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Kummer functions

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1: 13.6 Relations to Other Functions
§13.6(i) Elementary Functions
§13.6(ii) Incomplete Gamma Functions
§13.6(iv) Parabolic Cylinder Functions
§13.6(vi) Generalized Hypergeometric Functions
For representations of Coulomb functions in terms of Kummer functions see (33.2.4), (33.2.8) and (33.14.5).
2: 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 ) . …
3: 13.10 Integrals
§13.10(i) Indefinite Integrals
§13.10(ii) Laplace Transforms
§13.10(iii) Mellin Transforms
§13.10(iv) Fourier Transforms
4: 13.12 Products
§13.12 Products
13.12.1 M ( a , b , z ) M ( - a , - b , - z ) + a ( a - b ) z 2 b 2 ( 1 - b 2 ) M ( 1 + a , 2 + b , z ) M ( 1 - a , 2 - b , - z ) = 1 .
5: 13.2 Definitions and Basic Properties
§13.2(vi) Wronskians
Kummer’s Transformations
6: 13.11 Series
13.11.1 M ( a , b , z ) = Γ ( a - 1 2 ) e 1 2 z ( 1 4 z ) 1 2 - a s = 0 ( 2 a - 1 ) s ( 2 a - b ) s ( b ) s s ! ( a - 1 2 + s ) I a - 1 2 + s ( 1 2 z ) , a + 1 2 , b 0 , - 1 , - 2 , .
7: 13.31 Approximations
§13.31 Approximations
§13.31(i) Chebyshev-Series Expansions
13.31.3 z a U ( a , 1 + a - b , z ) = lim n A n ( z ) B n ( z ) .
8: 13.30 Tables
§13.30 Tables
9: 13.13 Addition and Multiplication Theorems
§13.13(i) Addition Theorems for M ( a , b , z )
The function M ( a , b , x + y ) has the following expansions: … The function U ( a , b , x + y ) has the following expansions: …
13.13.12 e y ( x + y x ) 1 - b n = 0 ( - y ) n n ! x n U ( a - n , b - n , x ) , | y | < | x | .
§13.13(iii) Multiplication Theorems for M ( a , b , z ) and U ( a , b , z )
10: 12.20 Approximations
Luke (1969b, pp. 25 and 35) gives Chebyshev-series expansions for the confluent hypergeometric functions U ( a , b , x ) and M ( a , b , x ) 13.2(i)) whose regions of validity include intervals with endpoints x = and x = 0 , respectively. …