Results of Struve and Related Functions

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DLMF Formula 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)=((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}] Successful Successful - -
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)}} 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
Fail
-.9119368045e-1-212.1906008*I <- {nu = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
23.48378801+14.39748233*I <- {nu = 2^(1/2)-I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
-183.2849891-97.24822718*I <- {nu = -2^(1/2)-I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
746.7960427+104.9552444*I <- {nu = -2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[-0.09119395476353123, -212.19060110903138] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ½, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[23.48378803135987, 14.397482376436031] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ½, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-183.28498965036638, -97.24822703421293] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ½, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[746.7960436976847, 104.95524615581924] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ½, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
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)}}} - 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}] Failure Failure Skip
Fail
Complex[0.09119395476353123, 212.19060110903138] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ½, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-23.48378803135987, -14.397482376436031] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ½, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[183.28498965036638, 97.24822703421293] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ½, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-746.7960436976847, -104.95524615581924] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ½, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
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)}} 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 - -
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)}} 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 - -
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)}}} 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
Fail
-.3292528958+.1261718930*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
1.117431897+5.592974705*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
2.886594176+29.83865170*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
2.791726027+2.817686588*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[-0.3292528958392771, 0.1261718930121784] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ½, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.1174318983838325, 0.0638795414755875] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ½, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-9.829307268242317, -7.250178123857234] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ½, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[3.2003851737933693, 1.557087772272263] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ½, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... 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), StruveH(nu, z) - BesselY(nu, z) w = StruveH[\[Nu], z], StruveH[\[Nu], z] - BesselY[\[Nu], z] Error Failure - 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)}}} 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
Fail
-3.157680020-2.702255231*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
-1.710995227+2.764547581*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
.58167052e-1+27.01022458*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
-.3670109651e-1-.1074053566e-1*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[-3.1576800205854676, -2.702255231734012] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ½, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-1.7109952263623578, -2.764547583270603] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ½, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-12.657734392988507, -10.078605248603424] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ½, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.3719580490471791, -1.2713393524739274] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ½, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... 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 = StruveL(nu, z), StruveL(nu, z) - BesselI(nu, z) w = StruveL[\[Nu], z], StruveL[\[Nu], z] - BesselI[\[Nu], z] Error Failure - 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) w = StruveH[\[Nu], x]+ A*BesselJ[\[Nu], x]+ B*BesselY[\[Nu], x] Failure Failure
Fail
1.813177648+3.357588143*I <- {A = 2^(1/2)+I*2^(1/2), B = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), x = 1}
-.3533489224+4.243660941*I <- {A = 2^(1/2)+I*2^(1/2), B = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), x = 2}
-3.044392825+1.991205432*I <- {A = 2^(1/2)+I*2^(1/2), B = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), x = 3}
