confluent hypergeometric function


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confluent hypergeometric function

[kən′flü·ənt ¦hī·pər‚jē·ə¦me‚trik ′fəŋk·shən]
(mathematics)
A solution to differential equation z (d 2 w/dz 2) + (ρ-z)(dw/dz)-α w = 0.
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where [[OMEGA].sub.d] = E([R.sup.2]) denotes the average power of desired information signal, while [F.sub.1](a,b,z) and [GAMMA](a) denote the confluent hypergeometric function [10, Eq.(9.210/1)] and Gamma function [10, Eq.
with 1F1 being the Kummer confluent hypergeometric function [6].
We define the confluent hypergeometric function with matrix parameters as
where [beta] - v > (m - 1)/2, [alpha] - v > (m - 1)/2, v > (m - 1)/2,[theta] > 0, [OMEGA] > 0, and [sub.1][F.sub.1] is the confluent hypergeometric function of the first kind of matrix argument (Gupta and Nagar [1]).
2 Close-to-convexity of the confluent hypergeometric function
where [GAMMA](x) is the gamma function, [sub.1][F.sub.1](a, b; x) is the confluent hypergeometric function, [w.sub.0] is the beam waist, p [greater than or equal to] -[absolute value of m] is a real valued parameter, m is the topological charge, p and m together specify the HyG mode, [chi] = [[rho].sup.2]/([w.sup.2.sub.0][[xi] + i]), and [xi] = z/[z.sub.R] with [z.sub.R] = [pi][w.sup.2.sub.0]/[lambda] the Rayleigh range.
then a confluent hypergeometric function can be shown to be a limiting case of a hypergeometric function as follows.
* Confluent hypergeometric function. This corresponds to v = -[alpha].
The Gini index of the exponential hierarchy can be expressed in terms of the confluent hypergeometric function, whose integral representation is given by (b [greater than] a):
The results are presented in terms of hypergeometric functions and confluent hypergeometric functions. It is of interest to note that the confluent hypergeometric function, M, yields the prior moment generating function of p from density (1).