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1: DLMF Project News
error generating summary
2: 15.20 Software
§15.20(ii) Real Parameters and Argument
§15.20(iii) Complex Parameters and Argument
3: 35.10 Methods of Computation
See Yan (1992) for the F 1 1 and F 1 2 functions of matrix argument in the case m = 2 , and Bingham et al. (1992) for Monte Carlo simulation on O ( m ) applied to a generalization of the integral (35.5.8). …
4: 15.4 Special Cases
§15.4(ii) Argument Unity
§15.4(iii) Other Arguments
5: 4.29 Graphics
§4.29(i) Real Arguments
See accompanying text
Figure 4.29.4: Principal values of arctanh x and arccoth x . ( arctanh x is complex when x < - 1 or x > 1 , and arccoth x is complex when - 1 < x < 1 .) Magnify
See accompanying text
Figure 4.29.6: Principal values of arccsch x and arcsech x . … Magnify
§4.29(ii) Complex Arguments
The conformal mapping w = sinh z is obtainable from Figure 4.15.7 by rotating both the w -plane and the z -plane through an angle 1 2 π , compare (4.28.8). …
6: 35.7 Gaussian Hypergeometric Function of Matrix Argument
35.7.2 P ν ( γ , δ ) ( T ) = Γ m ( γ + ν + 1 2 ( m + 1 ) ) Γ m ( γ + 1 2 ( m + 1 ) ) F 1 2 ( - ν , γ + δ + ν + 1 2 ( m + 1 ) γ + 1 2 ( m + 1 ) ; T ) , 0 < T < I ; γ , δ , ν ; ( γ ) > - 1 .
35.7.3 F 1 2 ( a , b c ; [ t 1 0 0 t 2 ] ) = k = 0 ( a ) k ( c - a ) k ( b ) k ( c - b ) k k ! ( c ) 2 k ( c - 1 2 ) k ( t 1 t 2 ) k F 1 2 ( a + k , b + k c + 2 k ; t 1 + t 2 - t 1 t 2 ) .
Let f : Ω (a) be orthogonally invariant, so that f ( T ) is a symmetric function of t 1 , , t m , the eigenvalues of the matrix argument T Ω ; (b) be analytic in t 1 , , t m in a neighborhood of T = 0 ; (c) satisfy f ( 0 ) = 1 . … Systems of partial differential equations for the F 1 0 (defined in §35.8) and F 1 1 functions of matrix argument can be obtained by applying (35.8.9) and (35.8.10) to (35.7.9). …
7: 35.8 Generalized Hypergeometric Functions of Matrix Argument
The generalized hypergeometric function F q p with matrix argument T 𝒮 , numerator parameters a 1 , , a p , and denominator parameters b 1 , , b q is … Multidimensional Mellin–Barnes integrals are established in Ding et al. (1996) for the functions F q p and F p p + 1 of matrix argument. A similar result for the F 1 0 function of matrix argument is given in Faraut and Korányi (1994, p. 346). …
8: 35.6 Confluent Hypergeometric Functions of Matrix Argument
35.6.2 Ψ ( a ; b ; T ) = 1 Γ m ( a ) Ω etr ( - T X ) | X | a - 1 2 ( m + 1 ) | I + X | b - a - 1 2 ( m + 1 ) d X , ( a ) > 1 2 ( m - 1 ) , T Ω .
35.6.3 L ν ( γ ) ( T ) = Γ m ( γ + ν + 1 2 ( m + 1 ) ) Γ m ( γ + 1 2 ( m + 1 ) ) F 1 1 ( - ν γ + 1 2 ( m + 1 ) ; T ) , ( γ ) , ( γ + ν ) > - 1 .
9: 14.2 Differential Equations
Unless stated otherwise in §§14.214.20 it is assumed that the arguments of the functions P ν μ ( x ) and Q ν μ ( x ) lie in the interval ( - 1 , 1 ) , and the arguments of the functions P ν μ ( x ) , Q ν μ ( x ) , and Q ν μ ( x ) lie in the interval ( 1 , ) . …
10: 10.17 Asymptotic Expansions for Large Argument
10.17.4 Y ν ( z ) ( 2 π z ) 1 2 ( sin ω k = 0 ( - 1 ) k a 2 k ( ν ) z 2 k + cos ω k = 0 ( - 1 ) k a 2 k + 1 ( ν ) z 2 k + 1 ) , | ph z | π - δ ,
10.17.14 | R ± ( ν , z ) | 2 | a ( ν ) | 𝒱 z , ± i ( t - ) exp ( | ν 2 - 1 4 | 𝒱 z , ± i ( t - 1 ) ) ,