1.813177648+.5291610183*I <- {A = 2^(1/2)+I*2^(1/2), B = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)-I*2^(1/2), x = 1}
... skip entries to safe data
 Skip
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 = 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
Fail
1.240425455+3.227234897*I <- {A = 2^(1/2)+I*2^(1/2), B = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), x = 1}
-1.194165497+3.657602257*I <- {A = 2^(1/2)+I*2^(1/2), B = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), x = 2}
-3.012593980+.934563193*I <- {A = 2^(1/2)+I*2^(1/2), B = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), x = 3}
1.240425455+.3988077733*I <- {A = 2^(1/2)+I*2^(1/2), B = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)-I*2^(1/2), x = 1}
... skip entries to safe data
 Skip
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*HankelH1(nu, z) w = StruveH[\[Nu], z]+ A*BesselJ[\[Nu], z]+ B*HankelH1[\[Nu], z] Failure Failure Skip Skip
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*HankelH2(nu, z) w = StruveH[\[Nu], z]+ A*BesselJ[\[Nu], z]+ B*HankelH2[\[Nu], z] Failure Failure Skip Skip
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 = 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 Skip Skip
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 = 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 Skip Skip
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 = 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 Skip Skip
11.4.E1 𝐊 n + 1 2 ⁑ ( z ) = ( 2 Ο€ ⁒ z ) 1 2 ⁒ βˆ‘ m = 0 n ( 2 ⁒ m ) ! ⁒  2 - 2 ⁒ m m ! ⁒ ( n - m ) ! ⁒ ( 1 2 ⁒ z ) n - 2 ⁒ m associated-Struve-K 𝑛 1 2 𝑧 superscript 2 πœ‹ 𝑧 1 2 superscript subscript π‘š 0 𝑛 2 π‘š superscript  2 2 π‘š π‘š 𝑛 π‘š superscript 1 2 𝑧 𝑛 2 π‘š {\displaystyle{\displaystyle\mathbf{K}_{n+\frac{1}{2}}\left(z\right)=\left(% \frac{2}{\pi z}\right)^{\frac{1}{2}}\sum_{m=0}^{n}\frac{(2m)!\,2^{-2m}}{m!\,(n% -m)!}\,(\tfrac{1}{2}z)^{n-2m}}} StruveH(n +(1)/(2), z) - BesselY(n +(1)/(2), z)=((2)/(Pi*z))^((1)/(2))* sum((factorial(2*m)*(2)^(- 2*m))/(factorial(m)*factorial(n - m))*((1)/(2)*z)^(n - 2*m), m = 0..n) StruveH[n +Divide[1,2], z] - BesselY[n +Divide[1,2], z]=(Divide[2,Pi*z])^(Divide[1,2])* Sum[Divide[(2*m)!*(2)^(- 2*m),(m)!*(n - m)!]*(Divide[1,2]*z)^(n - 2*m), {m, 0, n}] Failure Failure Skip Successful
