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21: 11.7 Integrals and Sums
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11.7.10 0 t Ξ½ 1 ⁒ 𝐇 Ξ½ ⁑ ( t ) ⁒ d t = Ο€ 2 Ξ½ + 1 ⁒ Ξ“ ⁑ ( Ξ½ + 1 ) , ⁑ Ξ½ > 3 2 ,
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§11.7(iii) Laplace Transforms
β–ΊThe following Laplace transforms of 𝐇 Ξ½ ⁑ ( t ) require ⁑ a > 0 for convergence, while those of 𝐋 Ξ½ ⁑ ( t ) require ⁑ a > 1 . … β–Ί
11.7.15 0 e a ⁒ t ⁒ 𝐋 0 ⁑ ( t ) ⁒ d t = 2 Ο€ ⁒ a 2 1 ⁒ arcsin ⁑ ( 1 a ) ,
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= 2 ⁒ a Ο€ ⁒ a 2 1 ⁒ arctan ⁑ ( 1 a 2 1 ) 2 Ο€ ⁒ a .
22: Bibliography T
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  • J. D. Talman (1983) LSFBTR: A subroutine for calculating spherical Bessel transforms. Comput. Phys. Comm. 30 (1), pp. 93–99.
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  • N. M. Temme (1985) Laplace type integrals: Transformation to standard form and uniform asymptotic expansions. Quart. Appl. Math. 43 (1), pp. 103–123.
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  • N. M. Temme (1992a) Asymptotic inversion of incomplete gamma functions. Math. Comp. 58 (198), pp. 755–764.
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  • N. M. Temme (1992b) Asymptotic inversion of the incomplete beta function. J. Comput. Appl. Math. 41 (1-2), pp. 145–157.
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  • F. Tu and Y. Yang (2013) Algebraic transformations of hypergeometric functions and automorphic forms on Shimura curves. Trans. Amer. Math. Soc. 365 (12), pp. 6697–6729.
  • 23: 10.22 Integrals
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    §10.22(v) Hankel Transform
    β–ΊThe Hankel transform (or Bessel transform) of a function f ⁑ ( x ) is defined as … β–ΊHankel’s inversion theorem is given by … β–ΊThe following two formulas are generalizations of the Hankel transform. …This is the Weber transform. …