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1: 10.39 Relations to Other Functions
Elementary Functions
Parabolic Cylinder Functions
Confluent Hypergeometric Functions
Generalized Hypergeometric Functions and Hypergeometric Function
2: 10.16 Relations to Other Functions
Elementary Functions
H 1 2 ( 2 ) ( z ) = i H 1 2 ( 2 ) ( z ) = i ( 2 π z ) 1 2 e i z .
Parabolic Cylinder Functions
Confluent Hypergeometric Functions
Generalized Hypergeometric Functions
3: 18.34 Bessel Polynomials
§18.34(i) Definitions and Recurrence Relation
18.34.1 y n ( x ; a ) = F 0 2 ( n , n + a 1 ; x 2 ) = ( n + a 1 ) n ( x 2 ) n F 1 1 ( n 2 n a + 2 ; 2 x ) = n ! ( 1 2 x ) n L n ( 1 a 2 n ) ( 2 x 1 ) = ( 1 2 x ) 1 1 2 a e 1 / x W 1 1 2 a , 1 2 ( a 1 ) + n ( 2 x 1 ) .
where 𝗄 n is a modified spherical Bessel function (10.49.9), and …
18.34.8 lim α P n ( α , a α 2 ) ( 1 + α x ) P n ( α , a α 2 ) ( 1 ) = y n ( x ; a ) .
4: 13.18 Relations to Other Functions
§13.18(iii) Modified Bessel Functions
5: 9.6 Relations to Other Functions
§9.6(i) Airy Functions as Bessel Functions, Hankel Functions, and Modified Bessel Functions
§9.6(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions as Airy Functions
9.6.20 H 2 / 3 ( 2 ) ( ζ ) = e 2 π i / 3 H 2 / 3 ( 2 ) ( ζ ) = e π i / 6 ( 3 / z ) ( Ai ( z ) + i Bi ( z ) ) .
6: 35.6 Confluent Hypergeometric Functions of Matrix Argument
§35.6(iii) Relations to Bessel Functions of Matrix Argument
7: 9.13 Generalized Airy Functions
Swanson and Headley (1967) define independent solutions A n ( z ) and B n ( z ) of (9.13.1) by …
A n ( 0 ) = p 1 / 2 B n ( 0 ) = p p Γ ( p ) .
8: 13.6 Relations to Other Functions
§13.6(iii) Modified Bessel Functions
9: 12.7 Relations to Other Functions
§12.7(iii) Modified Bessel Functions
10: 9.8 Modulus and Phase
(These definitions of θ ( x ) and ϕ ( x ) differ from Abramowitz and Stegun (1964, Chapter 10), and agree more closely with those used in Miller (1946) and Olver (1997b, Chapter 11).) …