LommelS1 - Maple Help

LommelS1

the Lommel function s

LommelS2

the Lommel function S

 Calling Sequence LommelS1(mu, nu, z) LommelS2(mu, nu, z)

Parameters

 mu - algebraic expression nu - algebraic expression z - algebraic expression

Description

 • The LommelS1(mu, nu, z) function is defined in terms of the hypergeometric function
 $\left[{\mathrm{LommelS1}}{}\left({a}{,}{b}{,}{z}\right){=}\frac{{{z}}^{{a}{+}{1}}{}{\mathrm{hypergeom}}{}\left(\left[{1}\right]{,}\left[\frac{{3}}{{2}}{-}\frac{{b}}{{2}}{+}\frac{{a}}{{2}}{,}\frac{{3}}{{2}}{+}\frac{{b}}{{2}}{+}\frac{{a}}{{2}}\right]{,}{-}\frac{{{z}}^{{2}}}{{4}}\right)}{\left({a}{-}{b}{+}{1}\right){}\left({a}{+}{b}{+}{1}\right)}{,}{-}{a}{+}{b}{-}{1}{\ne }{0}{\wedge }{a}{+}{b}{+}{1}{\ne }{0}{\wedge }\left(\frac{{3}}{{2}}{-}\frac{{b}}{{2}}{+}\frac{{a}}{{2}}\right){::}\left({¬}{\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }\left(\frac{{3}}{{2}}{+}\frac{{b}}{{2}}{+}\frac{{a}}{{2}}\right){::}\left({¬}{\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right)\right]$ (1)
 and LommelS2(mu, nu, z) is defined in terms of LommelS1(mu, nu, z) and Bessel functions.
 > LommelS2(mu,nu,z) = convert(LommelS2(mu,nu,z), LommelS1);
 ${\mathrm{LommelS2}}{}\left({\mathrm{\mu }}{,}{\mathrm{\nu }}{,}{z}\right){=}{\mathrm{LommelS1}}{}\left({\mathrm{\mu }}{,}{\mathrm{\nu }}{,}{z}\right){+}{{2}}^{{\mathrm{\mu }}{-}{1}}{}{\mathrm{\Gamma }}{}\left(\frac{{\mathrm{\mu }}}{{2}}{-}\frac{{\mathrm{\nu }}}{{2}}{+}\frac{{1}}{{2}}\right){}{\mathrm{\Gamma }}{}\left(\frac{{\mathrm{\mu }}}{{2}}{+}\frac{{\mathrm{\nu }}}{{2}}{+}\frac{{1}}{{2}}\right){}\left({\mathrm{sin}}{}\left(\frac{\left({\mathrm{\mu }}{-}{\mathrm{\nu }}\right){}{\mathrm{\pi }}}{{2}}\right){}{\mathrm{BesselJ}}{}\left({\mathrm{\nu }}{,}{z}\right){-}{\mathrm{cos}}{}\left(\frac{\left({\mathrm{\mu }}{-}{\mathrm{\nu }}\right){}{\mathrm{\pi }}}{{2}}\right){}{\mathrm{BesselY}}{}\left({\mathrm{\nu }}{,}{z}\right)\right)$ (2)
 • These functions solve the non-homogeneous linear differential equation of second order.
 > z^2*diff(f(z),\$(z,2))+z*diff(f(z),z)+(z^2-nu^2)*f(z) = z^(mu+1);
 ${{z}}^{{2}}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{z}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({z}\right)\right){+}{z}{}\left(\frac{{ⅆ}}{{ⅆ}{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({z}\right)\right){+}\left({-}{{\mathrm{\nu }}}^{{2}}{+}{{z}}^{{2}}\right){}{f}{}\left({z}\right){=}{{z}}^{{\mathrm{\mu }}{+}{1}}$ (3)
 The Lommel functions also solve the following third order linear homogeneous differential equation with polynomial coefficients.
 $\left[{f}{}\left({z}\right){=}{\mathrm{LommelS1}}{}\left({\mathrm{\mu }}{,}{\mathrm{\nu }}{,}{z}\right){,}\left[\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{z}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({z}\right){=}\frac{\left({\mathrm{\mu }}{-}{2}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{z}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({z}\right)\right)}{{z}}{+}\frac{\left({{\mathrm{\nu }}}^{{2}}{-}{{z}}^{{2}}{+}{\mathrm{\mu }}\right){}\left(\frac{{ⅆ}}{{ⅆ}{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({z}\right)\right)}{{{z}}^{{2}}}{+}\frac{\left(\left({\mathrm{\mu }}{-}{1}\right){}{{z}}^{{2}}{-}{{\mathrm{\nu }}}^{{2}}{}\left({\mathrm{\mu }}{+}{1}\right)\right){}{f}{}\left({z}\right)}{{{z}}^{{3}}}\right]\right]$ (4)

Examples

The AngerJ and WeberE, StruveH and StruveL functions can be viewed as particular cases of LommelS1.

