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confluent hypergeometric functions

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1: 35.6 Confluent Hypergeometric Functions of Matrix Argument
§35.6 Confluent Hypergeometric Functions of Matrix Argument
§35.6(i) Definitions
Laguerre Form
§35.6(ii) Properties
§35.6(iv) Asymptotic Approximations
2: 6.11 Relations to Other Functions
Confluent Hypergeometric Function
6.11.2 E 1 ( z ) = e z U ( 1 , 1 , z ) ,
3: 13.28 Physical Applications
§13.28(i) Exact Solutions of the Wave Equation
For potentials in quantum mechanics that are solvable in terms of confluent hypergeometric functions see Negro et al. (2000). …
§13.28(iii) Other Applications
4: 13.27 Mathematical Applications
§13.27 Mathematical Applications
Confluent hypergeometric functions are connected with representations of the group of third-order triangular matrices. … …
5: 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. …
6: 8.5 Confluent Hypergeometric Representations
§8.5 Confluent Hypergeometric Representations
For the confluent hypergeometric functions M , M , U , and the Whittaker functions M κ , μ and W κ , μ , see §§13.2(i) and 13.14(i). …
8.5.2 γ * ( a , z ) = e z M ( 1 , 1 + a , z ) = M ( a , 1 + a , z ) .
8.5.3 Γ ( a , z ) = e z U ( 1 a , 1 a , z ) = z a e z U ( 1 , 1 + a , z ) .
8.5.5 Γ ( a , z ) = e 1 2 z z 1 2 a 1 2 W 1 2 a 1 2 , 1 2 a ( z ) .
7: 13.6 Relations to Other Functions
§13.6(ii) Incomplete Gamma Functions
§13.6(iii) Modified Bessel Functions
§13.6(iv) Parabolic Cylinder Functions
Hermite Polynomials
Laguerre Polynomials
8: 13.18 Relations to Other Functions
§13.18(ii) Incomplete Gamma Functions
§13.18(iii) Modified Bessel Functions
§13.18(iv) Parabolic Cylinder Functions
Hermite Polynomials
Laguerre Polynomials
9: 13 Confluent Hypergeometric Functions
Chapter 13 Confluent Hypergeometric Functions
10: 13.2 Definitions and Basic Properties
13.2.34 𝒲 { M ( a , b , z ) , U ( a , b , z ) } = z b e z / Γ ( a ) ,
13.2.35 𝒲 { M ( a , b , z ) , e z U ( b a , b , e ± π i z ) } = e b π i z b e z / Γ ( b a ) ,
13.2.36 𝒲 { z 1 b M ( a b + 1 , 2 b , z ) , U ( a , b , z ) } = z b e z / Γ ( a b + 1 ) ,
Kummer’s Transformations