Struve and Related Functions - 11.9 Lommel Functions

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
11.9.E1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \deriv[2]{w}{z}+\frac{1}{z}\deriv{w}{z}+\left(1-\frac{\nu^{2}}{z^{2}}\right)w = z^{\mu-1}}
\deriv[2]{w}{z}+\frac{1}{z}\deriv{w}{z}+\left(1-\frac{\nu^{2}}{z^{2}}\right)w = z^{\mu-1}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
diff(w, [z$(2)])+(1)/(z)*diff(w, z)+(1 -((nu)^(2))/((z)^(2)))*w = (z)^(mu - 1)

D[w, {z, 2}]+Divide[1,z]*D[w, z]+(1 -Divide[\[Nu]^(2),(z)^(2)])*w == (z)^(\[Mu]- 1)
Failure Failure
Failed [300 / 300]
Result: -.7677724761+.5394693872e-1*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: 1.394855253+1.097179568*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.7677724760456721, 0.053946938885231305]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.9642783315232053, 1.0539469388852312]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
11.9.E2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle w = \Lommels{\mu}{\nu}@{z}+A\BesselJ{\nu}@{z}+B\BesselY{\nu}@{z}}
w = \Lommels{\mu}{\nu}@{z}+A\BesselJ{\nu}@{z}+B\BesselY{\nu}@{z}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(\nu+k+1)} > 0, \realpart@@{((-\nu)+k+1)} > 0} 
w = LommelS1(mu, nu, z)+ A*BesselJ(nu, z)+ B*BesselY(nu, z)

Error
Failure Missing Macro Error
Failed [300 / 300]
Result: .9752372341+.4710119144*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: 1.486411080-.4774576789*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 -
11.9.E3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Lommels{\mu}{\nu}@{z} = z^{\mu+1}\sum_{k=0}^{\infty}(-1)^{k}\frac{z^{2k}}{a_{k+1}(\mu,\nu)}}
\Lommels{\mu}{\nu}@{z} = z^{\mu+1}\sum_{k=0}^{\infty}(-1)^{k}\frac{z^{2k}}{a_{k+1}(\mu,\nu)}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
LommelS1(mu, nu, z) = (z)^(mu + 1)* sum((- 1)^(k)*((z)^(2*k))/(a[k + 1](mu , nu)), k = 0..infinity)

Error
Failure Missing Macro Error Error -
11.9.E4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle a_{k}(\mu,\nu) = \prod_{m=1}^{k}\left((\mu+2m-1)^{2}-\nu^{2}\right)}
a_{k}(\mu,\nu) = \prod_{m=1}^{k}\left((\mu+2m-1)^{2}-\nu^{2}\right)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
a[k](mu , nu) = product((mu + 2*m - 1)^(2)- (nu)^(2), m = 1..k)
 
Subscript[a, k][\[Mu], \[Nu]] == Product[(\[Mu]+ 2*m - 1)^(2)- \[Nu]^(2), {m, 1, k}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
11.9.E5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \LommelS{\mu}{\nu}@{z} = \Lommels{\mu}{\nu}@{z}+2^{\mu-1}\EulerGamma@{\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+\tfrac{1}{2}}\EulerGamma@{\tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2}}\*\left(\sin@{\tfrac{1}{2}(\mu-\nu)\pi}\,\BesselJ{\nu}@{z}-\cos@{\tfrac{1}{2}(\mu-\nu)\pi}\,\BesselY{\nu}@{z}\right)}
\LommelS{\mu}{\nu}@{z} = \Lommels{\mu}{\nu}@{z}+2^{\mu-1}\EulerGamma@{\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+\tfrac{1}{2}}\EulerGamma@{\tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2}}\*\left(\sin@{\tfrac{1}{2}(\mu-\nu)\pi}\,\BesselJ{\nu}@{z}-\cos@{\tfrac{1}{2}(\mu-\nu)\pi}\,\BesselY{\nu}@{z}\right)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(\nu+k+1)} > 0, \realpart@@{(\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+\tfrac{1}{2})} > 0, \realpart@@{(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2})} > 0, \realpart@@{((-\nu)+k+1)} > 0} 
LommelS2(mu, nu, z) = 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))*(sin((1)/(2)*(mu - nu)*Pi)*BesselJ(nu, z)- cos((1)/(2)*(mu - nu)*Pi)*BesselY(nu, z))

