Carlo Viola (auth.)'s An Introduction to Special Functions PDF

By Carlo Viola (auth.)

ISBN-10: 3319413449

ISBN-13: 9783319413440

ISBN-10: 3319413457

ISBN-13: 9783319413457

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.

Show description

Read or Download An Introduction to Special Functions PDF

Similar mathematics_1 books

Download e-book for iPad: The integration of functions of a single variable by G. H. Hardy

Famed for his achievements in quantity conception and mathematical research, G. H. Hardy ranks one of the 20th century's nice mathematicians and educators. during this vintage treatise, Hardy explores the combination of capabilities of a unmarried variable along with his attribute readability and precision. Following an creation, Hardy discusses basic features, their class and integration, and he provides a precis of effects.

Download e-book for kindle: Multiple Gaussian Hypergeometric Series by H. M. Srivastava

A a number of Gaussian hypergeometric sequence is a hypergeometric sequence in two
or extra variables which reduces to the common Gaussian hypergeometric
series, each time just one variable is non-zero. fascinating difficulties in the
theory of a number of Gaussian hypergeometric sequence consist in constructing
all detailed sequence and in setting up their areas of convergence. either of
these difficulties are fairly straight forward for unmarried sequence, and so they have
been thoroughly solved with regards to double sequence. This e-book is the 1st to
aim at providing a scientific (and thorough) dialogue of the complexity
of those difficulties while the measurement exceeds ; certainly, it supplies the
complete resolution of every of the issues in case of the triple Gaussian
hypergeometric sequence.

Extra resources for An Introduction to Special Functions

Sample text

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.

Download PDF sample

An Introduction to Special Functions by Carlo Viola (auth.)


by Steven
4.3

Rated 4.29 of 5 – based on 49 votes