entire functions
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11: 25.15 Dirichlet -functions
12: 15.16 Products
§15.16 Products
… โบwhere and , , are defined by the generating function โบ
15.16.2
.
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โบ
Generalized Legendre’s Relation
… โบFor 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
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โบSince (28.2.1) has no finite singularities its solutions are entire functions of .
Furthermore, a solution with given initial constant values of and at a point is an entire function of the three variables , , and .
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โบ
is an entire function of .
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14: 9.2 Differential Equation
15: Mathematical Introduction
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โบThis is because is akin to the notation used for Bessel functions (§10.2(ii)), inasmuch as is an entire function of each of its parameters , , and :โ this results in fewer restrictions and simpler equations.
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16: 8.2 Definitions and Basic Properties
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โบThe function
is entire in and .
When , is an entire function of , and is meromorphic with simple poles at , , with residue .
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17: 20.2 Definitions and Periodic Properties
18: 15.2 Definitions and Analytical Properties
19: 30.3 Eigenvalues
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โบWith , the spheroidal wave functions
are solutions of Equation (30.2.1) which are bounded on , or equivalently, which are of the form where is an entire function of .
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