By Carlo Viola (auth.)
The topics handled during this booklet were in particular selected to symbolize a bridge connecting the content material of a primary path at the straightforward idea of analytic capabilities with a rigorous remedy of a few of crucial distinctive features: the Euler gamma functionality, the Gauss hypergeometric functionality, and the Kummer confluent hypergeometric functionality. Such designated services are critical instruments in "higher calculus" and are usually encountered in just about all branches of natural and utilized arithmetic. the single wisdom assumed at the a part of the reader is an figuring out of easy recommendations to the extent of an hassle-free direction protecting the residue theorem, Cauchy's essential formulation, the Taylor and Laurent sequence expansions, poles and crucial singularities, department issues, and so forth. The booklet addresses the desires of complicated undergraduate and graduate scholars in arithmetic or physics.
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Extra resources for An Introduction to Special Functions
2k) is a rational multiple of π 2k and hence is transcendental by Lindemann’s theorem (1882) on the transcendence of π. For each k = 1, 2, 3, . . , ζ(2k + 1) is conjectured to be transcendental, but very little has been proved so far about the arithmetical nature of ζ(2k + 1). , for no explicitly given integer k ≥ 2 the irrationality of ζ(2k + 1) has been proved), and Zudilin (2004) proved that at least one of the four numbers ζ(5), ζ(7), ζ(9) and ζ(11) is irrational. 2 The Bernoulli polynomials are defined by Bn (x) = ∂ n z ex z ∂z n e z − 1 (n = 0, 1, 2, .
2), 1 ≥ 1 1 S2n+1 2n S2n+1 . 2), (2n)!! (2n − 1)!! 2 1 = 2n + 1 n k=1 2k 2k 2k − 1 2k + 1 = 1 π . 3) For n → ∞ we get Wallis’ formula: π = 2 ∞ 2 2 4 4 6 6 4k 2 = ··· . 3), (2n)!! (2n − 1)!! 2 = 1 π (2n + 1) 1 + O 2 n = πn 1 + O 1 n . 4) By Taylor’s formula, (1 + x)1/2 = 1 + 21 x + 1/2 x 2 + . . = 1 + O(x) (x → 0). 2 Hence 1 1/2 1 1+O (n → ∞). = 1+O n n Also (2n − 1)!! (2n)!! = (2n)! and (2n)!! 2 (2n)!! = = . (2n − 1)!! (2n)! (2n)! 1 Stirling’s Formula for n! 2 = πn 1 + O (2n)! n (n → ∞). 1 (Stirling’s ‘elementary’ formula) For n → ∞ we have log n!
To see this, note exp(|z|n+ε ), so that that |P(z)| |z|n , whence e P(z) = eRe P(z) ≤ e|P(z)| P(z) n n−1 ≤ n. If P(z) = a0 z + a1 z + . . + an (a0 = 0), and if z → ∞ along ord e one of the n half-lines defined by arg z = − n1 arg a0 , whence arg(a0 z n ) = arg a0 + n arg z = 0 so that a0 z n > 0, then Re P(z) − |z|n−ε = Re(a0 z n ) + O(|z|n−ε ) = |a0 ||z|n + O(|z|n−ε ) → +∞. This shows that lim sup z→∞ e P(z) = lim sup exp(Re P(z) − |z|n−ε ) = +∞, exp(|z|n−ε ) z→∞ whence ord e P(z) ≥ n. We aim at relating the order of an entire function with the density of its zeros.
An Introduction to Special Functions by Carlo Viola (auth.)