Bessel Functions - 11.2 Definitions

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
11.2.E1 𝐇 Ξ½ ⁑ ( z ) = ( 1 2 ⁒ z ) Ξ½ + 1 ⁒ βˆ‘ n = 0 ∞ ( - 1 ) n ⁒ ( 1 2 ⁒ z ) 2 ⁒ n Ξ“ ⁑ ( n + 3 2 ) ⁒ Ξ“ ⁑ ( n + Ξ½ + 3 2 ) Struve-H 𝜈 𝑧 superscript 1 2 𝑧 𝜈 1 superscript subscript 𝑛 0 superscript 1 𝑛 superscript 1 2 𝑧 2 𝑛 Euler-Gamma 𝑛 3 2 Euler-Gamma 𝑛 𝜈 3 2 {\displaystyle{\displaystyle\mathbf{H}_{\nu}\left(z\right)=(\tfrac{1}{2}z)^{% \nu+1}\sum_{n=0}^{\infty}\frac{(-1)^{n}(\tfrac{1}{2}z)^{2n}}{\Gamma\left(n+% \tfrac{3}{2}\right)\Gamma\left(n+\nu+\tfrac{3}{2}\right)}}}
\StruveH{\nu}@{z} = (\tfrac{1}{2}z)^{\nu+1}\sum_{n=0}^{\infty}\frac{(-1)^{n}(\tfrac{1}{2}z)^{2n}}{\EulerGamma@{n+\tfrac{3}{2}}\EulerGamma@{n+\nu+\tfrac{3}{2}}}		 β„œ ⁑ ( n + 3 2 ) > 0 , β„œ ⁑ ( n + Ξ½ + 3 2 ) > 0 formulae-sequence 𝑛 3 2 0 𝑛 𝜈 3 2 0 {\displaystyle{\displaystyle\Re(n+\tfrac{3}{2})>0,\Re(n+\nu+\tfrac{3}{2})>0}} 
StruveH(nu, z) = ((1)/(2)*z)^(nu + 1)* sum(((- 1)^(n)*((1)/(2)*z)^(2*n))/(GAMMA(n +(3)/(2))*GAMMA(n + nu +(3)/(2))), n = 0..infinity)

StruveH[\[Nu], z] == (Divide[1,2]*z)^(\[Nu]+ 1)* Sum[Divide[(- 1)^(n)*(Divide[1,2]*z)^(2*n),Gamma[n +Divide[3,2]]*Gamma[n + \[Nu]+Divide[3,2]]], {n, 0, Infinity}, GenerateConditions->None]
Successful Successful - Successful [Tested: 70]
11.2.E2 𝐋 Ξ½ ⁑ ( z ) = - i ⁒ e - 1 2 ⁒ Ο€ ⁒ i ⁒ Ξ½ ⁒ 𝐇 Ξ½ ⁑ ( i ⁒ z ) modified-Struve-L 𝜈 𝑧 𝑖 superscript 𝑒 1 2 πœ‹ 𝑖 𝜈 Struve-H 𝜈 𝑖 𝑧 {\displaystyle{\displaystyle\mathbf{L}_{\nu}\left(z\right)=-ie^{-\frac{1}{2}% \pi i\nu}\mathbf{H}_{\nu}\left(iz\right)}}
\modStruveL{\nu}@{z} = -ie^{-\frac{1}{2}\pi i\nu}\StruveH{\nu}@{iz}		 β„œ ⁑ ( n + Ξ½ + 3 2 ) > 0 𝑛 𝜈 3 2 0 {\displaystyle{\displaystyle\Re(n+\nu+\tfrac{3}{2})>0}} 
StruveL(nu, z) = - I*exp(-(1)/(2)*Pi*I*nu)*StruveH(nu, I*z)

StruveL[\[Nu], z] == - I*Exp[-Divide[1,2]*Pi*I*\[Nu]]*StruveH[\[Nu], I*z]
Failure Failure
Failed [8 / 70]
Result: 1.240284959+1.629557917*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}


