About the Project
NIST

Whittaker functions

AdvancedHelp

(0.002 seconds)

1—10 of 103 matching pages

1: 13.27 Mathematical Applications
§13.27 Mathematical Applications
The other group elements correspond to integral operators whose kernels can be expressed in terms of Whittaker functions. This identification can be used to obtain various properties of the Whittaker functions, including recurrence relations and derivatives. …
2: 13.28 Physical Applications
§13.28(ii) Coulomb Functions
3: 13.22 Zeros
§13.22 Zeros
4: 13.18 Relations to Other Functions
§13.18(i) Elementary Functions
§13.18(ii) Incomplete Gamma Functions
§13.18(iv) Parabolic Cylinder Functions
§13.18(v) Orthogonal Polynomials
For representations of Coulomb functions in terms of Whittaker functions see (33.2.3), (33.2.7), (33.14.4) and (33.14.7)
5: 13.24 Series
§13.24 Series
§13.24(i) Expansions in Series of Whittaker Functions
For expansions of arbitrary functions in series of M κ , μ ( z ) functions see Schäfke (1961b).
§13.24(ii) Expansions in Series of Bessel Functions
13.24.2 1 Γ ( 1 + 2 μ ) M κ , μ ( z ) = 2 2 μ z μ + 1 2 s = 0 p s ( μ ) ( z ) ( 2 κ z ) - 2 μ - s J 2 μ + s ( 2 κ z ) ,
6: 13.1 Special Notation
The main functions treated in this chapter are the Kummer functions M ( a , b , z ) and U ( a , b , z ) , Olver’s function M ( a , b , z ) , and the Whittaker functions M κ , μ ( z ) and W κ , μ ( z ) . …
7: 13.23 Integrals
§13.23(i) Laplace and Mellin Transforms
§13.23(ii) Fourier Transforms
§13.23(iii) Hankel Transforms
§13.23(iv) Integral Transforms in terms of Whittaker Functions
8: 13.25 Products
§13.25 Products
13.25.1 M κ , μ ( z ) M κ , - μ - 1 ( z ) + ( 1 2 + μ + κ ) ( 1 2 + μ - κ ) 4 μ ( 1 + μ ) ( 1 + 2 μ ) 2 M κ , μ + 1 ( z ) M κ , - μ ( z ) = 1 .
9: 13.19 Asymptotic Expansions for Large Argument
§13.19 Asymptotic Expansions for Large Argument
13.19.1 M κ , μ ( x ) Γ ( 1 + 2 μ ) Γ ( 1 2 + μ - κ ) e 1 2 x x - κ s = 0 ( 1 2 - μ + κ ) s ( 1 2 + μ + κ ) s s ! x - s , μ - κ - 1 2 , - 3 2 , .
13.19.2 M κ , μ ( z ) Γ ( 1 + 2 μ ) Γ ( 1 2 + μ - κ ) e 1 2 z z - κ s = 0 ( 1 2 - μ + κ ) s ( 1 2 + μ + κ ) s s ! z - s + Γ ( 1 + 2 μ ) Γ ( 1 2 + μ + κ ) e - 1 2 z ± ( 1 2 + μ - κ ) π i z κ s = 0 ( 1 2 + μ - κ ) s ( 1 2 - μ - κ ) s s ! ( - z ) - s , - 1 2 π + δ ± ph z 3 2 π - δ ,
10: 13.14 Definitions and Basic Properties
Standard solutions are: … The series …
§13.14(vi) Wronskians
§13.14(vii) Connection Formulas