LommelS1 - Maple Programming Help

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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
 $\left[{\mathrm{LommelS1}}{}\left({a}{,}{b}{,}{z}\right){=}\frac{{{z}}^{{a}{+}{1}}{}{\mathrm{hypergeom}}{}\left(\left[{1}\right]{,}\left[\frac{{3}}{{2}}{-}\frac{{1}}{{2}}{}{b}{+}\frac{{1}}{{2}}{}{a}{,}\frac{{3}}{{2}}{+}\frac{{1}}{{2}}{}{b}{+}\frac{{1}}{{2}}{}{a}\right]{,}{-}\frac{{1}}{{4}}{}{{z}}^{{2}}\right)}{\left({a}{-}{b}{+}{1}\right){}\left({a}{+}{b}{+}{1}\right)}{,}{\mathrm{And}}{}\left({-}{a}{+}{b}{-}{1}{\ne }{0}{,}{a}{+}{b}{+}{1}{\ne }{0}{,}\left(\frac{{3}}{{2}}{-}\frac{{1}}{{2}}{}{b}{+}\frac{{1}}{{2}}{}{a}\right){::}\left({\mathrm{Not}}{}\left({\mathrm{nonposint}}\right)\right){,}\left(\frac{{3}}{{2}}{+}\frac{{1}}{{2}}{}{b}{+}\frac{{1}}{{2}}{}{a}\right){::}\left({\mathrm{Not}}{}\left({\mathrm{nonposint}}\right)\right)\right)\right]$ (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);
 ${\mathrm{LommelS2}}{}\left({\mathrm{μ}}{,}{\mathrm{ν}}{,}{z}\right){=}{\mathrm{LommelS1}}{}\left({\mathrm{μ}}{,}{\mathrm{ν}}{,}{z}\right){+}{{2}}^{{\mathrm{μ}}{-}{1}}{}{\mathrm{Γ}}{}\left(\frac{{1}}{{2}}{}{\mathrm{μ}}{-}\frac{{1}}{{2}}{}{\mathrm{ν}}{+}\frac{{1}}{{2}}\right){}{\mathrm{Γ}}{}\left(\frac{{1}}{{2}}{}{\mathrm{μ}}{+}\frac{{1}}{{2}}{}{\mathrm{ν}}{+}\frac{{1}}{{2}}\right){}\left({\mathrm{sin}}{}\left(\frac{{1}}{{2}}{}\left({\mathrm{μ}}{-}{\mathrm{ν}}\right){}{\mathrm{π}}\right){}{\mathrm{BesselJ}}{}\left({\mathrm{ν}}{,}{z}\right){-}{\mathrm{cos}}{}\left(\frac{{1}}{{2}}{}\left({\mathrm{μ}}{-}{\mathrm{ν}}\right){}{\mathrm{π}}\right){}{\mathrm{BesselY}}{}\left({\mathrm{ν}}{,}{z}\right)\right)$ (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);
 ${{z}}^{{2}}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{z}}^{{2}}}{}{f}{}\left({z}\right)\right){+}{z}{}\left(\frac{{ⅆ}}{{ⅆ}{z}}{}{f}{}\left({z}\right)\right){+}\left({-}{{\mathrm{ν}}}^{{2}}{+}{{z}}^{{2}}\right){}{f}{}\left({z}\right){=}{{z}}^{{\mathrm{μ}}{+}{1}}$ (3)
 The Lommel functions also solve the following third order linear homogeneous differential equation with polynomial coefficients.
