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11: 5 Gamma Function
Chapter 5 Gamma Function
12: 4.34 Derivatives and Differential Equations
With a 0 , the general solutions of the differential equations …
4.34.12 w = ( 1 / a ) sinh ( a z + c ) ,
4.34.13 w = ( 1 / a ) cosh ( a z + c ) ,
4.34.14 w = ( 1 / a ) coth ( a z + c ) ,
13: Bibliography Y
  • A. Yu. Yeremin, I. E. Kaporin, and M. K. Kerimov (1985) The calculation of the Riemann zeta function in the complex domain. USSR Comput. Math. and Math. Phys. 25 (2), pp. 111–119.
  • A. Yu. Yeremin, I. E. Kaporin, and M. K. Kerimov (1988) Computation of the derivatives of the Riemann zeta-function in the complex domain. USSR Comput. Math. and Math. Phys. 28 (4), pp. 115–124.
  • T. Yoshida (1995) Computation of Kummer functions U ( a , b , x ) for large argument x by using the τ -method. Trans. Inform. Process. Soc. Japan 36 (10), pp. 2335–2342 (Japanese).
  • F. L. Yost, J. A. Wheeler, and G. Breit (1936) Coulomb wave functions in repulsive fields. Phys. Rev. 49 (2), pp. 174–189.
  • A. Young and A. Kirk (1964) Bessel Functions. Part IV: Kelvin Functions. Royal Society Mathematical Tables, Volume 10, Cambridge University Press, Cambridge-New York.
  • 14: 10.71 Integrals
    In the following equations f ν , g ν is any one of the four ordered pairs given in (10.63.1), and f ^ ν , g ^ ν is either the same ordered pair or any other ordered pair in (10.63.1).
    10.71.1 x 1 + ν f ν d x = - x 1 + ν 2 ( f ν + 1 - g ν + 1 ) = - x 1 + ν ( ν x g ν - g ν ) ,
    10.71.3 x ( f ν g ^ ν - g ν f ^ ν ) d x = x 2 2 ( f ^ ν ( f ν + 1 + g ν + 1 ) - g ^ ν ( f ν + 1 - g ν + 1 ) - f ν ( f ^ ν + 1 + g ^ ν + 1 ) + g ν ( f ^ ν + 1 - g ^ ν + 1 ) ) = 1 2 x ( f ν f ^ ν - f ν f ^ ν + g ν g ^ ν - g ν g ^ ν ) ,
    10.71.4 x ( f ν g ^ ν + g ν f ^ ν ) d x = 1 4 x 2 ( 2 f ν g ^ ν - f ν - 1 g ^ ν + 1 - f ν + 1 g ^ ν - 1 + 2 g ν f ^ ν - g ν - 1 f ^ ν + 1 - g ν + 1 f ^ ν - 1 ) .
    10.71.6 x f ν g ν d x = 1 4 x 2 ( 2 f ν g ν - f ν - 1 g ν + 1 - f ν + 1 g ν - 1 ) ,
    15: Software Index
    16: 23.18 Modular Transformations
    23.18.3 λ ( 𝒜 τ ) = λ ( τ ) ,
    23.18.5 η ( 𝒜 τ ) = ε ( 𝒜 ) ( - i ( c τ + d ) ) 1 / 2 η ( τ ) ,
    where the square root has its principal value and
    17: 8.16 Generalizations
    For a generalization of the incomplete gamma function, including asymptotic approximations, see Chaudhry and Zubair (1994, 2001) and Chaudhry et al. (1996). …
    18: 4.20 Derivatives and Differential Equations
    4.20.8 d n d z n cos z = cos ( z + 1 2 n π ) .
    With a 0 , the general solutions of the differential equations …
    4.20.12 w = A cos ( a z ) + B sin ( a z ) ,
    4.20.13 w = ( 1 / a ) sin ( a z + c ) ,
    4.20.14 w = ( 1 / a ) tan ( a z + c ) ,
    19: Viewing DLMF Interactive 3D Graphics
    Users can render a 3D scene and interactively rotate, scale, and otherwise explore a function surface. …
    20: Roderick S. C. Wong
    He is also the coauthor of Special Functions: A Graduate Text with Richard Beals, that will be published by Cambridge University Press in 2010. … Wong served as a Validator for the original release and publication in May 2010 of the NIST Digital Library of Mathematical Functions and the NIST Handbook of Mathematical Functions. …