Result: 33.65868914+29.08337177*I
Test Values: {nu = -1/2+1/2*I*3^(1/2), z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [8 / 70]
Result: Complex[1.2402849561066787, 1.6295579188731661]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[33.658689094091635, 29.08337174056143]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[Ξ½, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
11.2.E2 - i ⁒ e - 1 2 ⁒ Ο€ ⁒ i ⁒ Ξ½ ⁒ 𝐇 Ξ½ ⁑ ( i ⁒ z ) = ( 1 2 ⁒ z ) Ξ½ + 1 ⁒ βˆ‘ n = 0 ∞ ( 1 2 ⁒ z ) 2 ⁒ n Ξ“ ⁑ ( n + 3 2 ) ⁒ Ξ“ ⁑ ( n + Ξ½ + 3 2 ) 𝑖 superscript 𝑒 1 2 πœ‹ 𝑖 𝜈 Struve-H 𝜈 𝑖 𝑧 superscript 1 2 𝑧 𝜈 1 superscript subscript 𝑛 0 superscript 1 2 𝑧 2 𝑛 Euler-Gamma 𝑛 3 2 Euler-Gamma 𝑛 𝜈 3 2 {\displaystyle{\displaystyle-ie^{-\frac{1}{2}\pi i\nu}\mathbf{H}_{\nu}\left(iz% \right)=(\tfrac{1}{2}z)^{\nu+1}\sum_{n=0}^{\infty}\frac{(\tfrac{1}{2}z)^{2n}}{% \Gamma\left(n+\tfrac{3}{2}\right)\Gamma\left(n+\nu+\tfrac{3}{2}\right)}}}
-ie^{-\frac{1}{2}\pi i\nu}\StruveH{\nu}@{iz} = (\tfrac{1}{2}z)^{\nu+1}\sum_{n=0}^{\infty}\frac{(\tfrac{1}{2}z)^{2n}}{\EulerGamma@{n+\tfrac{3}{2}}\EulerGamma@{n+\nu+\tfrac{3}{2}}}		 β„œ ⁑ ( n + Ξ½ + 3 2 ) > 0 , β„œ ⁑ ( n + 3 2 ) > 0 formulae-sequence 𝑛 𝜈 3 2 0 𝑛 3 2 0 {\displaystyle{\displaystyle\Re(n+\nu+\tfrac{3}{2})>0,\Re(n+\tfrac{3}{2})>0}} 
- I*exp(-(1)/(2)*Pi*I*nu)*StruveH(nu, I*z) = ((1)/(2)*z)^(nu + 1)* sum((((1)/(2)*z)^(2*n))/(GAMMA(n +(3)/(2))*GAMMA(n + nu +(3)/(2))), n = 0..infinity)

- I*Exp[-Divide[1,2]*Pi*I*\[Nu]]*StruveH[\[Nu], I*z] == (Divide[1,2]*z)^(\[Nu]+ 1)* Sum[Divide[(Divide[1,2]*z)^(2*n),Gamma[n +Divide[3,2]]*Gamma[n + \[Nu]+Divide[3,2]]], {n, 0, Infinity}, GenerateConditions->None]
Failure Failure
Failed [8 / 70]
Result: -1.240284959-1.629557917*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}


Result: -33.65868914-29.08337177*I
Test Values: {nu = -1/2+1/2*I*3^(1/2), z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [8 / 70]
Result: Complex[-1.2402849561066787, -1.6295579188731661]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-33.658689094091635, -29.08337174056143]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[Ξ½, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
11.2.E5 𝐊 Ξ½ ⁑ ( z ) = 𝐇 Ξ½ ⁑ ( z ) - Y Ξ½ ⁑ ( z ) associated-Struve-K 𝜈 𝑧 Struve-H 𝜈 𝑧 Bessel-Y-Weber 𝜈 𝑧 {\displaystyle{\displaystyle\mathbf{K}_{\nu}\left(z\right)=\mathbf{H}_{\nu}% \left(z\right)-Y_{\nu}\left(z\right)}}
\StruveK{\nu}@{z} = \StruveH{\nu}@{z}-\BesselY{\nu}@{z}		 β„œ ⁑ ( Ξ½ + k + 1 ) > 0 , β„œ ⁑ ( ( - Ξ½ ) + k + 1 ) > 0 , β„œ ⁑ ( n + Ξ½ + 3 2 ) > 0 formulae-sequence 𝜈 π‘˜ 1 0 formulae-sequence 𝜈 π‘˜ 1 0 𝑛 𝜈 3 2 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0,\Re(n+\nu+\tfrac{% 3}{2})>0}} 
StruveH(nu, z) - BesselY(nu, z) = StruveH(nu, z)- BesselY(nu, z)