 $\left[{f}{}\left({z}\right){=}{\mathrm{LommelS1}}{}\left({\mathrm{μ}}{,}{\mathrm{ν}}{,}{z}\right){,}\left[\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{z}}^{{3}}}{}{f}{}\left({z}\right){=}\frac{\left({\mathrm{μ}}{-}{2}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{z}}^{{2}}}{}{f}{}\left({z}\right)\right)}{{z}}{+}\frac{\left({{\mathrm{ν}}}^{{2}}{-}{{z}}^{{2}}{+}{\mathrm{μ}}\right){}\left(\frac{{ⅆ}}{{ⅆ}{z}}{}{f}{}\left({z}\right)\right)}{{{z}}^{{2}}}{+}\frac{\left(\left({\mathrm{μ}}{-}{1}\right){}{{z}}^{{2}}{-}{{\mathrm{ν}}}^{{2}}{}\left({\mathrm{μ}}{+}{1}\right)\right){}{f}{}\left({z}\right)}{{{z}}^{{3}}}\right]\right]$ (4)

Examples

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

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

A MeijerG representation for the Lommel functions.

 > $\mathrm{LommelS1}\left(\mathrm{μ},\mathrm{ν},z\right)=\mathrm{convert}\left(\mathrm{LommelS1}\left(\mathrm{μ},\mathrm{ν},z\right),\mathrm{MeijerG}\right)$
 ${\mathrm{LommelS1}}{}\left({\mathrm{μ}}{,}{\mathrm{ν}}{,}{z}\right){=}{{2}}^{{\mathrm{μ}}{-}{1}}{}{\mathrm{Γ}}{}\left(\frac{{1}}{{2}}{}{\mathrm{μ}}{+}\frac{{1}}{{2}}{}{\mathrm{ν}}{+}\frac{{1}}{{2}}\right){}{\mathrm{Γ}}{}\left(\frac{{1}}{{2}}{}{\mathrm{μ}}{-}\frac{{1}}{{2}}{}{\mathrm{ν}}{+}\frac{{1}}{{2}}\right){}{\mathrm{MeijerG}}{}\left(\left[\left[\frac{{1}}{{2}}{}{\mathrm{μ}}{+}\frac{{1}}{{2}}\right]{,}\left[{}\right]\right]{,}\left[\left[\frac{{1}}{{2}}{}{\mathrm{μ}}{+}\frac{{1}}{{2}}\right]{,}\left[\frac{{1}}{{2}}{}{\mathrm{ν}}{,}{-}\frac{{1}}{{2}}{}{\mathrm{ν}}\right]\right]{,}\frac{{1}}{{4}}{}{{z}}^{{2}}\right)$ (9)
 > $\mathrm{LommelS2}\left(\mathrm{μ},\mathrm{ν},z\right)=\mathrm{convert}\left(\mathrm{LommelS2}\left(\mathrm{μ},\mathrm{ν},z\right),\mathrm{MeijerG}\right)$
 ${\mathrm{LommelS2}}{}\left({\mathrm{μ}}{,}{\mathrm{ν}}{,}{z}\right){=}\frac{{1}}{{2}}{}\frac{{\mathrm{MeijerG}}{}\left(\left[\left[\frac{{1}}{{2}}{}{\mathrm{μ}}{+}\frac{{1}}{{2}}\right]{,}\left[{}\right]\right]{,}\left[\left[\frac{{1}}{{2}}{}{\mathrm{μ}}{+}\frac{{1}}{{2}}{,}\frac{{1}}{{2}}{}{\mathrm{ν}}{,}{-}\frac{{1}}{{2}}{}{\mathrm{ν}}\right]{,}\left[{}\right]\right]{,}\frac{{1}}{{4}}{}{{z}}^{{2}}\right){}{{2}}^{{\mathrm{μ}}}}{{\mathrm{Γ}}{}\left({-}\frac{{1}}{{2}}{}{\mathrm{μ}}{+}\frac{{1}}{{2}}{}{\mathrm{ν}}{+}\frac{{1}}{{2}}\right){}{\mathrm{Γ}}{}\left({-}\frac{{1}}{{2}}{}{\mathrm{μ}}{-}\frac{{1}}{{2}}{}{\mathrm{ν}}{+}\frac{{1}}{{2}}\right)}$ (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)
