Struve and Related Functions - 11.9 Lommel Functions

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DLMF Formula Constraints Maple Mathematica Symbolic
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Mathematica 		Numeric
Maple 		Numeric
Mathematica
11.9.E1 d 2 w d z 2 + 1 z ⁒ d w d z + ( 1 - Ξ½ 2 z 2 ) ⁒ w = z ΞΌ - 1 derivative 𝑀 𝑧 2 1 𝑧 derivative 𝑀 𝑧 1 superscript 𝜈 2 superscript 𝑧 2 𝑀 superscript 𝑧 πœ‡ 1 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\frac{% 1}{z}\frac{\mathrm{d}w}{\mathrm{d}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}		 

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[ΞΌ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ξ½, 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[ΞΌ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ξ½, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
11.9.E2 w = s ΞΌ , Ξ½ ⁑ ( z ) + A ⁒ J Ξ½ ⁑ ( z ) + B ⁒ Y Ξ½ ⁑ ( z ) 𝑀 Lommel-s πœ‡ 𝜈 𝑧 𝐴 Bessel-J 𝜈 𝑧 𝐡 Bessel-Y-Weber 𝜈 𝑧 {\displaystyle{\displaystyle w=s_{{\mu},{\nu}}\left(z\right)+AJ_{\nu}\left(z% \right)+BY_{\nu}\left(z\right)}}
w = \Lommels{\mu}{\nu}@{z}+A\BesselJ{\nu}@{z}+B\BesselY{\nu}@{z}		 β„œ ⁑ ( Ξ½ + k + 1 ) > 0 , β„œ ⁑ ( ( - Ξ½ ) + k + 1 ) > 0 formulae-sequence 𝜈 π‘˜ 1 0 𝜈 π‘˜ 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((-\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 s ΞΌ , Ξ½ ⁑ ( z ) = z ΞΌ + 1 ⁒ βˆ‘ k = 0 ∞ ( - 1 ) k ⁒ z 2 ⁒ k a k + 1 ⁒ ( ΞΌ , Ξ½ ) Lommel-s πœ‡ 𝜈 𝑧 superscript 𝑧 πœ‡ 1 superscript subscript π‘˜ 0 superscript 1 π‘˜ superscript 𝑧 2 π‘˜ subscript π‘Ž π‘˜ 1 πœ‡ 𝜈 {\displaystyle{\displaystyle s_{{\mu},{\nu}}\left(z\right)=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)}		 

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 a k ⁒ ( ΞΌ , Ξ½ ) = ∏ m = 1 k ( ( ΞΌ + 2 ⁒ m - 1 ) 2 - Ξ½ 2 ) subscript π‘Ž π‘˜ πœ‡ 𝜈 superscript subscript product π‘š 1 π‘˜ superscript πœ‡ 2 π‘š 1 2 superscript 𝜈 2 {\displaystyle{\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)		 

