Digital Library of Mathematical Functions
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11 Struve and Related FunctionsStruve and Modified Struve Functions

§11.2 Definitions

Contents

§11.2(i) Power-Series Expansions

11.2.1 Hν(z) =(12z)ν+1n=0(-1)n(12z)2nΓ(n+32)Γ(n+ν+32),
11.2.2 Lν(z) =--12πνHν(z)
=(12z)ν+1n=0(12z)2nΓ(n+32)Γ(n+ν+32).

Principal values correspond to principal values of (12z)ν+1; compare §4.2(i).

The expansions (11.2.1) and (11.2.2) are absolutely convergent for all finite values of z. The functions z-ν-1Hν(z) and z-ν-1Lν(z) are entire functions of z and ν.

11.2.3 H0(z) =2π(z-z31232+z5123252-),
11.2.4 L0(z) =2π(z+z31232+z5123252+).
11.2.5 Kν(z) =Hν(z)-Yν(z),
11.2.6 Mν(z) =Lν(z)-Iν(z).

Principal values of Kν(z) and Mν(z) correspond to principal values of the functions on the right-hand sides of (11.2.5) and (11.2.6).

Unless indicated otherwise, Hν(z), Kν(z), Lν(z), and Mν(z) assume their principal values throughout the DLMF.

§11.2(ii) Differential Equations

Struve’s Equation

11.2.7 2wz2+1zwz+(1-ν2z2)w=(12z)ν-1πΓ(ν+12).

Particular solutions:

11.2.8 w=Hν(z),Kν(z).

Modified Struve’s Equation

11.2.9 2wz2+1zwz-(1+ν2z2)w=(12z)ν-1πΓ(ν+12).

Particular solutions:

11.2.10 w=Lν(z),Mν(z).

§11.2(iii) Numerically Satisfactory Solutions

In this subsection A and B are arbitrary constants.

When z=x, 0<x<, and ν0, numerically satisfactory general solutions of (11.2.7) are given by

11.2.11 w =Hν(x)+AJν(x)+BYν(x),
11.2.12 w =Kν(x)+AJν(x)+BYν(x).

(11.2.11) applies when x is bounded, and (11.2.12) applies when x is bounded away from the origin.

When z and ν0, numerically satisfactory general solutions of (11.2.7) are given by

11.2.13 w =Hν(z)+AJν(z)+BHν(1)(z),
11.2.14 w =Hν(z)+AJν(z)+BHν(2)(z),
11.2.15 w =Kν(z)+AHν(1)(z)+BHν(2)(z).

(11.2.13) applies when 0phzπ and |z| is bounded. (11.2.14) applies when -πphz0 and |z| is bounded. (11.2.15) applies when |phz|π and z is bounded away from the origin.

When ν0, numerically satisfactory general solutions of (11.2.9) are given by

11.2.16 w =Lν(z)+AKν(z)+BIν(z),
11.2.17 w =Mν(z)+AKν(z)+BIν(z).

(11.2.16) applies when |phz|12π with |z| bounded. (11.2.17) applies when |phz|12π with z bounded away from the origin.