 > $\mathrm{FunctionAdvisor}\left(\mathrm{relate},\mathrm{AngerJ},\mathrm{LommelS1}\right)$
 ${\mathrm{AngerJ}}{}\left({a}{,}{z}\right){=}\frac{{\mathrm{sin}}{}\left({\mathrm{\pi }}{}{a}\right){}\left({\mathrm{LommelS1}}{}\left({0}{,}{a}{,}{z}\right){-}{a}{}{\mathrm{LommelS1}}{}\left({-1}{,}{a}{,}{z}\right)\right)}{{\mathrm{\pi }}}$ (5)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{relate},\mathrm{WeberE},\mathrm{LommelS1}\right)$
 ${\mathrm{WeberE}}{}\left({a}{,}{z}\right){=}\frac{{-}{a}{}\left({1}{-}{\mathrm{cos}}{}\left({\mathrm{\pi }}{}{a}\right)\right){}{\mathrm{LommelS1}}{}\left({-1}{,}{a}{,}{z}\right){+}\left({-}{1}{-}{\mathrm{cos}}{}\left({\mathrm{\pi }}{}{a}\right)\right){}{\mathrm{LommelS1}}{}\left({0}{,}{a}{,}{z}\right)}{{\mathrm{\pi }}}$ (6)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{relate},\mathrm{StruveH},\mathrm{LommelS1}\right)$
 ${\mathrm{StruveH}}{}\left({a}{,}{z}\right){=}\frac{{2}{}{\mathrm{LommelS1}}{}\left({a}{,}{a}{,}{z}\right)}{{\mathrm{\Gamma }}{}\left({a}{+}\frac{{1}}{{2}}\right){}\sqrt{{\mathrm{\pi }}}{}{{2}}^{{a}}}$ (7)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{relate},\mathrm{StruveL},\mathrm{LommelS1}\right)$
 ${\mathrm{StruveL}}{}\left({a}{,}{z}\right){=}\frac{{-}{2}{}{I}{}{\mathrm{LommelS1}}{}\left({a}{,}{a}{,}{I}{}{z}\right){}{{z}}^{{a}}}{{\mathrm{\Gamma }}{}\left({a}{+}\frac{{1}}{{2}}\right){}\sqrt{{\mathrm{\pi }}}{}{\left({2}{}{I}{}{z}\right)}^{{a}}}$ (8)

A MeijerG representation for the Lommel functions.

 > $\mathrm{LommelS1}\left(\mathrm{\mu },\mathrm{\nu },z\right)=\mathrm{convert}\left(\mathrm{LommelS1}\left(\mathrm{\mu },\mathrm{\nu },z\right),\mathrm{MeijerG}\right)$
 ${\mathrm{LommelS1}}{}\left({\mathrm{\mu }}{,}{\mathrm{\nu }}{,}{z}\right){=}{{2}}^{{\mathrm{\mu }}{-}{1}}{}{\mathrm{\Gamma }}{}\left(\frac{{\mathrm{\mu }}}{{2}}{+}\frac{{\mathrm{\nu }}}{{2}}{+}\frac{{1}}{{2}}\right){}{\mathrm{\Gamma }}{}\left(\frac{{\mathrm{\mu }}}{{2}}{-}\frac{{\mathrm{\nu }}}{{2}}{+}\frac{{1}}{{2}}\right){}{\mathrm{MeijerG}}{}\left(\left[\left[\frac{{\mathrm{\mu }}}{{2}}{+}\frac{{1}}{{2}}\right]{,}\left[\right]\right]{,}\left[\left[\frac{{\mathrm{\mu }}}{{2}}{+}\frac{{1}}{{2}}\right]{,}\left[\frac{{\mathrm{\nu }}}{{2}}{,}{-}\frac{{\mathrm{\nu }}}{{2}}\right]\right]{,}\frac{{{z}}^{{2}}}{{4}}\right)$ (9)
 > $\mathrm{LommelS2}\left(\mathrm{\mu },\mathrm{\nu },z\right)=\mathrm{convert}\left(\mathrm{LommelS2}\left(\mathrm{\mu },\mathrm{\nu },z\right),\mathrm{MeijerG}\right)$
 ${\mathrm{LommelS2}}{}\left({\mathrm{\mu }}{,}{\mathrm{\nu }}{,}{z}\right){=}\frac{{\mathrm{MeijerG}}{}\left(\left[\left[\frac{{\mathrm{\mu }}}{{2}}{+}\frac{{1}}{{2}}\right]{,}\left[\right]\right]{,}\left[\left[\frac{{\mathrm{\mu }}}{{2}}{+}\frac{{1}}{{2}}{,}\frac{{\mathrm{\nu }}}{{2}}{,}{-}\frac{{\mathrm{\nu }}}{{2}}\right]{,}\left[\right]\right]{,}\frac{{{z}}^{{2}}}{{4}}\right){}{{2}}^{{\mathrm{\mu }}}}{{2}{}{\mathrm{\Gamma }}{}\left({-}\frac{{\mathrm{\mu }}}{{2}}{+}\frac{{\mathrm{\nu }}}{{2}}{+}\frac{{1}}{{2}}\right){}{\mathrm{\Gamma }}{}\left({-}\frac{{\mathrm{\mu }}}{{2}}{-}\frac{{\mathrm{\nu }}}{{2}}{+}\frac{{1}}{{2}}\right)}$ (10)

