About the Project

entire functions

AdvancedHelp

(0.004 seconds)

11—20 of 48 matching pages

11: 25.15 Dirichlet L -functions
โ–บIf ฯ‡ ฯ‡ 1 , then L โก ( s , ฯ‡ ) is an entire function of s . …
12: 15.16 Products
§15.16 Products
โ–บwhere A 0 = 1 and A s , s = 1 , 2 , , are defined by the generating function โ–บ
15.16.2 ( 1 z ) a + b c โข F โก ( 2 โข a , 2 โข b ; 2 โข c 1 ; z ) = s = 0 A s โข z s , | z | < 1 .
โ–บ
Generalized Legendre’s Relation
โ–บFor further results of this kind, and also series of products of hypergeometric functions, see Erdélyi et al. (1953a, §2.5.2).
13: 28.2 Definitions and Basic Properties
โ–บSince (28.2.1) has no finite singularities its solutions are entire functions of z . Furthermore, a solution w with given initial constant values of w and w at a point z 0 is an entire function of the three variables z , a , and q . … โ–บ cos โก ( ฯ€ โข ฮฝ ) is an entire function of a , q 2 . …
14: 9.2 Differential Equation
โ–บAll solutions are entire functions of z . …
15: Mathematical Introduction
โ–บThis is because ๐… is akin to the notation used for Bessel functions10.2(ii)), inasmuch as ๐… is an entire function of each of its parameters a , b , and c :โ€‹ this results in fewer restrictions and simpler equations. …
16: 8.2 Definitions and Basic Properties
โ–บThe function ฮณ โก ( a , z ) is entire in z and a . When z 0 , ฮ“ โก ( a , z ) is an entire function of a , and ฮณ โก ( a , z ) is meromorphic with simple poles at a = n , n = 0 , 1 , 2 , , with residue ( 1 ) n / n ! . …
17: 20.2 Definitions and Periodic Properties
โ–บFor fixed ฯ„ , each ฮธ j โก ( z | ฯ„ ) is an entire function of z with period 2 โข ฯ€ ; ฮธ 1 โก ( z | ฯ„ ) is odd in z and the others are even. …
18: 15.2 Definitions and Analytical Properties
โ–บThe principal branch of ๐… โก ( a , b ; c ; z ) is an entire function of a , b , and c . …
19: 30.3 Eigenvalues
โ–บWith ฮผ = m = 0 , 1 , 2 , , the spheroidal wave functions ๐–ฏ๐—Œ n m โก ( x , ฮณ 2 ) are solutions of Equation (30.2.1) which are bounded on ( 1 , 1 ) , or equivalently, which are of the form ( 1 x 2 ) 1 2 โข m โข g โก ( x ) where g โก ( z ) is an entire function of z . …
20: 16.5 Integral Representations and Integrals
โ–บIn the case p = q the left-hand side of (16.5.1) is an entire function, and the right-hand side supplies an integral representation valid when | ph โก ( z ) | < ฯ€ / 2 . …