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The function
 
The function
 +
\[
 +
{\bf H}_\nu (x) = \frac{2\left(\textstyle{\frac{x}{2}}\right)^\nu}{\Gamma \left(\nu +
 +
\textstyle{\frac{1}{2}}\right) \Gamma \left(\textstyle{\frac{1}{2}}\right)} \int_0^{\frac{\pi}{2}}
 +
\, \sin (x\, \cos \theta)\, \sin^{2\nu} \theta\, d\theta\, ,
 +
\]
 +
where $\nu$ is a complex parameter with ${\rm Re}\, \nu > \frac{1}{2}$ and $x$ a complex variable.
 +
It was introduced by H. Struve in {{Cite|S}} and it is therefore sometimes denoted by $S_\nu$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090700/s0907001.png" /></td> </tr></table>
+
The Struve function satisfies the inhomogeneous [[Bessel equation|Bessel equation]]:
 
+
\[
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090700/s0907002.png" /></td> </tr></table>
+
x^2 y'' + x y' + (x^2 - \nu^2) y = \frac{4 \left(\textstyle{\frac{x}{2}}\right)^{\nu+1}}{\Gamma \left(\nu + \textstyle{\frac{1}{2}}\right) \Gamma \left(\textstyle{\frac{1}{2}}\right)}
 
+
\]
that satisfies the inhomogeneous [[Bessel equation|Bessel equation]]:
+
(see 10.4 in {{Cite|Wa}}).
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090700/s0907003.png" /></td> </tr></table>
 
 
 
The power series expansion is:
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090700/s0907004.png" /></td> </tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090700/s0907005.png" /></td> </tr></table>
 
 
 
A Struve function of integral order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090700/s0907006.png" /> is related to a [[Weber function|Weber function]] by the following relations:
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090700/s0907007.png" /></td> </tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090700/s0907008.png" /></td> </tr></table>
 
 
 
A Struve function of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090700/s0907009.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090700/s09070010.png" /> an integer) is an elementary function, for example
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090700/s09070011.png" /></td> </tr></table>
 
 
 
When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090700/s09070012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090700/s09070013.png" /> the asymptotic expansion
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090700/s09070014.png" /></td> </tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090700/s09070015.png" /></td> </tr></table>
 
  
holds, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090700/s09070016.png" /> is a [[Neumann function|Neumann function]].
+
The Struve function has the expansion
 +
\begin{equation}\label{e:expansion}
 +
{\bf H}_\nu (x) = \left(\frac{x}{2}\right)^{\nu +1} \sum_{k=0}^\infty (-1)^k \frac{\left(\textstyle{\frac{x}{2}}\right)^{2k}}{\Gamma \left(k+ \textstyle{\frac{3}{2}}\right)
 +
\Gamma \left(\nu + k + \textstyle{\frac{3}{2}}\right)}\, .
 +
\end{equation}
 +
The Struve functions of integral order $n$ is related to the [[Weber function|Weber function]] ${\bf E}_n$ by the following relation:
 +
\[
 +
{\bf E}_n (x) = \sum_{m=1}^n \frac{e^{\frac{1}{2} (m-1) \pi i} \left(\textstyle{\frac{x}{2}}\right)^{n-k}}{\Gamma \left(1-\textstyle{\frac{m}{2}}\right) \Gamma \left(n+1 - \textstyle{\frac{m}{2}}\right)} - {\bf H}_n (x) \quad \mbox{for }\; n\geq 0
 +
\]
 +
\[
 +
{\bf E}_{-n} (x) = \frac{(-1)^{n+1}}{\pi} \sum_{0 \leq m < \frac{n}{2}} \frac{\Gamma \left( n - m -
 +
\textstyle{\frac{1}{2}}\right) \left(\frac{x}{2}\right)^{-n+2m+1}}{\Gamma \left(m + \textstyle{\frac{3}{2}}\right)} - {\bf H}_{-n} (x)\, \quad \mbox{for }\; n > 0\, ,
 +
\]
 +
(see 10.44 of {{Cite|Wa}}).  
  