StruveH[\[Nu], z] - BesselY[\[Nu], z] == StruveH[\[Nu], z]- BesselY[\[Nu], z]
Successful Successful - Successful [Tested: 70]
11.2.E6 𝐌 Ξ½ ⁑ ( z ) = 𝐋 Ξ½ ⁑ ( z ) - I Ξ½ ⁑ ( z ) associated-Struve-M 𝜈 𝑧 modified-Struve-L 𝜈 𝑧 modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle\mathbf{M}_{\nu}\left(z\right)=\mathbf{L}_{\nu}% \left(z\right)-I_{\nu}\left(z\right)}}
\modStruveM{\nu}@{z} = \modStruveL{\nu}@{z}-\modBesselI{\nu}@{z}		 β„œ ⁑ ( Ξ½ + k + 1 ) > 0 , β„œ ⁑ ( n + Ξ½ + 3 2 ) > 0 formulae-sequence 𝜈 π‘˜ 1 0 𝑛 𝜈 3 2 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re(n+\nu+\tfrac{3}{2})>0}} 
StruveL(nu, z) - BesselI(nu, z) = StruveL(nu, z)- BesselI(nu, z)

StruveL[\[Nu], z] - BesselI[\[Nu], z] == StruveL[\[Nu], z]- BesselI[\[Nu], z]
Successful Successful - Successful [Tested: 70]
11.2.E7 d 2 w d z 2 + 1 z ⁒ d w d z + ( 1 - Ξ½ 2 z 2 ) ⁒ w = ( 1 2 ⁒ z ) Ξ½ - 1 Ο€ ⁒ Ξ“ ⁑ ( Ξ½ + 1 2 ) derivative 𝑀 𝑧 2 1 𝑧 derivative 𝑀 𝑧 1 superscript 𝜈 2 superscript 𝑧 2 𝑀 superscript 1 2 𝑧 𝜈 1 πœ‹ Euler-Gamma 𝜈 1 2 {\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=% \frac{(\tfrac{1}{2}z)^{\nu-1}}{\sqrt{\pi}\Gamma\left(\nu+\tfrac{1}{2}\right)}}}
\deriv[2]{w}{z}+\frac{1}{z}\deriv{w}{z}+\left(1-\frac{\nu^{2}}{z^{2}}\right)w = \frac{(\tfrac{1}{2}z)^{\nu-1}}{\sqrt{\pi}\EulerGamma@{\nu+\tfrac{1}{2}}}		 β„œ ⁑ ( Ξ½ + 1 2 ) > 0 𝜈 1 2 0 {\displaystyle{\displaystyle\Re(\nu+\tfrac{1}{2})>0}} 
diff(w, [z$(2)])+(1)/(z)*diff(w, z)+(1 -((nu)^(2))/((z)^(2)))*w = (((1)/(2)*z)^(nu - 1))/(sqrt(Pi)*GAMMA(nu +(1)/(2)))

D[w, {z, 2}]+Divide[1,z]*D[w, z]+(1 -Divide[\[Nu]^(2),(z)^(2)])*w == Divide[(Divide[1,2]*z)^(\[Nu]- 1),Sqrt[Pi]*Gamma[\[Nu]+Divide[1,2]]]
Failure Failure
Failed [300 / 300]
Result: -.5630887369+.2307852889*I
Test Values: {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: 1.502962248+1.156533180*I
Test Values: {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[-0.563088736999922, 0.23078528896155245]
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[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.3603758852198513, 0.9342077190875079]
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[Ξ½, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}