 > $\mathrm{Series}\left(\mathrm{LommelS1}\left(\mathrm{μ},\mathrm{ν},z\right),z,4\right)$
 ${{z}}^{{\mathrm{μ}}}{}\left(\frac{{1}}{\left({\mathrm{μ}}{-}{\mathrm{ν}}{+}{1}\right){}\left({\mathrm{μ}}{+}{\mathrm{ν}}{+}{1}\right)}{}{z}{-}\frac{{1}}{\left({\mathrm{μ}}{-}{\mathrm{ν}}{+}{1}\right){}\left({\mathrm{μ}}{-}{\mathrm{ν}}{+}{3}\right){}\left({\mathrm{μ}}{+}{\mathrm{ν}}{+}{1}\right){}\left({\mathrm{μ}}{+}{\mathrm{ν}}{+}{3}\right)}{}{{z}}^{{3}}{+}{\mathrm{O}}\left({{z}}^{{5}}\right)\right){,}{\mathrm{And}}{}\left(\left({\mathrm{μ}}{+}{\mathrm{ν}}\right){::}\left({\mathrm{Not}}{}\left({\mathrm{And}}{}\left({\mathrm{negint}}{,}{\mathrm{odd}}\right)\right)\right){,}\left({\mathrm{μ}}{-}{\mathrm{ν}}\right){::}\left({\mathrm{Not}}{}\left({\mathrm{And}}{}\left({\mathrm{negint}}{,}{\mathrm{odd}}\right)\right)\right)\right)$ (12)
 > $\mathrm{Series}\left(\mathrm{LommelS2}\left(\mathrm{μ},\mathrm{ν},z\right),z,4\right)$
 ${{z}}^{{-}{\mathrm{ν}}}{}\left(\frac{{1}}{{2}}{}\frac{{{4}}^{{\mathrm{ν}}}{}{\mathrm{Γ}}{}\left(\frac{{1}}{{2}}{}{\mathrm{μ}}{-}\frac{{1}}{{2}}{}{\mathrm{ν}}{+}\frac{{1}}{{2}}\right){}{\mathrm{Γ}}{}\left(\frac{{1}}{{2}}{}{\mathrm{μ}}{+}\frac{{1}}{{2}}{}{\mathrm{ν}}{+}\frac{{1}}{{2}}\right){}{\mathrm{csc}}{}\left({\mathrm{π}}{}{\mathrm{ν}}\right){}{\mathrm{cos}}{}\left(\frac{{1}}{{2}}{}\left({\mathrm{μ}}{-}{\mathrm{ν}}\right){}{\mathrm{π}}\right){}{{2}}^{{\mathrm{μ}}{-}{\mathrm{ν}}}}{{\mathrm{Γ}}{}\left({1}{-}{\mathrm{ν}}\right)}{-}\frac{{1}}{{8}}{}\frac{{{4}}^{{\mathrm{ν}}}{}{\mathrm{Γ}}{}\left(\frac{{1}}{{2}}{}{\mathrm{μ}}{-}\frac{{1}}{{2}}{}{\mathrm{ν}}{+}\frac{{1}}{{2}}\right){}{\mathrm{Γ}}{}\left(\frac{{1}}{{2}}{}{\mathrm{μ}}{+}\frac{{1}}{{2}}{}{\mathrm{ν}}{+}\frac{{1}}{{2}}\right){}{\mathrm{csc}}{}\left({\mathrm{π}}{}{\mathrm{ν}}\right){}{\mathrm{cos}}{}\left(\frac{{1}}{{2}}{}\left({\mathrm{μ}}{-}{\mathrm{ν}}\right){}{\mathrm{π}}\right){}{{2}}^{{\mathrm{μ}}{-}{\mathrm{ν}}}}{{\mathrm{Γ}}{}\left({-}{\mathrm{ν}}{+}{2}\right)}{}{{z}}^{{2}}{+}{\mathrm{O}}\left({{