A modified Struve function is the function
+
The Struve function of order $n + \frac{1}{2}$ with integer $n$ can be expressed in terms of elementary functions. For instance
 +
\begin{align*}
 +
{\bf H}_{1/2} (x) &= \left(\frac{2}{\pi x}\right)^{\frac{1}{2}} (1-\cos x)\\
 +
{\bf H}_{3/2} (x) &= \left(\frac{x}{2\pi}\right)^{\frac{1}{2}} \left(1+\frac{2}{x^2}\right)
 +
- \left(\frac{2}{\pi x}\right)^{\frac{1}{2}} \left(\sin x + \frac{\cos x}{x}\right)
 +
\end{align*}
 +
(cf. 10.42 of {{Cite|Wa}}).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090700/s09070017.png" /></td> </tr></table>
+
For $|{\rm arg}\, x| < \pi$ and $|x|$ large we have the asymptotic expansion
 +
\[
 +
{\bf H}_\nu (x) = Y_\nu (x) + \frac{1}{\pi} \sum_{m=0}^{k-1} \frac{\Gamma \left(m +  \textstyle{\frac{1}{2}}\right)}{\Gamma \left(\nu +  \textstyle{\frac{1}{2}} - m\right)  \left(\textstyle{\frac{x}{2}}\right)^{2m - \nu +1}} + O \left(|x|^{\nu - 2k -1}\right)\, ,
 +
\]
 +
where $Y_\nu$ is the [[Neumann function|Neumann function]].
  
Its series expansion is:
+
The modified Struve function is given by
 +
\[
 +
{\bf L}_\nu (x) = \left\{\begin{array}{ll}
 +
&e^{-\frac{1}{2} \nu \pi i} {\bf H}_\nu ( ix) \quad &\mbox{if } -\pi < {\rm arg}\, z \leq \frac{\pi}{2}\\
 +
&e^{\frac{3}{2} \nu \pi i} {\bf H}_\nu ( -ix) \quad &\mbox{if } \frac{\pi}{2} < {\rm arg}\, z \leq \pi
 +
\end{array}\right.
 +
\]
 +
and thus bears the same relation to the Struwe function ${\bf H}_\nu (x)$ as the modified
 +
Bessel function $I_\nu$ bears to the Bessel function $J_\nu$ (see [[Cylinder functions]]).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090700/s09070018.png" /></td> </tr></table>
+
The expansion \eqref{e:expansion} translates into a corresponding expansion for the modified Struve function. We have moreover the interesting relation
 +
\[
 +
{\bf L}_\nu (x) = I_{-\nu} (x) - \frac{2 \left(\textstyle{\frac{x}{2}}\right)^\nu}{\Gamma \left( \nu + \textstyle{\frac{1}{2}}\right) \Gamma \left(\textstyle{\frac{1}{2}}\right)} \int_0^\infty \sin (xu)\, (1+ u^2)^{-\nu - \frac{1}{2}}\, du
 +
\]
 +
which leads to the asymptotic expansion
 +
\[
 +
{\bf L}_\nu (x) = I_{-\nu} (x) - \frac{\left(\textstyle{\frac{x}{2}}\right)^{\nu-1}}{\sqrt{\pi}\,\Gamma \left(\nu + \textstyle{\frac{1}{2}}\right)}
 +
\]
 +
for $|x|$ large.
  
For large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090700/s09070019.png" />, the asymptotic expansion
+
The  Struve function can be expressed in terms of a [[Hypergeometric  function|hypergeometric function]] of type $_1 F_2$, cf. {{Cite|AS}}, formula (7.5).
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090700/s09070020.png" /></td> </tr></table>
 
 
 
holds, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090700/s09070021.png" /> is a modified Bessel function (cf. [[Bessel functions|Bessel functions]]).
 
 
 
A Struve function is sometimes denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090700/s09070022.png" />. Introduced by H. Struve [[#References|[1]]].
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Struve,  ''Ann. Physik Chemie'' , '''17'''  (1882)  pp. 1008–1016</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Jahnke,  F. Emde,  F. Lösch,  "Tafeln höheren Funktionen" , Teubner  (1966)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Abramowitz,  I.A. Stegun,  "Handbook of mathematical functions" , Dover, reprint  (1970)</TD></TR></table>
+
{|
 +
|-
 +
|valign="top"|{{Ref|AS}}||valign="top"| M. Abramowitz,  I.A. Stegun,  "Handbook of mathematical functions" , Dover, reprint  (1970)
  