11.4.E2 𝐋 n + 1 2 ⁑ ( z ) = I - n - 1 2 ⁑ ( z ) - ( 2 Ο€ ⁒ z ) 1 2 ⁒ βˆ‘ m = 0 n ( - 1 ) m ⁒ ( 2 ⁒ m ) ! ⁒  2 - 2 ⁒ m m ! ⁒ ( n - m ) ! ⁒ ( 1 2 ⁒ z ) n - 2 ⁒ m modified-Struve-L 𝑛 1 2 𝑧 modified-Bessel-first-kind 𝑛 1 2 𝑧 superscript 2 πœ‹ 𝑧 1 2 superscript subscript π‘š 0 𝑛 superscript 1 π‘š 2 π‘š superscript  2 2 π‘š π‘š 𝑛 π‘š superscript 1 2 𝑧 𝑛 2 π‘š {\displaystyle{\displaystyle\mathbf{L}_{n+\frac{1}{2}}\left(z\right)=I_{-n-% \frac{1}{2}}\left(z\right)-\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}\sum_{m=0% }^{n}\frac{(-1)^{m}(2m)!\,2^{-2m}}{m!\,(n-m)!}\,(\tfrac{1}{2}z)^{n-2m}}} StruveL(n +(1)/(2), z)= BesselI(- n -(1)/(2), z)-((2)/(Pi*z))^((1)/(2))* sum(((- 1)^(m)*factorial(2*m)*(2)^(- 2*m))/(factorial(m)*factorial(n - m))*((1)/(2)*z)^(n - 2*m), m = 0..n) StruveL[n +Divide[1,2], z]= BesselI[- n -Divide[1,2], z]-(Divide[2,Pi*z])^(Divide[1,2])* Sum[Divide[(- 1)^(m)*(2*m)!*(2)^(- 2*m),(m)!*(n - m)!]*(Divide[1,2]*z)^(n - 2*m), {m, 0, n}] Failure Failure Skip Successful
11.4.E3 𝐇 - n - 1 2 ⁑ ( z ) = ( - 1 ) n ⁒ J n + 1 2 ⁑ ( z ) Struve-H 𝑛 1 2 𝑧 superscript 1 𝑛 Bessel-J 𝑛 1 2 𝑧 {\displaystyle{\displaystyle\mathbf{H}_{-n-\frac{1}{2}}\left(z\right)=(-1)^{n}% J_{n+\frac{1}{2}}\left(z\right)}} StruveH(- n -(1)/(2), z)=(- 1)^(n)* BesselJ(n +(1)/(2), z) StruveH[- n -Divide[1,2], z]=(- 1)^(n)* BesselJ[n +Divide[1,2], z] Failure Failure Successful Successful
11.4.E4 𝐋 - n - 1 2 ⁑ ( z ) = I n + 1 2 ⁑ ( z ) modified-Struve-L 𝑛 1 2 𝑧 modified-Bessel-first-kind 𝑛 1 2 𝑧 {\displaystyle{\displaystyle\mathbf{L}_{-n-\frac{1}{2}}\left(z\right)=I_{n+% \frac{1}{2}}\left(z\right)}} StruveL(- n -(1)/(2), z)= BesselI(n +(1)/(2), z) StruveL[- n -Divide[1,2], z]= BesselI[n +Divide[1,2], z] Failure Failure Successful Successful
11.4.E5 𝐇 1 2 ⁑ ( z ) = ( 2 Ο€ ⁒ z ) 1 2 ⁒ ( 1 - cos ⁑ z ) Struve-H 1 2 𝑧 superscript 2 πœ‹ 𝑧 1 2 1 𝑧 {\displaystyle{\displaystyle\mathbf{H}_{\frac{1}{2}}\left(z\right)=\left(\frac% {2}{\pi z}\right)^{\frac{1}{2}}(1-\cos z)}} StruveH((1)/(2), z)=((2)/(Pi*z))^((1)/(2))*(1 - cos(z)) StruveH[Divide[1,2], z]=(Divide[2,Pi*z])^(Divide[1,2])*(1 - Cos[z]) Failure Failure Successful Successful
11.4.E6 𝐇 - 1 2 ⁑ ( z ) = ( 2 Ο€ ⁒ z ) 1 2 ⁒ sin ⁑ z Struve-H 1 2 𝑧 superscript 2 πœ‹ 𝑧 1 2 𝑧 {\displaystyle{\displaystyle\mathbf{H}_{-\frac{1}{2}}\left(z\right)=\left(% \frac{2}{\pi z}\right)^{\frac{1}{2}}\sin z}} StruveH(-(1)/(2), z)=((2)/(Pi*z))^((1)/(2))* sin(z) StruveH[-Divide[1,2], z]=(Divide[2,Pi*z])^(Divide[1,2])* Sin[z] Failure Failure Successful Successful
11.4.E7 𝐋 1 2 ⁑ ( z ) = ( 2 Ο€ ⁒ z ) 1 2 ⁒ ( cosh ⁑ z - 1 ) modified-Struve-L 1 2 𝑧 superscript 2 πœ‹ 𝑧 1 2 𝑧 1 {\displaystyle{\displaystyle\mathbf{L}_{\frac{1}{2}}\left(z\right)=\left(\frac% {2}{\pi z}\right)^{\frac{1}{2}}(\cosh z-1)}} StruveL((1)/(2), z)=((2)/(Pi*z))^((1)/(2))*(cosh(z)- 1) StruveL[Divide[1,2], z]=(Divide[2,Pi*z])^(Divide[1,2])*(Cosh[z]- 1) Failure Failure Successful Successful