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 S ΞΌ , Ξ½ ⁑ ( z ) = s ΞΌ , Ξ½ ⁑ ( z ) + 2 ΞΌ - 1 ⁒ Ξ“ ⁑ ( 1 2 ⁒ ΞΌ + 1 2 ⁒ Ξ½ + 1 2 ) ⁒ Ξ“ ⁑ ( 1 2 ⁒ ΞΌ - 1 2 ⁒ Ξ½ + 1 2 ) ⁒ ( sin ⁑ ( 1 2 ⁒ ( ΞΌ - Ξ½ ) ⁒ Ο€ ) ⁒ J Ξ½ ⁑ ( z ) - cos ⁑ ( 1 2 ⁒ ( ΞΌ - Ξ½ ) ⁒ Ο€ ) ⁒ Y Ξ½ ⁑ ( z ) ) Lommel-S πœ‡ 𝜈 𝑧 Lommel-s πœ‡ 𝜈 𝑧 superscript 2 πœ‡ 1 Euler-Gamma 1 2 πœ‡ 1 2 𝜈 1 2 Euler-Gamma 1 2 πœ‡ 1 2 𝜈 1 2 1 2 πœ‡ 𝜈 πœ‹ Bessel-J 𝜈 𝑧 1 2 πœ‡ 𝜈 πœ‹ Bessel-Y-Weber 𝜈 𝑧 {\displaystyle{\displaystyle S_{{\mu},{\nu}}\left(z\right)=s_{{\mu},{\nu}}% \left(z\right)+2^{\mu-1}\Gamma\left(\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+\tfrac{1}{% 2}\right)\Gamma\left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2}\right)\*% \left(\sin\left(\tfrac{1}{2}(\mu-\nu)\pi\right)\,J_{\nu}\left(z\right)-\cos% \left(\tfrac{1}{2}(\mu-\nu)\pi\right)\,Y_{\nu}\left(z\right)\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)		 β„œ ⁑ ( Ξ½ + k + 1 ) > 0 , β„œ ⁑ ( 1 2 ⁒ ΞΌ + 1 2 ⁒ Ξ½ + 1 2 ) > 0 , β„œ ⁑ ( 1 2 ⁒ ΞΌ - 1 2 ⁒ Ξ½ + 1 2 ) > 0 , β„œ ⁑ ( ( - Ξ½ ) + k + 1 ) > 0 formulae-sequence 𝜈 π‘˜ 1 0 formulae-sequence 1 2 πœ‡ 1 2 𝜈 1 2 0 formulae-sequence 1 2 πœ‡ 1 2 𝜈 1 2 0 𝜈 π‘˜ 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re(\tfrac{1}{2}\mu+\tfrac{1}{2}\nu% +\tfrac{1}{2})>0,\Re(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2})>0,\Re((-\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 s ΞΌ , - Ξ½ ⁑ ( z ) = s ΞΌ , Ξ½ ⁑ ( z ) Lommel-s πœ‡ 𝜈 𝑧 Lommel-s πœ‡ 𝜈 𝑧 {\displaystyle{\displaystyle s_{{\mu},{-\nu}}\left(z\right)=s_{{\mu},{\nu}}% \left(z\right)}}
\Lommels{\mu}{-\nu}@{z} = \Lommels{\mu}{\nu}@{z}		 

LommelS1(mu, - nu, z) = LommelS1(mu, nu, z)

Error
Successful Missing Macro Error - -
11.9#Ex2 S ΞΌ , - Ξ½ ⁑ ( z ) = S ΞΌ , Ξ½ ⁑ ( z ) Lommel-S πœ‡ 𝜈 𝑧 Lommel-S πœ‡ 𝜈 𝑧 {\displaystyle{\displaystyle S_{{\mu},{-\nu}}\left(z\right)=S_{{\mu},{\nu}}% \left(z\right)}}
\LommelS{\mu}{-\nu}@{z} = \LommelS{\mu}{\nu}@{z}		 β„œ ⁑ ( ( - Ξ½ ) + k + 1 ) > 0 , β„œ ⁑ ( Ξ½ + k + 1 ) > 0 , β„œ ⁑ ( 1 2 ⁒ ΞΌ + 1 2 ⁒ ( - Ξ½ ) + 1 2 ) > 0 , β„œ ⁑ ( 1 2 ⁒ ΞΌ + 1 2 ⁒ Ξ½ + 1 2 ) > 0 , β„œ ⁑ ( 1 2 ⁒ ΞΌ - 1 2 ⁒ ( - Ξ½ ) + 1 2 ) > 0 , β„œ ⁑ ( 1 2 ⁒ ΞΌ - 1 2 ⁒ Ξ½ + 1 2 ) > 0 , β„œ ⁑ ( ( - ( - Ξ½ ) ) + k + 1 ) > 0 formulae-sequence 𝜈 π‘˜ 1 0 formulae-sequence 𝜈 π‘˜ 1 0 formulae-sequence 1 2 πœ‡ 1 2 𝜈 1 2 0 formulae-sequence 1 2 πœ‡ 1 2 𝜈 1 2 0 formulae-sequence 1 2 πœ‡ 1 2 𝜈 1 2 0 formulae-sequence 1 2 πœ‡ 1 2 𝜈 1 2 0 𝜈 π‘˜ 1 0 {\displaystyle{\displaystyle\Re((-\nu)+k+1)>0,\Re(\nu+k+1)>0,\Re(\tfrac{1}{2}% \mu+\tfrac{1}{2}(-\nu)+\tfrac{1}{2})>0,\Re(\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+% \tfrac{1}{2})>0,\Re(\tfrac{1}{2}\mu-\tfrac{1}{2}(-\nu)+\tfrac{1}{2})>0,\Re(% \tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2})>0,\Re((-(-\nu))+k+1)>0}} 
LommelS2(mu, - nu, z) = LommelS2(mu, nu, z)

