11 Struve and Related Functions Struve and Modified Struve Functions 11.1 Special Notation 11.3 Graphics

§11.2 Definitions

Keywords:: Struve functions and modified Struve functions
Permalink:: http://dlmf.nist.gov/11.2

§11.2(i) Power-Series Expansions
§11.2(ii) Differential Equations
§11.2(iii) Numerically Satisfactory Solutions

§11.2(i) Power-Series Expansions

Notes:: See Watson (1944, pp. 328–329) and Olver (1997b, pp. 274–276). The notation $\mathop{\mathbf{M}_{{\nu}}\/}\nolimits\!\left(z\right)$ is new and this function has been introduced to play a similar role to $\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)$ that $\mathop{\mathbf{K}_{{\nu}}\/}\nolimits\!\left(z\right)$ does to $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ .
Keywords:: Struve functions and modified Struve functions, power series, principal values
Referenced by:: §10.22(i), §10.43(i), §11.13(iv)
Permalink:: http://dlmf.nist.gov/11.2.i

11.2.1 $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)=(\tfrac{1}{2}z)^{{\nu+1% }}\sum_{{n=0}}^{\infty}\frac{(-1)^{{n}}(\tfrac{1}{2}z)^{{2n}}}{\mathop{\Gamma% \/}\nolimits\!\left(n+\tfrac{3}{2}\right)\mathop{\Gamma\/}\nolimits\!\left(n+% \nu+\tfrac{3}{2}\right)},$

Defines:: $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function
Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : complex variable, $\nu$ : real or complex order and : integer order
A&S Ref:: 12.1.3
Referenced by:: §11.10(vi), §11.13(ii), §11.2(i), §11.4(iii), §11.6(ii), §11.6(ii)
Permalink:: http://dlmf.nist.gov/11.2.E1
Encodings:: TeX, pMML, png

11.2.2 $\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)=-ie^{{-\frac{1}{2}\pi i% \nu}}\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(iz\right)=(\tfrac{1}{2}z)^{% {\nu+1}}\sum_{{n=0}}^{\infty}\frac{(\tfrac{1}{2}z)^{{2n}}}{\mathop{\Gamma\/}% \nolimits\!\left(n+\tfrac{3}{2}\right)\mathop{\Gamma\/}\nolimits\!\left(n+\nu+% \tfrac{3}{2}\right)}.$

Defines:: $\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Struve function
Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, : base of exponential function, : complex variable, $\nu$ : real or complex order and : integer order
A&S Ref:: 12.2.1
Referenced by:: §11.13(ii), §11.2(i), §11.4(i), §11.4(iii), ¶ ‣ §11.5(ii), §11.5(i), §11.6(ii), §11.6(ii)
Permalink:: http://dlmf.nist.gov/11.2.E2
Encodings:: TeX, pMML, png

Principal values correspond to principal values of $(\tfrac{1}{2}z)^{{\nu+1}}$ ; compare §4.2(i).

The expansions (11.2.1) and (11.2.2) are absolutely convergent for all finite values of . The functions $z^{{-\nu-1}}\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ and $z^{{-\nu-1}}\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)$ are entire functions of and $\nu$ .

11.2.3 $\mathop{\mathbf{H}_{{0}}\/}\nolimits\!\left(z\right)=\frac{2}{\pi}\left(z-% \frac{z^{3}}{1^{2}\cdot 3^{2}}+\frac{z^{5}}{1^{2}\cdot 3^{2}\cdot 5^{2}}-% \cdots\right),$

Symbols:: $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function and : complex variable
A&S Ref:: 12.1.4
Permalink:: http://dlmf.nist.gov/11.2.E3
Encodings:: TeX, pMML, png

11.2.4 $\mathop{\mathbf{L}_{{0}}\/}\nolimits\!\left(z\right)=\frac{2}{\pi}\left(z+% \frac{z^{3}}{1^{2}\cdot 3^{2}}+\frac{z^{5}}{1^{2}\cdot 3^{2}\cdot 5^{2}}+% \cdots\right).$

Symbols:: $\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Struve function and : complex variable
Permalink:: http://dlmf.nist.gov/11.2.E4
Encodings:: TeX, pMML, png

11.2.5 $\mathop{\mathbf{K}_{{\nu}}\/}\nolimits\!\left(z\right)=\mathop{\mathbf{H}_{{% \nu}}\/}\nolimits\!\left(z\right)-\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right),$

