By Carlo Viola (auth.)

The topics handled during this publication were in particular selected to symbolize a bridge connecting the content material of a primary direction at the simple idea of analytic features with a rigorous therapy of a few of crucial precise services: the Euler gamma functionality, the Gauss hypergeometric functionality, and the Kummer confluent hypergeometric functionality. Such distinctive capabilities are fundamental 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 knowing of uncomplicated techniques to the extent of an effortless direction masking the residue theorem, Cauchy's necessary formulation, the Taylor and Laurent sequence expansions, poles and crucial singularities, department issues, and so forth. The e-book addresses the wishes of complex undergraduate and graduate scholars in arithmetic or physics.

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2n + 1)!! © Springer International Publishing Switzerland 2016 C. 2) 49 50 5 Summation Formulae with the usual notation (2n − 1)!! = 1 · 3 · 5 · · · (2n − 1), (2n)!! = 2 · 4 · 6 · · · (2n) for the semifactorial N !! of a positive integer N . For 0 ≤ x ≤ π/2 we have 0 ≤ sin x ≤ 1, whence sin x ≥ sin2 x ≥ sin3 x ≥ . . Therefore, integrating from 0 to π/2, S1 ≥ S2 ≥ S3 ≥ . . 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 = ··· .

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 have z log E , p zn z ≤ log E , p zn ∞ 1 z = − k zn k= p+1 z ≤ zn p+1 p z = log 1 − + zn k ∞ ≤ k= p+1 1 1 1 + + 2 + ... 9), log E |z n |≥2|z| z , p zn |z n |−( p+1) ≤ 2 |z| p+1 |z n |≥2|z| |z| p+1 . 9), we have p ≤ β ≤ p + 1. 23) yields z |z|β . 23) log E |z n |≥2|z| z , p zn ≤ 2 |z|β+ε |z n |≥2|z| β+ε 2 |z| 2 p+1−β−ε ≤ In either case log E |z n |≥2|z| If |z/z n | > log E 1 2 |z n |−( p+1) |z| p+1−β−ε |z n |−(β+ε) |z|β+ε .

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! 6) and √ n! = 2πn n e n 1+O 1 n . 7) are equivalent since, by Taylor’s formula, exp O( n1 ) = 1 + O( n1 ) and log(1 + O( n1 )) = O( n1 ).

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