... skip entries to safe data
11.2.E8 w = 𝐇 Ξ½ ⁑ ( z ) , 𝐊 Ξ½ ⁑ ( z ) 𝑀 Struve-H 𝜈 𝑧 associated-Struve-K 𝜈 𝑧 {\displaystyle{\displaystyle w=\mathbf{H}_{\nu}\left(z\right),\mathbf{K}_{\nu}% \left(z\right)}}
w = \StruveH{\nu}@{z},\StruveK{\nu}@{z}		 β„œ ⁑ ( n + Ξ½ + 3 2 ) > 0 , β„œ ⁑ ( Ξ½ + k + 1 ) > 0 , β„œ ⁑ ( ( - Ξ½ ) + k + 1 ) > 0 formulae-sequence 𝑛 𝜈 3 2 0 formulae-sequence 𝜈 π‘˜ 1 0 𝜈 π‘˜ 1 0 {\displaystyle{\displaystyle\Re(n+\nu+\tfrac{3}{2})>0,\Re(\nu+k+1)>0,\Re((-\nu% )+k+1)>0}} 
w = StruveH(nu, z); StruveH(nu, z) - BesselY(nu, z)

w == StruveH[\[Nu], z]
 StruveH[\[Nu], z] - BesselY[\[Nu], z]
Failure Failure Error Error
11.2.E9 d 2 w d z 2 + 1 z ⁒ d w d z - ( 1 + Ξ½ 2 z 2 ) ⁒ w = ( 1 2 ⁒ z ) Ξ½ - 1 Ο€ ⁒ Ξ“ ⁑ ( Ξ½ + 1 2 ) derivative 𝑀 𝑧 2 1 𝑧 derivative 𝑀 𝑧 1 superscript 𝜈 2 superscript 𝑧 2 𝑀 superscript 1 2 𝑧 𝜈 1 πœ‹ Euler-Gamma 𝜈 1 2 {\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=% \frac{(\tfrac{1}{2}z)^{\nu-1}}{\sqrt{\pi}\Gamma\left(\nu+\tfrac{1}{2}\right)}}}
\deriv[2]{w}{z}+\frac{1}{z}\deriv{w}{z}-\left(1+\frac{\nu^{2}}{z^{2}}\right)w = \frac{(\tfrac{1}{2}z)^{\nu-1}}{\sqrt{\pi}\EulerGamma@{\nu+\tfrac{1}{2}}}		 β„œ ⁑ ( Ξ½ + 1 2 ) > 0 𝜈 1 2 0 {\displaystyle{\displaystyle\Re(\nu+\tfrac{1}{2})>0}} 
diff(w, [z$(2)])+(1)/(z)*diff(w, z)-(1 +((nu)^(2))/((z)^(2)))*w = (((1)/(2)*z)^(nu - 1))/(sqrt(Pi)*GAMMA(nu +(1)/(2)))

D[w, {z, 2}]+Divide[1,z]*D[w, z]-(1 +Divide[\[Nu]^(2),(z)^(2)])*w == Divide[(Divide[1,2]*z)^(\[Nu]- 1),Sqrt[Pi]*Gamma[\[Nu]+Divide[1,2]]]
Failure Failure
Failed [300 / 300]
Result: -2.295139545-.7692147111*I
Test Values: {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: -.2290885595+.1565331804*I
Test Values: {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[-2.2951395445687996, -0.7692147110384474]
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[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-2.0924266927887287, -0.06579228091249201]
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[Ξ½, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}

