! ( s ) 2 k 1 a 1 s 2 k + 1 Γ ( s ) 0 ( 1 e x 1 1 x + 1 2 k = 1 n B 2 k ( 2 k ) ! x 2 k 1 ) x s 1 e a x d x , s > ( 2 n + 1 ) , s 1 , a > 0 .
25.11.35 n = 0 ( 1 ) n ( n + a ) s = 1 Γ ( s ) 0 x s 1 e a x 1 + e x d x = 2 s ( ζ ( s , 1 2 a ) ζ ( s , 1 2 ( 1 + a ) ) ) , a > 0 , s > 0 ; or a = 0 , a 0 , 0 < s < 1 .
7: 14.20 Conical (or Mehler) Functions
§14.20(iv) Integral Representation
From (14.20.9) or (14.20.10) it is evident that 𝖯 1 2 + i τ ( cos θ ) is positive for real θ . …
14.20.12 g ( x ) = 0 P 1 2 + i τ μ ( x ) f ( τ ) d τ .
§14.20(x) Zeros and Integrals
For integrals with respect to τ involving 𝖯 1 2 + i τ ( x ) , see Prudnikov et al. (1990, pp. 218–228).
8: 25.5 Integral Representations
25.5.2 ζ ( s ) = 1 Γ ( s + 1 ) 0 e x x s ( e x 1 ) 2 d x , s > 1 .
25.5.4 ζ ( s ) = 1 ( 1 2 1 s ) Γ ( s + 1 ) 0 e x x s ( e x + 1 ) 2 d x , s > 0 .
25.5.6 ζ ( s ) = 1 2 + 1 s 1 + 1 Γ ( s ) 0 ( 1 e x 1 1 x + 1 2 ) x s 1 e x d x , s > 1 .
25.5.7 ζ ( s ) = 1 2 + 1 s 1 + m = 1 n B 2 m ( 2 m ) ! ( s ) 2 m 1 + 1 Γ ( s ) 0 ( 1 e x 1 1 x + 1 2 m = 1 n B 2 m ( 2 m ) ! x 2 m 1 ) x s 1 e x d x , s > ( 2 n + 1 ) , n = 1 , 2 , 3 , .
9: 6.7 Integral Representations
§6.7 Integral Representations
§6.7(ii) Sine and Cosine Integrals
§6.7(iii) Auxiliary Functions
For collections of integral representations see Bierens de Haan (1939, pp. 56–59, 72–73, 82–84, 121, 133–136, 155, 179–181, 223, 225–227, 230, 259–260, 374, 377, 397–398, 408, 416, 424, 431, 438–439, 442–444, 488, 496–500, 567–571, 585, 602, 638, 675–677), Corrington (1961), Erdélyi et al. (1954a, vol. 1, pp. 267–270), Geller and Ng (1969), Nielsen (1906b), Oberhettinger (1974, pp. 244–246), Oberhettinger and Badii (1973, pp. 364–371), and Watrasiewicz (1967).
10: 12.14 The Function W ( a , x )
Other expansions, involving cos ( 1 4 x 2 ) and sin ( 1 4 x 2 ) , can be obtained from (12.4.3) to (12.4.6) by replacing a by i a and z by x e π i / 4 ; see Miller (1955, p. 80), and also (12.14.15) and (12.14.16).
§12.14(vi) Integral Representations
These follow from the contour integrals of §12.5(ii), which are valid for general complex values of the argument z and parameter a . … follows from (12.2.3), and has solutions W ( 1 2 μ 2 , ± μ t 2 ) . …In the following expansions, obtained from Olver (1959), μ is large and positive, and δ is again an arbitrary small positive constant. …