11.4.E8 𝐋 - 1 2 ⁑ ( z ) = ( 2 Ο€ ⁒ z ) 1 2 ⁒ sinh ⁑ z modified-Struve-L 1 2 𝑧 superscript 2 πœ‹ 𝑧 1 2 𝑧 {\displaystyle{\displaystyle\mathbf{L}_{-\frac{1}{2}}\left(z\right)=\left(% \frac{2}{\pi z}\right)^{\frac{1}{2}}\sinh z}} StruveL(-(1)/(2), z)=((2)/(Pi*z))^((1)/(2))* sinh(z) StruveL[-Divide[1,2], z]=(Divide[2,Pi*z])^(Divide[1,2])* Sinh[z] Failure Failure Successful Successful
11.4.E9 𝐇 3 2 ⁑ ( z ) = ( z 2 ⁒ Ο€ ) 1 2 ⁒ ( 1 + 2 z 2 ) - ( 2 Ο€ ⁒ z ) 1 2 ⁒ ( sin ⁑ z + cos ⁑ z z ) Struve-H 3 2 𝑧 superscript 𝑧 2 πœ‹ 1 2 1 2 superscript 𝑧 2 superscript 2 πœ‹ 𝑧 1 2 𝑧 𝑧 𝑧 {\displaystyle{\displaystyle\mathbf{H}_{\frac{3}{2}}\left(z\right)=\left(\frac% {z}{2\pi}\right)^{\frac{1}{2}}\left(1+\frac{2}{z^{2}}\right)-\left(\frac{2}{% \pi z}\right)^{\frac{1}{2}}\left(\sin z+\frac{\cos z}{z}\right)}} StruveH((3)/(2), z)=((z)/(2*Pi))^((1)/(2))*(1 +(2)/((z)^(2)))-((2)/(Pi*z))^((1)/(2))*(sin(z)+(cos(z))/(z)) StruveH[Divide[3,2], z]=(Divide[z,2*Pi])^(Divide[1,2])*(1 +Divide[2,(z)^(2)])-(Divide[2,Pi*z])^(Divide[1,2])*(Sin[z]+Divide[Cos[z],z]) Failure Failure Successful Successful
11.4.E10 𝐇 - 3 2 ⁑ ( z ) = ( 2 Ο€ ⁒ z ) 1 2 ⁒ ( cos ⁑ z - sin ⁑ z z ) Struve-H 3 2 𝑧 superscript 2 πœ‹ 𝑧 1 2 𝑧 𝑧 𝑧 {\displaystyle{\displaystyle\mathbf{H}_{-\frac{3}{2}}\left(z\right)=\left(% \frac{2}{\pi z}\right)^{\frac{1}{2}}\left(\cos z-\frac{\sin z}{z}\right)}} StruveH(-(3)/(2), z)=((2)/(Pi*z))^((1)/(2))*(cos(z)-(sin(z))/(z)) StruveH[-Divide[3,2], z]=(Divide[2,Pi*z])^(Divide[1,2])*(Cos[z]-Divide[Sin[z],z]) Failure Failure Successful Successful
11.4.E11 𝐋 3 2 ⁑ ( z ) = - ( z 2 ⁒ Ο€ ) 1 2 ⁒ ( 1 - 2 z 2 ) + ( 2 Ο€ ⁒ z ) 1 2 ⁒ ( sinh ⁑ z - cosh ⁑ z z ) modified-Struve-L 3 2 𝑧 superscript 𝑧 2 πœ‹ 1 2 1 2 superscript 𝑧 2 superscript 2 πœ‹ 𝑧 1 2 𝑧 𝑧 𝑧 {\displaystyle{\displaystyle\mathbf{L}_{\frac{3}{2}}\left(z\right)=-\left(% \frac{z}{2\pi}\right)^{\frac{1}{2}}\left(1-\frac{2}{z^{2}}\right)+\left(\frac{% 2}{\pi z}\right)^{\frac{1}{2}}\left(\sinh z-\frac{\cosh z}{z}\right)}} StruveL((3)/(2), z)= -((z)/(2*Pi))^((1)/(2))*(1 -(2)/((z)^(2)))+((2)/(Pi*z))^((1)/(2))*(sinh(z)-(cosh(z))/(z)) StruveL[Divide[3,2], z]= -(Divide[z,2*Pi])^(Divide[1,2])*(1 -Divide[2,(z)^(2)])+(Divide[2,Pi*z])^(Divide[1,2])*(Sinh[z]-Divide[Cosh[z],z]) Failure Failure Successful Successful
11.4.E12 𝐋 - 3 2 ⁑ ( z ) = ( 2 Ο€ ⁒ z ) 1 2 ⁒ ( cosh ⁑ z - sinh ⁑ z z ) modified-Struve-L 3 2 𝑧 superscript 2 πœ‹ 𝑧 1 2 𝑧 𝑧 𝑧 {\displaystyle{\displaystyle\mathbf{L}_{-\frac{3}{2}}\left(z\right)=\left(% \frac{2}{\pi z}\right)^{\frac{1}{2}}\left(\cosh z-\frac{\sinh z}{z}\right)}} StruveL(-(3)/(2), z)=((2)/(Pi*z))^((1)/(2))*(cosh(z)-(sinh(z))/(z)) StruveL[-Divide[3,2], z]=(Divide[2,Pi*z])^(Divide[1,2])*(Cosh[z]-Divide[Sinh[z],z]) Failure Failure Successful Successful
