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11 Struve and Related FunctionsStruve and Modified Struve Functions

§11.2 Definitions

Contents
  1. §11.2(i) Power-Series Expansions
  2. §11.2(ii) Differential Equations
  3. §11.2(iii) Numerically Satisfactory Solutions

§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) =ie12πiν𝐇ν(iz)=(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 zν1𝐇ν(z) and zν1𝐋ν(z) are entire functions of z and ν.

11.2.3 𝐇0(z) =2π(zz31232+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 d2wdz2+1zdwdz+(1ν2z2)w=(12z)ν1πΓ(ν+12).

Particular solutions:

11.2.8 w=𝐇ν(z),𝐊ν(z).

Modified Struve’s Equation

11.2.9 d2wdz2+1zdwdz(1+ν2z2)w=(12z)ν1πΓ(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.