... skip entries to safe data
11.2.E10 w = 𝐋 Ξ½ ⁑ ( z ) , 𝐌 Ξ½ ⁑ ( z ) 𝑀 modified-Struve-L 𝜈 𝑧 associated-Struve-M 𝜈 𝑧 {\displaystyle{\displaystyle w=\mathbf{L}_{\nu}\left(z\right),\mathbf{M}_{\nu}% \left(z\right)}}
w = \modStruveL{\nu}@{z},\modStruveM{\nu}@{z}		 β„œ ⁑ ( Ξ½ + k + 1 ) > 0 , β„œ ⁑ ( n + Ξ½ + 3 2 ) > 0 formulae-sequence 𝜈 π‘˜ 1 0 𝑛 𝜈 3 2 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re(n+\nu+\tfrac{3}{2})>0}} 
w = StruveL(nu, z); StruveL(nu, z) - BesselI(nu, z)

w == StruveL[\[Nu], z]
 StruveL[\[Nu], z] - BesselI[\[Nu], z]
Failure Failure Error Error
11.2.E11 w = 𝐇 Ξ½ ⁑ ( x ) + A ⁒ J Ξ½ ⁑ ( x ) + B ⁒ Y Ξ½ ⁑ ( x ) 𝑀 Struve-H 𝜈 π‘₯ 𝐴 Bessel-J 𝜈 π‘₯ 𝐡 Bessel-Y-Weber 𝜈 π‘₯ {\displaystyle{\displaystyle w=\mathbf{H}_{\nu}\left(x\right)+AJ_{\nu}\left(x% \right)+BY_{\nu}\left(x\right)}}
w = \StruveH{\nu}@{x}+A\BesselJ{\nu}@{x}+B\BesselY{\nu}@{x}		 β„œ ⁑ ( Ξ½ + k + 1 ) > 0 , β„œ ⁑ ( ( - Ξ½ ) + k + 1 ) > 0 , β„œ ⁑ ( n + Ξ½ + 3 2 ) > 0 formulae-sequence 𝜈 π‘˜ 1 0 formulae-sequence 𝜈 π‘˜ 1 0 𝑛 𝜈 3 2 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0,\Re(n+\nu+\tfrac{% 3}{2})>0}} 
w = StruveH(nu, x)+ A*BesselJ(nu, x)+ B*BesselY(nu, x)

w == StruveH[\[Nu], x]+ A*BesselJ[\[Nu], x]+ B*BesselY[\[Nu], x]
Failure Failure
Failed [300 / 300]
Result: -.568729179e-1+1.004857129*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 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, x = 3/2}


Result: 1.306236381+1.613216681*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 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, x = 1/2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.056872918319905263, 1.0048571288175818]
Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-1.7531990546092198, -1.6096988531229037]
Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[Ξ½, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
11.2.E12 w = 𝐊 Ξ½ ⁑ ( x ) + A ⁒ J Ξ½ ⁑ ( x ) + B ⁒ Y Ξ½ ⁑ ( x ) 𝑀 associated-Struve-K 𝜈 π‘₯ 𝐴 Bessel-J 𝜈 π‘₯ 𝐡 Bessel-Y-Weber 𝜈 π‘₯ {\displaystyle{\displaystyle w=\mathbf{K}_{\nu}\left(x\right)+AJ_{\nu}\left(x% \right)+BY_{\nu}\left(x\right)}}
w = \StruveK{\nu}@{x}+A\BesselJ{\nu}@{x}+B\BesselY{\nu}@{x}		 β„œ ⁑ ( Ξ½ + k + 1 ) > 0 , β„œ ⁑ ( ( - Ξ½ ) + k + 1 ) > 0 , β„œ ⁑ ( n + Ξ½ + 3 2 ) > 0 formulae-sequence 𝜈 π‘˜ 1 0 formulae-sequence 𝜈 π‘˜ 1 0 𝑛 𝜈 3 2 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0,\Re(n+\nu+\tfrac{% 3}{2})>0}} 
w = StruveH(nu, x) - BesselY(nu, x)+ A*BesselJ(nu, x)+ B*BesselY(nu, x)

w == StruveH[\[Nu], x] - BesselY[\[Nu], x]+ A*BesselJ[\[Nu], x]+ B*BesselY[\[Nu], x]
Failure Failure
Failed [300 / 300]
Result: -.4449553305+.6668360043*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 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, x = 3/2}