 
+
|-
 
+
|valign="top"|{{Ref|BE}}||valign="top"| H. Bateman (ed.)  A. Erdélyi (ed.)  et al. (ed.) , ''Higher transcendental functions'' , '''2. Bessel functions, parabolic cylinder functions, orthogonal polynomials''' , McGraw-Hill  (1953)
====Comments====
+
|-
The Struve function can be expressed in terms of a [[Hypergeometric function|hypergeometric function]] of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090700/s09070023.png" />, cf. [[#References|[a1]]], formula (7.5).
+
|valign="top"|{{Ref|JES}}||valign="top"| E. Jahnke,  F. Emde,  F. Lösch,  "Tafeln höheren Funktionen" , Teubner  (1966)
 
+
|-
====References====
+
|valign="top"|{{Ref|S}}||valign="top"| H. Struve,  ''Ann. Physik Chemie'' , '''17'''  (1882)  pp. 1008–1016
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"H. Bateman (ed.)  A. Erdélyi (ed.)  et al. (ed.) , ''Higher transcendental functions'' , '''2. Bessel functions, parabolic cylinder functions, orthogonal polynomials''' , McGraw-Hill  (1953)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> G.N. Watson,  "A treatise on the theory of Bessel functions" , '''1''' , Cambridge Univ. Press  (1952)</TD></TR></table>
+
|-
 +
|valign="top"|{{Ref|Wa}}||valign="top"| G.N. Watson,  "A treatise on the theory of Bessel functions", '''1–2''', Cambridge Univ. Press  (1952) {{MR|1349110}} {{MR|1570252}} {{MR|0010746}}  {{MR|1520278}}    {{ZBL|0849.33001}} {{ZBL|0174.36202}}  {{ZBL|0063.08184}}
 +
|-
 +
|}

Revision as of 08:16, 22 February 2014

2020 Mathematics Subject Classification: Primary: 34-XX [MSN][ZBL]

The function \[ {\bf H}_\nu (x) = \frac{2\left(\textstyle{\frac{x}{2}}\right)^\nu}{\Gamma \left(\nu + \textstyle{\frac{1}{2}}\right) \Gamma \left(\textstyle{\frac{1}{2}}\right)} \int_0^{\frac{\pi}{2}} \, \sin (x\, \cos \theta)\, \sin^{2\nu} \theta\, d\theta\, , \] where $\nu$ is a complex parameter with ${\rm Re}\, \nu > \frac{1}{2}$ and $x$ a complex variable. It was introduced by H. Struve in [S] and it is therefore sometimes denoted by $S_\nu$.

The Struve function satisfies the inhomogeneous Bessel equation: \[ x^2 y'' + x y' + (x^2 - \nu^2) y = \frac{4 \left(\textstyle{\frac{x}{2}}\right)^{\nu+1}}{\Gamma \left(\nu + \textstyle{\frac{1}{2}}\right) \Gamma \left(\textstyle{\frac{1}{2}}\right)} \] (see 10.4 in [Wa]).

The Struve function has the expansion \begin{equation}\label{e:expansion} {\bf H}_\nu (x) = \left(\frac{x}{2}\right)^{\nu +1} \sum_{k=0}^\infty (-1)^k \frac{\left(\textstyle{\frac{x}{2}}\right)^{2k}}{\Gamma \left(k+ \textstyle{\frac{3}{2}}\right) \Gamma \left(\nu + k + \textstyle{\frac{3}{2}}\right)}\, . \end{equation} The Struve functions of integral order $n$ is related to the Weber function ${\bf E}_n$ by the following relation: \[ {\bf E}_n (x) = \sum_{m=1}^n \frac{e^{\frac{1}{2} (m-1) \pi i} \left(\textstyle{\frac{x}{2}}\right)^{n-k}}{\Gamma \left(1-\textstyle{\frac{m}{2}}\right) \Gamma \left(n+1 - \textstyle{\frac{m}{2}}\right)} - {\bf H}_n (x) \quad \mbox{for }\; n\geq 0 \] \[ {\bf E}_{-n} (x) = \frac{(-1)^{n+1}}{\pi} \sum_{0 \leq m < \frac{n}{2}} \frac{\Gamma \left( n - m - \textstyle{\frac{1}{2}}\right) \left(\frac{x}{2}\right)^{-n+2m+1}}{\Gamma \left(m + \textstyle{\frac{3}{2}}\right)} - {\bf H}_{-n} (x)\, \quad \mbox{for }\; n > 0\, , \] (see 10.44 of [Wa]).