Error
Successful Missing Macro Error - -
11.9#Ex1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Lommels{\mu}{-\nu}@{z} = \Lommels{\mu}{\nu}@{z}}
\Lommels{\mu}{-\nu}@{z} = \Lommels{\mu}{\nu}@{z}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
LommelS1(mu, - nu, z) = LommelS1(mu, nu, z)

Error
Successful Missing Macro Error - -
11.9#Ex2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \LommelS{\mu}{-\nu}@{z} = \LommelS{\mu}{\nu}@{z}}
\LommelS{\mu}{-\nu}@{z} = \LommelS{\mu}{\nu}@{z}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{((-\nu)+k+1)} > 0, \realpart@@{(\nu+k+1)} > 0, \realpart@@{(\tfrac{1}{2}\mu+\tfrac{1}{2}(-\nu)+\tfrac{1}{2})} > 0, \realpart@@{(\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+\tfrac{1}{2})} > 0, \realpart@@{(\tfrac{1}{2}\mu-\tfrac{1}{2}(-\nu)+\tfrac{1}{2})} > 0, \realpart@@{(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2})} > 0, \realpart@@{((-(-\nu))+k+1)} > 0} 
LommelS2(mu, - nu, z) = LommelS2(mu, nu, z)

Error
Successful Missing Macro Error - -
11.9.E7 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Lommels{\mu}{\nu}@{z} = 2^{\mu+1}\sum_{k=0}^{\infty}\*\frac{(2k+\mu+1)\EulerGamma@{k+\mu+1}}{k!(2k+\mu-\nu+1)(2k+\mu+\nu+1)}\BesselJ{2k+\mu+1}@{z}}
\Lommels{\mu}{\nu}@{z} = 2^{\mu+1}\sum_{k=0}^{\infty}\*\frac{(2k+\mu+1)\EulerGamma@{k+\mu+1}}{k!(2k+\mu-\nu+1)(2k+\mu+\nu+1)}\BesselJ{2k+\mu+1}@{z}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{((2k+\mu+1)+k+1)} > 0, \realpart@@{(k+\mu+1)} > 0} 
LommelS1(mu, nu, z) = (2)^(mu + 1)* sum(*((2*k + mu + 1)*GAMMA(k + mu + 1))/(factorial(k)*(2*k + mu - nu + 1)*(2*k + mu + nu + 1))*BesselJ(2*k + mu + 1, z), k = 0..infinity)

Error
Error Missing Macro Error - -
11.9.E8 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Lommels{\mu}{\nu}@{z} = 2^{(\mu+\nu-1)/2}\EulerGamma@{\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+\tfrac{1}{2}}z^{(\mu+1-\nu)/2}\*\sum_{k=0}^{\infty}\frac{(\tfrac{1}{2}z)^{k}}{k!(2k+\mu-\nu+1)}\BesselJ{k+\frac{1}{2}(\mu+\nu+1)}@{z}}
\Lommels{\mu}{\nu}@{z} = 2^{(\mu+\nu-1)/2}\EulerGamma@{\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+\tfrac{1}{2}}z^{(\mu+1-\nu)/2}\*\sum_{k=0}^{\infty}\frac{(\tfrac{1}{2}z)^{k}}{k!(2k+\mu-\nu+1)}\BesselJ{k+\frac{1}{2}(\mu+\nu+1)}@{z}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{((k+\frac{1}{2}(\mu+\nu+1))+k+1)} > 0, \realpart@@{(\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+\tfrac{1}{2})} > 0} 
LommelS1(mu, nu, z) = (2)^((mu + nu - 1)/2)* GAMMA((1)/(2)*mu +(1)/(2)*nu +(1)/(2))*(z)^((mu + 1 - nu)/2)* sum((((1)/(2)*z)^(k))/(factorial(k)*(2*k + mu - nu + 1))*BesselJ(k +(1)/(2)*(mu + nu + 1), z), k = 0..infinity)

Error
Failure Missing Macro Error Skipped - Because timed out -
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