z}}^{{4}}\right)\right){+}{{z}}^{{\mathrm{ν}}}{}\left({-}\frac{{1}}{{2}}{}\frac{{\mathrm{Γ}}{}\left(\frac{{1}}{{2}}{}{\mathrm{μ}}{-}\frac{{1}}{{2}}{}{\mathrm{ν}}{+}\frac{{1}}{{2}}\right){}{\mathrm{Γ}}{}\left(\frac{{1}}{{2}}{}{\mathrm{μ}}{+}\frac{{1}}{{2}}{}{\mathrm{ν}}{+}\frac{{1}}{{2}}\right){}{\mathrm{csc}}{}\left({\mathrm{π}}{}{\mathrm{ν}}\right){}{\mathrm{cos}}{}\left(\frac{{1}}{{2}}{}{\mathrm{π}}{}\left({\mathrm{μ}}{+}{\mathrm{ν}}\right)\right){}{{2}}^{{\mathrm{μ}}{-}{\mathrm{ν}}}}{{\mathrm{Γ}}{}\left({\mathrm{ν}}{+}{1}\right)}{+}\frac{{1}}{{8}}{}\frac{{\mathrm{Γ}}{}\left(\frac{{1}}{{2}}{}{\mathrm{μ}}{-}\frac{{1}}{{2}}{}{\mathrm{ν}}{+}\frac{{1}}{{2}}\right){}{\mathrm{Γ}}{}\left(\frac{{1}}{{2}}{}{\mathrm{μ}}{+}\frac{{1}}{{2}}{}{\mathrm{ν}}{+}\frac{{1}}{{2}}\right){}{\mathrm{csc}}{}\left({\mathrm{π}}{}{\mathrm{ν}}\right){}{\mathrm{cos}}{}\left(\frac{{1}}{{2}}{}{\mathrm{π}}{}\left({\mathrm{μ}}{+}{\mathrm{ν}}\right)\right){}{{2}}^{{\mathrm{μ}}{-}{\mathrm{ν}}}}{{\mathrm{Γ}}{}\left({\mathrm{ν}}{+}{2}\right)}{}{{z}}^{{2}}{+}{\mathrm{O}}\left({{z}}^{{4}}\right)\right){+}{{z}}^{{\mathrm{μ}}}{}\left(\frac{{1}}{\left({\mathrm{μ}}{-}{\mathrm{ν}}{+}{1}\right){}\left({\mathrm{μ}}{+}{\mathrm{ν}}{+}{1}\right)}{}{z}{-}\frac{{1}}{\left({\mathrm{μ}}{-}{\mathrm{ν}}{+}{1}\right){}\left({\mathrm{μ}}{-}{\mathrm{ν}}{+}{3}\right){}\left({\mathrm{μ}}{+}{\mathrm{ν}}{+}{1}\right){}\left({\mathrm{μ}}{+}{\mathrm{ν}}{+}{3}\right)}{}{{z}}^{{3}}{+}{\mathrm{O}}\left({{z}}^{{5}}\right)\right){,}{\mathrm{And}}{}\left({\mathrm{ν}}{::}\left({\mathrm{Not}}{}\left({\mathrm{integer}}\right)\right){,}\left(\frac{{1}}{{2}}{}{\mathrm{μ}}{-}\frac{{1}}{{2}}{}{\mathrm{ν}}{+}\frac{{1}}{{2}}\right){::}\left({\mathrm{Not}}{}\left({\mathrm{nonposint}}\right)\right){,}\left(\frac{{1}}{{2}}{}{\mathrm{μ}}{+}\frac{{1}}{{2}}{}{\mathrm{ν}}{+}\frac{{1}}{{2}}\right){::}\left({\mathrm{Not}}{}\left({\mathrm{nonposint}}\right)\right){,}\left(\frac{{1}}{{2}}{}{\mathrm{μ}}{-}\frac{{1}}{{2}}{}{\mathrm{ν}}{+}\frac{{3}}{{2}}\right){::}\left({\mathrm{Not}}{}\left({\mathrm{nonposint}}\right)\right){,}\left(\frac{{1}}{{2}}{}{\mathrm{μ}}{+}\frac{{1}}{{2}}{}{\mathrm{ν}}{+}\frac{{3}}{{2}}\right){::}\left({\mathrm{Not}}{}\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.