11.4.E13 𝐇 Ξ½ ⁑ ( x ) β‰₯ 0 Struve-H 𝜈 π‘₯ 0 {\displaystyle{\displaystyle\mathbf{H}_{\nu}\left(x\right)>=0}} StruveH(nu, x)> = 0 StruveH[\[Nu], x]> = 0 Failure Failure Skip Successful
11.4.E14 𝐇 Ξ½ ⁑ ( z ) = 2 ⁒ ( 1 2 ⁒ z ) Ξ½ + 1 Ο€ ⁒ Ξ“ ⁑ ( Ξ½ + 3 2 ) ⁒ ( 1 + Ο‘ ) Struve-H 𝜈 𝑧 2 superscript 1 2 𝑧 𝜈 1 πœ‹ Euler-Gamma 𝜈 3 2 1 italic-Ο‘ {\displaystyle{\displaystyle\mathbf{H}_{\nu}\left(z\right)=\frac{2(\tfrac{1}{2% }z)^{\nu+1}}{\sqrt{\pi}\Gamma\left(\nu+\tfrac{3}{2}\right)}(1+\vartheta)}} StruveH(nu, z)=(2*((1)/(2)*z)^(nu + 1))/(sqrt(Pi)*GAMMA(nu +(3)/(2)))*(1 + vartheta) StruveH[\[Nu], z]=Divide[2*(Divide[1,2]*z)^(\[Nu]+ 1),Sqrt[Pi]*Gamma[\[Nu]+Divide[3,2]]]*(1 + \[CurlyTheta]) Failure Failure
Fail
-.1259323815-.6376916086*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), vartheta = 2^(1/2)+I*2^(1/2)}
-.5771716954+.6854347047e-1*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), vartheta = 2^(1/2)-I*2^(1/2)}
.1290633837+.5197827843*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), vartheta = -2^(1/2)-I*2^(1/2)}
.5803026975-.1864522947*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), vartheta = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Skip
11.4.E15 | Ο‘ | < 2 3 ⁒ exp ⁑ ( 1 4 ⁒ | z | 2 | Ξ½ 0 + 3 2 | - 1 ) italic-Ο‘ 2 3 1 4 superscript 𝑧 2 subscript 𝜈 0 3 2 1 {\displaystyle{\displaystyle|\vartheta|<\frac{2}{3}\exp\left(\frac{\tfrac{1}{4% }|z|^{2}}{|\nu_{0}+\tfrac{3}{2}|}-1\right)}} abs(vartheta)<(2)/(3)*exp(((1)/(4)*(abs(z))^(2))/(abs(nu[0]+(3)/(2)))- 1) Abs[\[CurlyTheta]]<Divide[2,3]*Exp[Divide[Divide[1,4]*(Abs[z])^(2),Abs[Subscript[\[Nu], 0]+Divide[3,2]]]- 1] Failure Failure
Fail
2. < .3339546072 <- {z = 2^(1/2)+I*2^(1/2), vartheta = 2^(1/2)+I*2^(1/2), nu[0] = 2^(1/2)+I*2^(1/2)}
2. < .3339546072 <- {z = 2^(1/2)+I*2^(1/2), vartheta = 2^(1/2)+I*2^(1/2), nu[0] = 2^(1/2)-I*2^(1/2)}
2. < .4967563064 <- {z = 2^(1/2)+I*2^(1/2), vartheta = 2^(1/2)+I*2^(1/2), nu[0] = -2^(1/2)-I*2^(1/2)}
2. < .4967563064 <- {z = 2^(1/2)+I*2^(1/2), vartheta = 2^(1/2)+I*2^(1/2), nu[0] = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Skip
11.4.E16 𝐇 Ξ½ ⁑ ( z ⁒ e m ⁒ Ο€ ⁒ i ) = e m ⁒ Ο€ ⁒ i ⁒ ( Ξ½ + 1 ) ⁒ 𝐇 Ξ½ ⁑ ( z ) Struve-H 𝜈 𝑧 superscript 𝑒 π‘š πœ‹ 𝑖 superscript 𝑒 π‘š πœ‹ 𝑖 𝜈 1 Struve-H 𝜈 𝑧 {\displaystyle{\displaystyle\mathbf{H}_{\nu}\left(ze^{m\pi i}\right)=e^{m\pi i% (\nu+1)}\mathbf{H}_{\nu}\left(z\right)}} StruveH(nu, z*exp(m*Pi*I))= exp(m*Pi*I*(nu + 1))*StruveH(nu, z) StruveH[\[Nu], z*Exp[m*Pi*I]]= Exp[m*Pi*I*(\[Nu]+ 1)]*StruveH[\[Nu], z] Failure Failure
Fail
13.93052781-18.31382455*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 3}
