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11: 4.8 Identities
If a 0 and a z has its general value, then …
12: 16.2 Definition and Analytic Properties
Unless indicated otherwise it is assumed that in the DLMF generalized hypergeometric functions assume their principal values. … In general the series (16.2.1) diverges for all nonzero values of z . …Note that if m is the value of the numerically largest a j that is a nonpositive integer, then the identity …
13: 33.13 Complex Variable and Parameters
The functions F ( η , ρ ) , G ( η , ρ ) , and H ± ( η , ρ ) may be extended to noninteger values of by generalizing ( 2 + 1 ) ! = Γ ( 2 + 2 ) , and supplementing (33.6.5) by a formula derived from (33.2.8) with U ( a , b , z ) expanded via (13.2.42). …
14: 16.4 Argument Unity
The function F q q + 1 with argument unity and general values of the parameters is discussed in Bühring (1992). …
15: 22.15 Inverse Functions
The principal values satisfy …and unless stated otherwise it is assumed that the inverse functions assume their principal values. The general solutions of (22.15.1), (22.15.2), (22.15.3) are, respectively, …
16: 1.9 Calculus of a Complex Variable
Equations (1.9.18) and (1.9.20) hold for general values of the phases, but not necessarily for the principal values. …
17: 35.8 Generalized Hypergeometric Functions of Matrix Argument
Value at 𝐓 = 𝟎
18: 12.14 The Function W ( a , x )
These follow from the contour integrals of §12.5(ii), which are valid for general complex values of the argument z and parameter a . …
19: 35.10 Methods of Computation
§35.10 Methods of Computation
For small values of 𝐓 the zonal polynomial expansion given by (35.8.1) can be summed numerically. … 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). … These algorithms are extremely efficient, converge rapidly even for large values of m , and have complexity linear in m .
20: 14.20 Conical (or Mehler) Functions
Lastly, for the range 1 < x < , P 1 2 + i τ μ ( x ) is a real-valued solution of (14.20.1); in terms of Q 1 2 ± i τ μ ( x ) (which are complex-valued in general): …