The Struve function of order $n + \frac{1}{2}$ with integer $n$ can be expressed in terms of elementary functions. For instance \begin{align*} {\bf H}_{1/2} (x) &= \left(\frac{2}{\pi x}\right)^{\frac{1}{2}} (1-\cos x)\\ {\bf H}_{3/2} (x) &= \left(\frac{x}{2\pi}\right)^{\frac{1}{2}} \left(1+\frac{2}{x^2}\right) - \left(\frac{2}{\pi x}\right)^{\frac{1}{2}} \left(\sin x + \frac{\cos x}{x}\right) \end{align*} (cf. 10.42 of [Wa]).

For $|{\rm arg}\, x| < \pi$ and $|x|$ large we have the asymptotic expansion \[ {\bf H}_\nu (x) = Y_\nu (x) + \frac{1}{\pi} \sum_{m=0}^{k-1} \frac{\Gamma \left(m + \textstyle{\frac{1}{2}}\right)}{\Gamma \left(\nu + \textstyle{\frac{1}{2}} - m\right) \left(\textstyle{\frac{x}{2}}\right)^{2m - \nu +1}} + O \left(|x|^{\nu - 2k -1}\right)\, , \] where $Y_\nu$ is the Neumann function.

The modified Struve function is given by \[ {\bf L}_\nu (x) = \left\{\begin{array}{ll} &e^{-\frac{1}{2} \nu \pi i} {\bf H}_\nu ( ix) \quad &\mbox{if } -\pi < {\rm arg}\, z \leq \frac{\pi}{2}\\ &e^{\frac{3}{2} \nu \pi i} {\bf H}_\nu ( -ix) \quad &\mbox{if } \frac{\pi}{2} < {\rm arg}\, z \leq \pi \end{array}\right. \] and thus bears the same relation to the Struwe function ${\bf H}_\nu (x)$ as the modified Bessel function $I_\nu$ bears to the Bessel function $J_\nu$ (see Cylinder functions).

The expansion \eqref{e:expansion} translates into a corresponding expansion for the modified Struve function. We have moreover the interesting relation \[ {\bf L}_\nu (x) = I_{-\nu} (x) - \frac{2 \left(\textstyle{\frac{x}{2}}\right)^\nu}{\Gamma \left( \nu + \textstyle{\frac{1}{2}}\right) \Gamma \left(\textstyle{\frac{1}{2}}\right)} \int_0^\infty \sin (xu)\, (1+ u^2)^{-\nu - \frac{1}{2}}\, du \] which leads to the asymptotic expansion \[ {\bf L}_\nu (x) = I_{-\nu} (x) - \frac{\left(\textstyle{\frac{x}{2}}\right)^{\nu-1}}{\sqrt{\pi}\,\Gamma \left(\nu + \textstyle{\frac{1}{2}}\right)} \] for $|x|$ large.

The Struve function can be expressed in terms of a hypergeometric function of type $_1 F_2$, cf. [AS], formula (7.5).

References

[AS] M. Abramowitz, I.A. Stegun, "Handbook of mathematical functions" , Dover, reprint (1970)
[BE] H. Bateman (ed.) A. Erdélyi (ed.) et al. (ed.) , Higher transcendental functions , 2. Bessel functions, parabolic cylinder functions, orthogonal polynomials , McGraw-Hill (1953)
[JES] E. Jahnke, F. Emde, F. Lösch, "Tafeln höheren Funktionen" , Teubner (1966)
[S] H. Struve, Ann. Physik Chemie , 17 (1882) pp. 1008–1016
[Wa] G.N. Watson, "A treatise on the theory of Bessel functions", 1–2, Cambridge Univ. Press (1952) MR1349110 MR1570252 MR0010746 MR1520278 Zbl 0849.33001 Zbl 0174.36202 Zbl 0063.08184
How to Cite This Entry:
Struve function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Struve_function&oldid=31327
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article