Defines:: $\mathop{\mathbf{K}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function
Symbols:: $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, : complex variable and $\nu$ : real or complex order
Referenced by:: §11.13(iii), §11.13(iv), §11.2(i), §11.4(i), §11.5(i), §11.6(i), §11.6(i), §11.6(iii)
Permalink:: http://dlmf.nist.gov/11.2.E5
Encodings:: TeX, pMML, png

11.2.6 $\mathop{\mathbf{M}_{{\nu}}\/}\nolimits\!\left(z\right)=\mathop{\mathbf{L}_{{% \nu}}\/}\nolimits\!\left(z\right)-\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right).$

Defines:: $\mathop{\mathbf{M}_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Struve function
Symbols:: $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Struve function, : complex variable and $\nu$ : real or complex order
Referenced by:: §11.13(iii), §11.13(iv), §11.2(i), §11.6(i), §11.6(i)
Permalink:: http://dlmf.nist.gov/11.2.E6
Encodings:: TeX, pMML, png

Principal values of $\mathop{\mathbf{K}_{{\nu}}\/}\nolimits\!\left(z\right)$ and $\mathop{\mathbf{M}_{{\nu}}\/}\nolimits\!\left(z\right)$ correspond to principal values of the functions on the right-hand sides of (11.2.5) and (11.2.6).

Unless indicated otherwise, $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ , $\mathop{\mathbf{K}_{{\nu}}\/}\nolimits\!\left(z\right)$ , $\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)$ , and $\mathop{\mathbf{M}_{{\nu}}\/}\nolimits\!\left(z\right)$ assume their principal values throughout the DLMF.

§11.2(ii) Differential Equations

Notes:: See Watson (1944, pp. 328–329) and Olver (1997b, pp. 274–276).
Keywords:: Bessel’s equation, Struve functions and modified Struve functions, differential equations
Permalink:: http://dlmf.nist.gov/11.2.ii

¶ Struve’s Equation

Keywords:: Struve functions and modified Struve functions, differential equations

11.2.7 $\frac{{d}^{2}w}{{dz}^{2}}+\frac{1}{z}\frac{dw}{dz}+\left(1-\frac{\nu^{2}}{z^{2% }}\right)w=\frac{(\tfrac{1}{2}z)^{{\nu-1}}}{\sqrt{\pi}\mathop{\Gamma\/}% \nolimits\!\left(\nu+\tfrac{1}{2}\right)}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\frac{df}{dx}$ : derivative of with respect to , : complex variable and $\nu$ : real or complex order
Referenced by:: §11.13(iv), §11.2(iii), §11.2(iii), §11.9(i)
Permalink:: http://dlmf.nist.gov/11.2.E7
Encodings:: TeX, pMML, png

Particular solutions:

11.2.8 $w=\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right),\mathop{\mathbf{K}_{{% \nu}}\/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, $\mathop{\mathbf{K}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, : complex variable and $\nu$ : real or complex order
Permalink:: http://dlmf.nist.gov/11.2.E8
Encodings:: TeX, pMML, png

¶ Modified Struve’s Equation

Keywords:: Struve functions and modified Struve functions, differential equations, modified Bessel’s equation

11.2.9 $\frac{{d}^{2}w}{{dz}^{2}}+\frac{1}{z}\frac{dw}{dz}-\left(1+\frac{\nu^{2}}{z^{2% }}\right)w=\frac{(\tfrac{1}{2}z)^{{\nu-1}}}{\sqrt{\pi}\mathop{\Gamma\/}% \nolimits\!\left(\nu+\tfrac{1}{2}\right)}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\frac{df}{dx}$ : derivative of with respect to , : complex variable and $\nu$ : real or complex order
Referenced by:: §11.13(iv), §11.2(iii)
Permalink:: http://dlmf.nist.gov/11.2.E9
Encodings:: TeX, pMML, png

Particular solutions:

11.2.10 $w=\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right),\mathop{\mathbf{M}_{{% \nu}}\/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{\mathbf{M}_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Struve function, $\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Struve function, : complex variable and $\nu$ : real or complex order
Permalink:: http://dlmf.nist.gov/11.2.E10
Encodings:: TeX, pMML, png

§11.2(iii) Numerically Satisfactory Solutions

Notes:: See §2.7(iv) and Olver (1997b, pp. 274–277). The last reference restricts (11.2.15) to the sector $\left|\mathop{\mathrm{ph}\/}\nolimits z\right|\leq\tfrac{1}{2}\pi$ , and instead covers the sector $\tfrac{1}{2}\pi\leq\mathop{\mathrm{ph}\/}\nolimits z\leq\tfrac{3}{2}\pi$ with another set of solutions. (Similarly for the conjugate sector $-\tfrac{3}{2}\pi\leq\mathop{\mathrm{ph}\/}\nolimits z\leq-\tfrac{1}{2}\pi$ .)
Keywords:: Struve functions and modified Struve functions, differential equations
Permalink:: http://dlmf.nist.gov/11.2.iii

In this subsection and are arbitrary constants.