Result: .1477245032+1.196204678*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 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, x = 1/2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.4449553308212987, 0.6668360040225405]
Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.2518593906559602, -2.1242453536287655]
Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[Ξ½, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
11.2.E13 w = 𝐇 Ξ½ ⁑ ( z ) + A ⁒ J Ξ½ ⁑ ( z ) + B ⁒ H Ξ½ ( 1 ) ⁑ ( z ) 𝑀 Struve-H 𝜈 𝑧 𝐴 Bessel-J 𝜈 𝑧 𝐡 Hankel-H-1-Bessel-third-kind 𝜈 𝑧 {\displaystyle{\displaystyle w=\mathbf{H}_{\nu}\left(z\right)+AJ_{\nu}\left(z% \right)+B{H^{(1)}_{\nu}}\left(z\right)}}
w = \StruveH{\nu}@{z}+A\BesselJ{\nu}@{z}+B\HankelH{1}{\nu}@{z}		 β„œ ⁑ ( Ξ½ + k + 1 ) > 0 , β„œ ⁑ ( n + Ξ½ + 3 2 ) > 0 formulae-sequence 𝜈 π‘˜ 1 0 𝑛 𝜈 3 2 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re(n+\nu+\tfrac{3}{2})>0}} 
w = StruveH(nu, z)+ A*BesselJ(nu, z)+ B*HankelH1(nu, z)

w == StruveH[\[Nu], z]+ A*BesselJ[\[Nu], z]+ B*HankelH1[\[Nu], z]
Failure Failure
Failed [300 / 300]
Result: -.4180841979+.8728935730*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 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: 1.928541044+.4861253769*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 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[-0.4180841980733331, 0.8728935728522607]
Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-2.285405641595042, -1.3320778184897675]
Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, 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
11.2.E14 w = 𝐇 Ξ½ ⁑ ( z ) + A ⁒ J Ξ½ ⁑ ( z ) + B ⁒ H Ξ½ ( 2 ) ⁑ ( z ) 𝑀 Struve-H 𝜈 𝑧 𝐴 Bessel-J 𝜈 𝑧 𝐡 Hankel-H-2-Bessel-third-kind 𝜈 𝑧 {\displaystyle{\displaystyle w=\mathbf{H}_{\nu}\left(z\right)+AJ_{\nu}\left(z% \right)+B{H^{(2)}_{\nu}}\left(z\right)}}
w = \StruveH{\nu}@{z}+A\BesselJ{\nu}@{z}+B\HankelH{2}{\nu}@{z}		 β„œ ⁑ ( Ξ½ + k + 1 ) > 0 , β„œ ⁑ ( n + Ξ½ + 3 2 ) > 0 formulae-sequence 𝜈 π‘˜ 1 0 𝑛 𝜈 3 2 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re(n+\nu+\tfrac{3}{2})>0}} 
w = StruveH(nu, z)+ A*BesselJ(nu, z)+ B*HankelH2(nu, z)

w == StruveH[\[Nu], z]+ A*BesselJ[\[Nu], z]+ B*HankelH2[\[Nu], z]
Failure Failure
Failed [300 / 300]
Result: .1098269700-.5965662020*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 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: .3171413600-.3710144720*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 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[0.109826969919957, -0.5965662019254474]
Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.9889109079558663, -0.015623729667162342]
Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, 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
11.2.E15 w = 𝐊 Ξ½ ⁑ ( z ) + A ⁒ H Ξ½ ( 1 ) ⁑ ( z ) + B ⁒ H Ξ½ ( 2 ) ⁑ ( z ) 𝑀 associated-Struve-K 𝜈 𝑧 𝐴 Hankel-H-1-Bessel-third-kind 𝜈 𝑧 𝐡 Hankel-H-2-Bessel-third-kind 𝜈 𝑧 {\displaystyle{\displaystyle w=\mathbf{K}_{\nu}\left(z\right)+A{H^{(1)}_{\nu}}% \left(z\right)+B{H^{(2)}_{\nu}}\left(z\right)}}
w = \StruveK{\nu}@{z}+A\HankelH{1}{\nu}@{z}+B\HankelH{2}{\nu}@{z}		 β„œ ⁑ ( Ξ½ + k + 1 ) > 0 , β„œ ⁑ ( ( - Ξ½ ) + k + 1 ) > 0 , β„œ ⁑ ( n + Ξ½ + 3 2 ) > 0 formulae-sequence 𝜈 π‘˜ 1 0 formulae-sequence 𝜈 π‘˜ 1 0 𝑛 𝜈 3 2 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0,\Re(n+\nu+\tfrac{% 3}{2})>0}} 
w = StruveH(nu, z) - BesselY(nu, z)+ A*HankelH1(nu, z)+ B*HankelH2(nu, z)