The series expansion of the Lommel functions is not computable using the series command because it would involve factoring out abstract powers, leading to a result of the form z^mu1*series_1 + z^mu2*series_2 + .... This type of extended series expansion, however, can be computed using the Series command of the MathematicalFunctions package.

 > $\mathrm{with}\left(\mathrm{MathematicalFunctions},\mathrm{Series}\right)$
 $\left[{\mathrm{Series}}\right]$ (11)
 > $\mathrm{Series}\left(\mathrm{LommelS1}\left(\mathrm{\mu },\mathrm{\nu },z\right),z,4\right)$
 ${{z}}^{{\mathrm{\mu }}}{}\left(\frac{{1}}{\left({\mathrm{\mu }}{-}{\mathrm{\nu }}{+}{1}\right){}\left({\mathrm{\mu }}{+}{\mathrm{\nu }}{+}{1}\right)}{}{z}{-}\frac{{1}}{\left({\mathrm{\mu }}{-}{\mathrm{\nu }}{+}{1}\right){}\left({\mathrm{\mu }}{-}{\mathrm{\nu }}{+}{3}\right){}\left({\mathrm{\mu }}{+}{\mathrm{\nu }}{+}{1}\right){}\left({\mathrm{\mu }}{+}{\mathrm{\nu }}{+}{3}\right)}{}{{z}}^{{3}}{+}{O}{}\left({{z}}^{{5}}\right)\right){,}\left({\mathrm{\mu }}{+}{\mathrm{\nu }}\right){::}\left({¬}\left({\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{'}{\mathrm{negint}}{'}\right]\right){\wedge }{\mathrm{odd}}\right)\right){\wedge }\left({\mathrm{\mu }}{-}{\mathrm{\nu }}\right){::}\left({¬}\left({\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{'}{\mathrm{negint}}{'}\right]\right){\wedge }{\mathrm{odd}}\right)\right)$ (12)
 > $\mathrm{Series}\left(\mathrm{LommelS2}\left(\mathrm{\mu },\mathrm{\nu },z\right),z,4\right)$
 ${{z}}^{{-}{\mathrm{\nu }}}{}\left(\frac{{{4}}^{{\mathrm{\nu }}}{}{\mathrm{csc}}{}\left({\mathrm{\pi }}{}{\mathrm{\nu }}\right){}{\mathrm{cos}}{}\left(\frac{\left({\mathrm{\mu }}{-}{\mathrm{\nu }}\right){}{\mathrm{\pi }}}{{2}}\right){}{\mathrm{\Gamma }}{}\left(\frac{{\mathrm{\mu }}}{{2}}{+}\frac{{\mathrm{\nu }}}{{2}}{+}\frac{{1}}{{2}}\right){}{\mathrm{\Gamma }}{}\left(\frac{{\mathrm{\mu }}}{{2}}{-}\frac{{\mathrm{\nu }}}{{2}}{+}\frac{{1}}{{2}}\right){}{{2}}^{{\mathrm{\mu }}{-}{\mathrm{\nu }}}}{{2}{}{\mathrm{\Gamma }}{}\left({-}{\mathrm{\nu }}{+}{1}\right)}{-}\frac{{1}}{{8}}{}\frac{{{4}}^{{\mathrm{\nu }}}{}{\mathrm{csc}}{}\left({\mathrm{\pi }}{}{\mathrm{\nu }}\right){}{\mathrm{cos}}{}\left(\frac{\left({\mathrm{\mu }}{-}{\mathrm{\nu }}\right){}{\mathrm{\pi }}}{{2}}\right){}{\mathrm{\Gamma }}{}\left(\frac{{\mathrm{\mu }}}{{2}}{+}\frac{{\mathrm{\nu }}}{{2}}{+}\frac{{1}}{{2}}\right){}{\mathrm{\Gamma }}{}\left(\frac{{\mathrm{\mu }}}{{2}}{-}\frac{{\mathrm{\nu }}}{{2}}{+}\frac{{1}}{{2}}\right){}{{2}}^{{\mathrm{\mu }}{-}{\mathrm{\nu }}}}{{\mathrm{\Gamma }}{}\left({-}{\mathrm{\nu }}{+}{2}\right)}{}{{z}}^{{2}}{+}{O}{}\left({{z}}^{{4}}\right)\right){+}{{z}}^{{\mathrm{\nu }}}{}\left({-}\frac{{\mathrm{csc}}{}\left({\mathrm{\pi }}{}{\mathrm{\nu }}\right){}{\mathrm{cos}}{}\left(\frac{{\mathrm{\pi }}{}\left({\mathrm{\mu }}{+}{\mathrm{\nu }}\right)}{{2}}\right){}{\mathrm{\Gamma }}{}\left(\frac{{\mathrm{\mu }}}{{2}}{+}\frac{{\mathrm{\nu }}}{{2}}{+}\frac{{1}}{{2}}\right){}{\mathrm{\Gamma }}{}\left(\frac{{\mathrm{\mu }}}{{2}}{-}\frac{{\mathrm{\nu }}}{{2}}{+}\frac{{1}}{{2}}\right){}{{2}}^{{\mathrm{\mu }}{-}{\mathrm{\nu }}}}{{2}{}{\mathrm{\Gamma }}{}\left({\mathrm{\nu }}{+}{1}\right)}{+}\frac{{1}}{{8}}{}\frac{{\mathrm{csc}}{}\left({\mathrm{\pi }}{}{\mathrm{\nu }}\right){}{\mathrm{cos}}{}\left(\frac{{\mathrm{\pi }}{}\left({\mathrm{\mu }}{+}{\mathrm{\nu }}\right)}{{2}}\right){}{\mathrm{\Gamma }}{}\left(\frac{{\mathrm{\mu }}}{{2}}{+}\frac{{\mathrm{\nu }}}{{2}}{+}\frac{{1}}{{2}}\right){}{\mathrm{\Gamma }}{}\left(\frac{{\mathrm{\mu }}}{{2}}{-}\frac{{\mathrm{\nu }}}{{2}}{+}\frac{{1}}{{2}}\right){}{{2}}^{{\mathrm{\mu }}{-}{\mathrm{\nu }}}}{{\mathrm{\Gamma }}{}\left({\mathrm{\nu }}{+}{2}\right)}{}{{z}}^{{2}}{+}{O}{}\left({{z}}^{{4}}\right)\right){+}{{z}}^{{\mathrm{\mu }}}{}\left(\frac{{1}}{\left({\mathrm{\mu }}{-}{\mathrm{\nu }}{+}{1}\right){}\left({\mathrm{\mu }}{+}{\mathrm{\nu }}{+}{1}\right)}{}{z}{-}\frac{{1}}{\left({\mathrm{\mu }}{-}{\mathrm{\nu }}{+}{1}\right){}\left({\mathrm{\mu }}{-}{\mathrm{\nu }}{+}{3}\right){}\left({\mathrm{\mu }}{+}{\mathrm{\nu }}{+}{1}\right){}\left({\mathrm{\mu }}{+}{\mathrm{\nu }}{+}{3}\right)}{}{{z}}^{{3}}{+}{O}{}\left({{z}}^{{5}}\right)\right){,}{\mathrm{\nu }}{::}\left({¬}{\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{'}{\mathrm{integer}}{'}\right]\right)\right){\wedge }\left(\frac{{\mathrm{\mu }}}{{2}}{-}\frac{{\mathrm{\nu }}}{{2}}{+}\frac{{1}}{{2}}\right){::}\left({¬}{\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }\left(\frac{{\mathrm{\mu }}}{{2}}{+}\frac{{\mathrm{\nu }}}{{2}}{+}\frac{{1}}{{2}}\right){::}\left({¬}{\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }\left(\frac{{\mathrm{\mu }}}{{2}}{-}\frac{{\mathrm{\nu }}}{{2}}{+}\frac{{3}}{{2}}\right){::}\left({¬}{\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }\left(\frac{{\mathrm{\mu }}}{{2}}{+}\frac{{\mathrm{\nu }}}{{2}}{+}\frac{{3}}{{2}}\right){::}\left({¬}{\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right)$ (13)

References

 Abramowitz, M., and Stegun, I., eds. Handbook of Mathematical Functions. New York: Dover publications.
 Gradshteyn, and Ryzhik. Table of Integrals, Series and Products. 5th ed. Academic Press.
 Luke, Y. The Special Functions and Their Approximations. Vol. 1 Chap. 6.