Error
Successful Missing Macro Error - -
11.9.E7 s ΞΌ , Ξ½ ⁑ ( z ) = 2 ΞΌ + 1 ⁒ βˆ‘ k = 0 ∞ ⁒ ( 2 ⁒ k + ΞΌ + 1 ) ⁒ Ξ“ ⁑ ( k + ΞΌ + 1 ) k ! ⁒ ( 2 ⁒ k + ΞΌ - Ξ½ + 1 ) ⁒ ( 2 ⁒ k + ΞΌ + Ξ½ + 1 ) ⁒ J 2 ⁒ k + ΞΌ + 1 ⁑ ( z ) Lommel-s πœ‡ 𝜈 𝑧 superscript 2 πœ‡ 1 superscript subscript π‘˜ 0 2 π‘˜ πœ‡ 1 Euler-Gamma π‘˜ πœ‡ 1 π‘˜ 2 π‘˜ πœ‡ 𝜈 1 2 π‘˜ πœ‡ 𝜈 1 Bessel-J 2 π‘˜ πœ‡ 1 𝑧 {\displaystyle{\displaystyle s_{{\mu},{\nu}}\left(z\right)=2^{\mu+1}\sum_{k=0}% ^{\infty}\*\frac{(2k+\mu+1)\Gamma\left(k+\mu+1\right)}{k!(2k+\mu-\nu+1)(2k+\mu% +\nu+1)}J_{2k+\mu+1}\left(z\right)}}
\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}		 β„œ ⁑ ( ( 2 ⁒ k + ΞΌ + 1 ) + k + 1 ) > 0 , β„œ ⁑ ( k + ΞΌ + 1 ) > 0 formulae-sequence 2 π‘˜ πœ‡ 1 π‘˜ 1 0 π‘˜ πœ‡ 1 0 {\displaystyle{\displaystyle\Re((2k+\mu+1)+k+1)>0,\Re(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 s ΞΌ , Ξ½ ⁑ ( z ) = 2 ( ΞΌ + Ξ½ - 1 ) / 2 ⁒ Ξ“ ⁑ ( 1 2 ⁒ ΞΌ + 1 2 ⁒ Ξ½ + 1 2 ) ⁒ z ( ΞΌ + 1 - Ξ½ ) / 2 ⁒ βˆ‘ k = 0 ∞ ( 1 2 ⁒ z ) k k ! ⁒ ( 2 ⁒ k + ΞΌ - Ξ½ + 1 ) ⁒ J k + 1 2 ⁒ ( ΞΌ + Ξ½ + 1 ) ⁑ ( z ) Lommel-s πœ‡ 𝜈 𝑧 superscript 2 πœ‡ 𝜈 1 2 Euler-Gamma 1 2 πœ‡ 1 2 𝜈 1 2 superscript 𝑧 πœ‡ 1 𝜈 2 superscript subscript π‘˜ 0 superscript 1 2 𝑧 π‘˜ π‘˜ 2 π‘˜ πœ‡ 𝜈 1 Bessel-J π‘˜ 1 2 πœ‡ 𝜈 1 𝑧 {\displaystyle{\displaystyle s_{{\mu},{\nu}}\left(z\right)=2^{(\mu+\nu-1)/2}% \Gamma\left(\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+\tfrac{1}{2}\right)z^{(\mu+1-\nu)/% 2}\*\sum_{k=0}^{\infty}\frac{(\tfrac{1}{2}z)^{k}}{k!(2k+\mu-\nu+1)}J_{k+\frac{% 1}{2}(\mu+\nu+1)}\left(z\right)}}
\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}		 β„œ ⁑ ( ( k + 1 2 ⁒ ( ΞΌ + Ξ½ + 1 ) ) + k + 1 ) > 0 , β„œ ⁑ ( 1 2 ⁒ ΞΌ + 1 2 ⁒ Ξ½ + 1 2 ) > 0 formulae-sequence π‘˜ 1 2 πœ‡ 𝜈 1 π‘˜ 1 0 1 2 πœ‡ 1 2 𝜈 1 2 0 {\displaystyle{\displaystyle\Re((k+\frac{1}{2}(\mu+\nu+1))+k+1)>0,\Re(\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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