By Chuang C.T. (ed.)

ISBN-10: 0821850504

ISBN-13: 9780821850503

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**Extra info for Analytic Functions of One Complex Variable**

**Sample text**

By Miller [309], in the so-called class A of good functions introduced by Lighthill [257, p. 15]. One says that a function f (x) is a good function, if it is everywhere differentiable any number of times and if it and all its derivatives are O x−N as x increases without limit, for all N . Examples of good functions are f (x) = P (x) exp(−γx), where P (x) is a polynomial, γ > 0. 1), the so-called two-dimensional Erd´elyiKober operators of fractional integration of orders α > 0, β > 0: xη y δ K η,α K δ,β f (x, y) = Γ(α)Γ(β) ∞ ∞ t−η−α τ −δ−β (t − x)α−1 (τ − y)β−1 f (t, τ )dtdτ, x y have been considered by different authors in the modified class A of good functions: f (x, y) – differentiable any number of times and having partial derivatives which are O |x|−ε1 , |y|−ε2 as x → ∞, y → ∞, for all ε1 > 0, ε2 > 0.

L)): I α,β,η −1 = I −α,−β,α+η (cf. Saigo [415], [420], [421]). 19). However, we point out that such differintegral inversion formulas have not been considered as new objects, namely, as generalized fractional derivatives. This interpretation, for special cases, will be done in the next section. 6. Properties and examples of generalized (multiple) Erd´ elyi-Kober derivatives. 19). 1. 1) (η1 +···+ηm ) preserve the power functions of Cα (they are simultaneosly of the subspace Cα ): (γk ),(δk ) Dβ,m {xp } = xp m Γ γ +δ + p +1 k k β , p Γ γ + + 1 k k =1 β p > α.

Ii) m = 2. 13) dx dx δ1 > 0, δ2 > 0; α0 , α1 , α2 are determined by β, γ1 , γ2 . Such differential operators are related to the differential equations satisfied by the Gauss, Bessel and Wright functions and the polynomials of mathematical physics. j-j ) (see Raina [387], Raina and Kiryakova [390]). f) where β = 2 − (α0 + α1 + α2 ) > 0, γ = α1 +βα2 −1 , γ2 = αβ2 . f ) related to the Bessel functions Jν (x). g. McBride [289], SprinkhnizenKuiper [457], and also Chapter 3 (for m := 2). g. d n α +n,β −n,η −n α ,β ,η I := I , dx where n is the smallest integer such that 0 < α + n ≤ 1.

### Analytic Functions of One Complex Variable by Chuang C.T. (ed.)

by Steven

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