StruveH - Maple Help

StruveH, StruveL

Struve functions

 Calling Sequence StruveH(v, x) StruveL(v, x)

Parameters

 v - algebraic expression (order or index) x - algebraic expression (argument)

Description

 • The Struve function StruveH(v, x) solves the inhomogeneous Bessel equation

${x}^{2}\mathrm{y\text{'}\text{'}}+x\mathrm{y\text{'}}+\left(-{v}^{2}+{x}^{2}\right)y=\frac{4{\left(\frac{x}{2}\right)}^{v+1}}{\sqrt{\mathrm{\pi }}\mathrm{\Gamma }\left(v+\frac{1}{2}\right)}$

 • The modified Struve function StruveL(v, x) solves the inhomogeneous Bessel equation

${x}^{2}\mathrm{y\text{'}\text{'}}+x\mathrm{y\text{'}}-\left({v}^{2}+{x}^{2}\right)y=\frac{4{\left(\frac{x}{2}\right)}^{v+1}}{\sqrt{\mathrm{\pi }}\mathrm{\Gamma }\left(v+\frac{1}{2}\right)}$

Examples

 > $\mathrm{StruveH}\left(0,0\right)$
 ${0}$ (1)
 > $\mathrm{StruveL}\left(1.5-I,2.6+3I\right)$
 ${-6.136109206}{-}{0.3149238903}{}{I}$ (2)
 > $\mathrm{series}\left(\mathrm{StruveH}\left(\frac{1}{3},x\right),x,4\right)$
 $\frac{{3}{}{{2}}^{{2}}{{3}}}{}{{x}}^{{4}}{{3}}}}{{5}{}\sqrt{{\mathrm{\pi }}}{}{\mathrm{\Gamma }}{}\left(\frac{{5}}{{6}}\right)}{-}\frac{{3}{}{{2}}^{{2}}{{3}}}{}{{x}}^{{10}}{{3}}}}{{55}{}\sqrt{{\mathrm{\pi }}}{}{\mathrm{\Gamma }}{}\left(\frac{{5}}{{6}}\right)}{+}{\mathrm{O}}{}\left({{x}}^{{4}}\right)$ (3)
 > $\mathrm{StruveH}\left(\frac{3}{2},x\right)$
 ${\mathrm{StruveH}}{}\left(\frac{{3}}{{2}}{,}{x}\right)$ (4)
 > $\mathrm{diff}\left(\mathrm{StruveL}\left(v,x\right),x\right)$
 ${\mathrm{StruveL}}{}\left({v}{+}{1}{,}{x}\right){+}\frac{{v}{}{\mathrm{StruveL}}{}\left({v}{,}{x}\right)}{{x}}{+}\frac{{\left(\frac{{x}}{{2}}\right)}^{{v}}}{\sqrt{{\mathrm{\pi }}}{}{\mathrm{\Gamma }}{}\left({v}{+}\frac{{3}}{{2}}\right)}$ (5)

References

 Abramowitz, M., and Stegun, I., eds. Handbook of Mathematical Functions. New York: Dover, 1972.