-.5904691739e-2-.3463835767e-1*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), m = 3}
.2512942813+.1005768782*I <- {nu = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2), m = 3}
-2.971927621-.3003885072*I <- {nu = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2), m = 3}
... skip entries to safe data
 Error
11.4.E17 𝐋 Ξ½ ⁑ ( z ⁒ e m ⁒ Ο€ ⁒ i ) = e m ⁒ Ο€ ⁒ i ⁒ ( Ξ½ + 1 ) ⁒ 𝐋 Ξ½ ⁑ ( z ) modified-Struve-L 𝜈 𝑧 superscript 𝑒 π‘š πœ‹ 𝑖 superscript 𝑒 π‘š πœ‹ 𝑖 𝜈 1 modified-Struve-L 𝜈 𝑧 {\displaystyle{\displaystyle\mathbf{L}_{\nu}\left(ze^{m\pi i}\right)=e^{m\pi i% (\nu+1)}\mathbf{L}_{\nu}\left(z\right)}} StruveL(nu, z*exp(m*Pi*I))= exp(m*Pi*I*(nu + 1))*StruveL(nu, z) StruveL[\[Nu], z*Exp[m*Pi*I]]= Exp[m*Pi*I*(\[Nu]+ 1)]*StruveL[\[Nu], z] Failure Failure
Fail
23.48202217-14.39410521*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 3}
-.1507858067e-1-.2518674877e-1*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), m = 3}
.2367740036+.2211567113*I <- {nu = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2), m = 3}
-2.405166395+.6652992785*I <- {nu = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2), m = 3}
... skip entries to safe data
 Error
11.4.E18 𝐇 Ξ½ ⁑ ( z ) = 4 Ο€ 1 / 2 ⁒ Ξ“ ⁑ ( Ξ½ + 1 2 ) ⁒ βˆ‘ k = 0 ∞ ( 2 ⁒ k + Ξ½ + 1 ) ⁒ Ξ“ ⁑ ( k + Ξ½ + 1 ) k ! ⁒ ( 2 ⁒ k + 1 ) ⁒ ( 2 ⁒ k + 2 ⁒ Ξ½ + 1 ) ⁒ J 2 ⁒ k + Ξ½ + 1 ⁑ ( z ) Struve-H 𝜈 𝑧 4 superscript πœ‹ 1 2 Euler-Gamma 𝜈 1 2 superscript subscript π‘˜ 0 2 π‘˜ 𝜈 1 Euler-Gamma π‘˜ 𝜈 1 π‘˜ 2 π‘˜ 1 2 π‘˜ 2 𝜈 1 Bessel-J 2 π‘˜ 𝜈 1 𝑧 {\displaystyle{\displaystyle\mathbf{H}_{\nu}\left(z\right)=\frac{4}{\pi^{1/2}% \Gamma\left(\nu+\tfrac{1}{2}\right)}\*\sum_{k=0}^{\infty}\frac{(2k+\nu+1)% \Gamma\left(k+\nu+1\right)}{k!(2k+1)(2k+2\nu+1)}J_{2k+\nu+1}\left(z\right)}} StruveH(nu, z)=(4)/((Pi)^(1/ 2)* GAMMA(nu +(1)/(2)))* sum(((2*k + nu + 1)* GAMMA(k + nu + 1))/(factorial(k)*(2*k + 1)*(2*k + 2*nu + 1))*BesselJ(2*k + nu + 1, z), k = 0..infinity) StruveH[\[Nu], z]=Divide[4,(Pi)^(1/ 2)* Gamma[\[Nu]+Divide[1,2]]]* Sum[Divide[(2*k + \[Nu]+ 1)* Gamma[k + \[Nu]+ 1],(k)!*(2*k + 1)*(2*k + 2*\[Nu]+ 1)]*BesselJ[2*k + \[Nu]+ 1, z], {k, 0, Infinity}] Failure Failure Skip Error
11.4.E19 𝐇 Ξ½ ⁑ ( z ) = ( z 2 ⁒ Ο€ ) 1 / 2 ⁒ βˆ‘ k = 0 ∞ ( 1 2 ⁒ z ) k k ! ⁒ ( k + 1 2 ) ⁒ J k + Ξ½ + 1 2 ⁑ ( z ) Struve-H 𝜈 𝑧 superscript 𝑧 2 πœ‹ 1 2 superscript subscript π‘˜ 0 superscript 1 2 𝑧 π‘˜ π‘˜ π‘˜ 1 2 Bessel-J π‘˜ 𝜈 1 2 𝑧 {\displaystyle{\displaystyle\mathbf{H}_{\nu}\left(z\right)=\left(\frac{z}{2\pi% }\right)^{1/2}\sum_{k=0}^{\infty}\frac{(\tfrac{1}{2}z)^{k}}{k!(k+\tfrac{1}{2})% }J_{k+\nu+\frac{1}{2}}\left(z\right)}} StruveH(nu, z)=((z)/(2*Pi))^(1/ 2)* sum((((1)/(2)*z)^(k))/(factorial(k)*(k +(1)/(2)))*BesselJ(k + nu +(1)/(2), z), k = 0..infinity) StruveH[\[Nu], z]=(Divide[z,2*Pi])^(1/ 2)* Sum[Divide[(Divide[1,2]*z)^(k),(k)!*(k +Divide[1,2])]*BesselJ[k + \[Nu]+Divide[1,2], z], {k, 0, Infinity}] Failure Failure Skip Error