When , $0<x<\infty$ , and $\realpart{\nu}\geq 0$ , numerically satisfactory general solutions of (11.2.7) are given by

11.2.11 $w=\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(x\right)+A\mathop{J_{{\nu}}\/}% \nolimits\!\left(x\right)+B\mathop{Y_{{\nu}}\/}\nolimits\!\left(x\right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, : real variable, $\nu$ : real or complex order, : arbitrary constant and : arbitrary constant
Referenced by:: §11.2(iii)
Permalink:: http://dlmf.nist.gov/11.2.E11
Encodings:: TeX, pMML, png

11.2.12 $w=\mathop{\mathbf{K}_{{\nu}}\/}\nolimits\!\left(x\right)+A\mathop{J_{{\nu}}\/}% \nolimits\!\left(x\right)+B\mathop{Y_{{\nu}}\/}\nolimits\!\left(x\right).$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{\mathbf{K}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, : real variable, $\nu$ : real or complex order, : arbitrary constant and : arbitrary constant
Referenced by:: §11.2(iii)
Permalink:: http://dlmf.nist.gov/11.2.E12
Encodings:: TeX, pMML, png

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

When $z\in\Complex$ and $\realpart{\nu}\geq 0$ , numerically satisfactory general solutions of (11.2.7) are given by

11.2.13 $w=\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)+A\mathop{J_{{\nu}}\/}% \nolimits\!\left(z\right)+B\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z% \right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, : complex variable, $\nu$ : real or complex order, : arbitrary constant and : arbitrary constant
Referenced by:: §11.2(iii)
Permalink:: http://dlmf.nist.gov/11.2.E13
Encodings:: TeX, pMML, png

11.2.14 $w=\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)+A\mathop{J_{{\nu}}\/}% \nolimits\!\left(z\right)+B\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z% \right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, : complex variable, $\nu$ : real or complex order, : arbitrary constant and : arbitrary constant
Referenced by:: §11.2(iii)
Permalink:: http://dlmf.nist.gov/11.2.E14
Encodings:: TeX, pMML, png

11.2.15 $w=\mathop{\mathbf{K}_{{\nu}}\/}\nolimits\!\left(z\right)+A\mathop{{H^{{(1)}}_{% {\nu}}}\/}\nolimits\!\left(z\right)+B\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!% \left(z\right).$

Symbols:: $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $\mathop{\mathbf{K}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, : complex variable, $\nu$ : real or complex order, : arbitrary constant and : arbitrary constant
Referenced by:: §11.2(iii), §11.2(iii)
Permalink:: http://dlmf.nist.gov/11.2.E15
Encodings:: TeX, pMML, png

(11.2.13) applies when $0\leq\mathop{\mathrm{ph}\/}\nolimits z\leq\pi$ and is bounded. (11.2.14) applies when $-\pi\leq\mathop{\mathrm{ph}\/}\nolimits z\leq 0$ and is bounded. (11.2.15) applies when $\left|\mathop{\mathrm{ph}\/}\nolimits z\right|\leq\pi$ and is bounded away from the origin.

When $\realpart{\nu}\geq 0$ , numerically satisfactory general solutions of (11.2.9) are given by

11.2.16 $w=\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)+A\mathop{K_{{\nu}}\/}% \nolimits\!\left(z\right)+B\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right),$

Symbols:: $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Struve function, : complex variable, $\nu$ : real or complex order, : arbitrary constant and : arbitrary constant
Referenced by:: §11.2(iii)
Permalink:: http://dlmf.nist.gov/11.2.E16
Encodings:: TeX, pMML, png

11.2.17 $w=\mathop{\mathbf{M}_{{\nu}}\/}\nolimits\!\left(z\right)+A\mathop{K_{{\nu}}\/}% \nolimits\!\left(z\right)+B\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{\mathbf{M}_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Struve function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, : complex variable, $\nu$ : real or complex order, : arbitrary constant and : arbitrary constant
Referenced by:: §11.2(iii)
Permalink:: http://dlmf.nist.gov/11.2.E17
Encodings:: TeX, pMML, png

(11.2.16) applies when $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{1}{2}\pi$ with bounded. (11.2.17) applies when $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{1}{2}\pi$ with bounded away from the origin.