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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: 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
7: 13 Confluent Hypergeometric Functions
Chapter 13 Confluent Hypergeometric Functions
8: 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 ) . …
9: 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 ) .
10: 13.10 Integrals
13.10.6 0 e - z t - t 2 t 2 b - 2 M ( a , b , t 2 ) d t = 1 2 π - 1 2 Γ ( b - 1 2 ) U ( b - 1 2 , a + 1 2 , 1 4 z 2 ) , b > 1 2 , z > 0 ,
13.10.10 0 t λ - 1 M ( a , b , - t ) d t = Γ ( λ ) Γ ( a - λ ) Γ ( a ) Γ ( b - λ ) , 0 < λ < a ,
13.10.12 0 cos ( 2 x t ) M ( a , b , - t 2 ) d t = π 2 Γ ( a ) x 2 a - 1 e - x 2 U ( b - 1 2 , a + 1 2 , x 2 ) , a > 0 .
13.10.14 0 e - t t 1 2 ν M ( a , b , t ) J ν ( 2 x t ) d t = x 1 2 ν e - x Γ ( b - a ) U ( a , a - b + ν + 2 , x ) , x > 0 , - 1 < ν < 2 ( b - a ) - 1 2 ,
13.10.16 0 e - t t 1 2 ν U ( a , b , t ) J ν ( 2 x t ) d t = Γ ( ν - b + 2 ) x 1 2 ν e - x M ( a , a - b + ν + 2 , x ) , x > 0 , max ( b - 2 , - 1 ) < ν .