By Carlo Viola (auth.)
The matters handled during this ebook were particularly selected to symbolize a bridge connecting the content material of a primary direction at the hassle-free idea of analytic services with a rigorous remedy of a few of crucial unique services: the Euler gamma functionality, the Gauss hypergeometric functionality, and the Kummer confluent hypergeometric functionality. Such distinct services are integral instruments in "higher calculus" and are often encountered in just about all branches of natural and utilized arithmetic. the one wisdom assumed at the a part of the reader is an figuring out of simple ideas to the extent of an user-friendly direction protecting the residue theorem, Cauchy's vital formulation, the Taylor and Laurent sequence expansions, poles and crucial singularities, department issues, and so on. The booklet addresses the wishes of complicated undergraduate and graduate scholars in arithmetic or physics.
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Additional resources for An Introduction to Special Functions
The Taylor expansion of G(z) implies that G(z) is a polynomial of degree μ ≤ ν = α . 1, F(z) = e G(z) has order μ, whence α = μ ≤ α . Thus α = μ = α ∈ N, and deg G(z) = μ = α. 5 Let z 1 , z 2 , . . → ∞ be any sequence of non-zero complex numbers with exponent of convergence β < +∞. The canonical product z , p = zn E n p 1− n z 1 z exp zn k zn k=1 k is an entire function of order β. 11). 2 we know that β ≤ α. 3), z , p zn E n ε exp(|z|β+ε ) for every ε > 0. 22) with ε/2 in place of ε one gets z , p zn E log n ε |z|β+ε/2 = o(|z|β+ε ) ≤ ϑ|z|β+ε with 0 < ϑ < 1, whence E n If |z/z n | ≤ 1 2 z , p zn ≤ exp(ϑ|z|β+ε ) = o exp(|z|β+ε ) .
We define the exponent of convergence of the sequence (z n ) as the infimum β of the exponents B > 0 such that ∞ |z n |−B < +∞. 7) n=1 ∞ If |z n |−B = +∞ for every B > 0, the exponent of convergence of (z n ) is +∞. 2 If an entire function F(z) satisfying F(0) = 0 has order α, and if the sequence z 1 , z 2 , . . of the zeros of F(z) has exponent of convergence β, then β ≤ α. 6) with r = |z n | we get n |z n |−(α+2ε) n −(α+2ε)/(α+ε) = n −(1+ε1 ) , for a suitable ε1 > 0. Therefore ∞ |z n |−(α+2ε) n=1 ∞ n −(1+ε1 ) < +∞.
Of respective multiplicities m 1 , m 2 , . . Then the product g(z)ϕ(z) has no poles, and therefore is an entire function h(z), whence g(z) = h(z)/ϕ(z) with h(z) and ϕ(z) entire functions. If g(z) has a pole at z = 0 of multiplicity m, then g(z) : = z m g(z) is regular at z = 0, whence g(z) = h(z)/ϕ(z) with entire h(z) and ϕ(z), and g(z) = h(z)/(z m ϕ(z)) with entire h(z) and z m ϕ(z). 2 (Weierstrass) Let F(z) be meromorphic, and regular and = 0 at z = 0. Let z 1 , z 2 , . . be the zeros and poles of F(z) with respective multiplicities |m 1 |, |m 2 |, .
An Introduction to Special Functions by Carlo Viola (auth.)