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LommelS1 - the Lommel function s
LommelS2 - the Lommel function S
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Calling Sequence
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LommelS1(mu, nu, z)
LommelS2(mu, nu, z)
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Parameters
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mu
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algebraic expression
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nu
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algebraic expression
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z
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algebraic expression
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Description
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The LommelS1(mu, nu, z) function is defined in terms of the hypergeometric function
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FunctionAdvisor( definition, LommelS1);
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![[LommelS1(a, b, z) = z^(a+1)*hypergeom([1], [3/2-(1/2)*b+(1/2)*a, 3/2+(1/2)*b+(1/2)*a], -(1/4)*z^2)/((a-b+1)*(a+b+1)), And(-a+b-1 <> 0, a+b+1 <> 0, (3/2-(1/2)*b+(1/2)*a)::(Not(nonposint)), (3/2+(1/2)*b+(1/2)*a)::(Not(nonposint)))]](/support/helpjp/helpview.aspx?si=6311/file00692/math82.png)
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and LommelS2(mu, nu, z) is defined in terms of LommelS1(mu, nu, z) and Bessel functions.
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LommelS2(mu,nu,z) = convert(LommelS2(mu,nu,z), LommelS1);
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These functions solve the non-homogeneous linear differential equation of second order.
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z^2*diff(f(z),`$`(z,2))+z*diff(f(z),z)+(z^2-nu^2)*f(z) = z^(mu+1);
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The Lommel functions also solve the following third order linear homogeneous differential equation with polynomial coefficients.
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FunctionAdvisor( DE, LommelS1(mu,nu,z));
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![[f(z) = LommelS1(mu, nu, z), [diff(f(z), `$`(z, 3)) = (mu-2)*(diff(f(z), `$`(z, 2)))/z+(-z^2+mu+nu^2)*(diff(f(z), z))/z^2+((mu-1)*z^2-nu^2*(mu+1))*f(z)/z^3]]](/support/helpjp/helpview.aspx?si=6311/file00692/math116.png)
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Examples
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The AngerJ and WeberE, StruveH and StruveL functions can be viewed as particular cases of LommelS1.
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A MeijerG representation for the Lommel functions.
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![LommelS1(mu, nu, z) = 2^(mu-1)*GAMMA((1/2)*mu+(1/2)*nu+1/2)*GAMMA((1/2)*mu-(1/2)*nu+1/2)*MeijerG([[(1/2)*mu+1/2], []], [[(1/2)*mu+1/2], [(1/2)*nu, -(1/2)*nu]], (1/4)*z^2)](/support/helpjp/helpview.aspx?si=6311/file00692/math176.png)
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![LommelS2(mu, nu, z) = (1/2)*MeijerG([[(1/2)*mu+1/2], []], [[(1/2)*mu+1/2, (1/2)*nu, -(1/2)*nu], []], (1/4)*z^2)*2^mu/(GAMMA(-(1/2)*mu+(1/2)*nu+1/2)*GAMMA(-(1/2)*mu-(1/2)*nu+1/2))](/support/helpjp/helpview.aspx?si=6311/file00692/math183.png)
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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.
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![[Series]](/support/helpjp/helpview.aspx?si=6311/file00692/math203.png)
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References
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Abramowitz, M., and Stegun, I., eds. Handbook of Mathematical Functions. New York: Dover publications.
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Gradshteyn, and Ryzhik. Table of Integrals, Series and Products. 5th ed. Academic Press.
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Luke, Y. The Special Functions and Their Approximations. Vol. 1 Chap. 6.
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