w == StruveH[\[Nu], z] - BesselY[\[Nu], z]+ A*HankelH1[\[Nu], z]+ B*HankelH2[\[Nu], z]
Failure Failure
Failed [300 / 300]
Result: -.9224011534+.2769363875*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 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: 1.154538681+.9695969456*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 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[-0.9224011534734378, 0.27693638794598185]
Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-1.3912406162671118, -1.5643629838862487]
Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, 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
11.2.E16 w = 𝐋 Ξ½ ⁑ ( z ) + A ⁒ K Ξ½ ⁑ ( z ) + B ⁒ I Ξ½ ⁑ ( z ) 𝑀 modified-Struve-L 𝜈 𝑧 𝐴 modified-Bessel-second-kind 𝜈 𝑧 𝐡 modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle w=\mathbf{L}_{\nu}\left(z\right)+AK_{\nu}\left(z% \right)+BI_{\nu}\left(z\right)}}
w = \modStruveL{\nu}@{z}+A\modBesselK{\nu}@{z}+B\modBesselI{\nu}@{z}		 β„œ ⁑ ( Ξ½ + k + 1 ) > 0 , β„œ ⁑ ( n + Ξ½ + 3 2 ) > 0 formulae-sequence 𝜈 π‘˜ 1 0 𝑛 𝜈 3 2 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re(n+\nu+\tfrac{3}{2})>0}} 
w = StruveL(nu, z)+ A*BesselK(nu, z)+ B*BesselI(nu, z)

w == StruveL[\[Nu], z]+ A*BesselK[\[Nu], z]+ B*BesselI[\[Nu], z]
Failure Failure
Failed [300 / 300]
Result: -.4427134717+.1412701443*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 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: .8499113341+3.412421345*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 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[-0.4427134718200613, 0.1412701442672558]
Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-1.8647663358395983, -0.37009195882490975]
Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, 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
11.2.E17 w = 𝐌 Ξ½ ⁑ ( z ) + A ⁒ K Ξ½ ⁑ ( z ) + B ⁒ I Ξ½ ⁑ ( z ) 𝑀 associated-Struve-M 𝜈 𝑧 𝐴 modified-Bessel-second-kind 𝜈 𝑧 𝐡 modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle w=\mathbf{M}_{\nu}\left(z\right)+AK_{\nu}\left(z% \right)+BI_{\nu}\left(z\right)}}
w = \modStruveM{\nu}@{z}+A\modBesselK{\nu}@{z}+B\modBesselI{\nu}@{z}		 β„œ ⁑ ( Ξ½ + k + 1 ) > 0 , β„œ ⁑ ( n + Ξ½ + 3 2 ) > 0 formulae-sequence 𝜈 π‘˜ 1 0 𝑛 𝜈 3 2 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re(n+\nu+\tfrac{3}{2})>0}} 
w = StruveL(nu, z) - BesselI(nu, z)+ A*BesselK(nu, z)+ B*BesselI(nu, z)

w == StruveL[\[Nu], z] - BesselI[\[Nu], z]+ A*BesselK[\[Nu], z]+ B*BesselI[\[Nu], z]
Failure Failure
Failed [300 / 300]
Result: .876284277e-1+.1517241441*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 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: .9234962821+3.599925727*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 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[0.08762842754807953, 0.15172414402816306]
Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.09828151494898707, -0.22324970290386212]
Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, 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
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