11.4.E20 𝐇 Ξ½ ⁑ ( z ) = ( 1 2 ⁒ z ) Ξ½ + 1 2 Ξ“ ⁑ ( Ξ½ + 1 2 ) ⁒ βˆ‘ k = 0 ∞ ( 1 2 ⁒ z ) k k ! ⁒ ( k + Ξ½ + 1 2 ) ⁒ J k + 1 2 ⁑ ( z ) Struve-H 𝜈 𝑧 superscript 1 2 𝑧 𝜈 1 2 Euler-Gamma 𝜈 1 2 superscript subscript π‘˜ 0 superscript 1 2 𝑧 π‘˜ π‘˜ π‘˜ 𝜈 1 2 Bessel-J π‘˜ 1 2 𝑧 {\displaystyle{\displaystyle\mathbf{H}_{\nu}\left(z\right)=\frac{(\tfrac{1}{2}% z)^{\nu+\frac{1}{2}}}{\Gamma\left(\nu+\tfrac{1}{2}\right)}\sum_{k=0}^{\infty}% \frac{(\tfrac{1}{2}z)^{k}}{k!(k+\nu+\tfrac{1}{2})}J_{k+\frac{1}{2}}\left(z% \right)}} StruveH(nu, z)=(((1)/(2)*z)^(nu +(1)/(2)))/(GAMMA(nu +(1)/(2)))*sum((((1)/(2)*z)^(k))/(factorial(k)*(k + nu +(1)/(2)))*BesselJ(k +(1)/(2), z), k = 0..infinity) StruveH[\[Nu], z]=Divide[(Divide[1,2]*z)^(\[Nu]+Divide[1,2]),Gamma[\[Nu]+Divide[1,2]]]*Sum[Divide[(Divide[1,2]*z)^(k),(k)!*(k + \[Nu]+Divide[1,2])]*BesselJ[k +Divide[1,2], z], {k, 0, Infinity}] Failure Failure Skip Error
11.4.E21 𝐇 0 ⁑ ( z ) = 4 Ο€ ⁒ βˆ‘ k = 0 ∞ J 2 ⁒ k + 1 ⁑ ( z ) 2 ⁒ k + 1 Struve-H 0 𝑧 4 πœ‹ superscript subscript π‘˜ 0 Bessel-J 2 π‘˜ 1 𝑧 2 π‘˜ 1 {\displaystyle{\displaystyle\mathbf{H}_{0}\left(z\right)=\frac{4}{\pi}\sum_{k=% 0}^{\infty}\frac{J_{2k+1}\left(z\right)}{2k+1}}} StruveH(0, z)=(4)/(Pi)*sum((BesselJ(2*k + 1, z))/(2*k + 1), k = 0..infinity) StruveH[0, z]=Divide[4,Pi]*Sum[Divide[BesselJ[2*k + 1, z],2*k + 1], {k, 0, Infinity}] Failure Successful Skip -
11.4.E21 4 Ο€ ⁒ βˆ‘ k = 0 ∞ J 2 ⁒ k + 1 ⁑ ( z ) 2 ⁒ k + 1 = 2 ⁒ βˆ‘ k = 0 ∞ ( - 1 ) k ⁒ J k + 1 2 2 ⁑ ( 1 2 ⁒ z ) 4 πœ‹ superscript subscript π‘˜ 0 Bessel-J 2 π‘˜ 1 𝑧 2 π‘˜ 1 2 superscript subscript π‘˜ 0 superscript 1 π‘˜ Bessel-J π‘˜ 1 2 2 1 2 𝑧 {\displaystyle{\displaystyle\frac{4}{\pi}\sum_{k=0}^{\infty}\frac{J_{2k+1}% \left(z\right)}{2k+1}=2\sum_{k=0}^{\infty}(-1)^{k}{J_{k+\frac{1}{2}}^{2}}\left% (\tfrac{1}{2}z\right)}} (4)/(Pi)*sum((BesselJ(2*k + 1, z))/(2*k + 1), k = 0..infinity)= 2*sum((- 1)^(k)* (BesselJ(k +(1)/(2), (1)/(2)*z))^(2), k = 0..infinity) Divide[4,Pi]*Sum[Divide[BesselJ[2*k + 1, z],2*k + 1], {k, 0, Infinity}]= 2*Sum[(- 1)^(k)* (BesselJ[k +Divide[1,2], Divide[1,2]*z])^(2), {k, 0, Infinity}] Failure Failure Skip Error
11.4.E22 𝐇 1 ⁑ ( z ) = 2 Ο€ ⁒ ( 1 - J 0 ⁑ ( z ) ) + 4 Ο€ ⁒ βˆ‘ k = 1 ∞ J 2 ⁒ k ⁑ ( z ) 4 ⁒ k 2 - 1 Struve-H 1 𝑧 2 πœ‹ 1 Bessel-J 0 𝑧 4 πœ‹ superscript subscript π‘˜ 1 Bessel-J 2 π‘˜ 𝑧 4 superscript π‘˜ 2 1 {\displaystyle{\displaystyle\mathbf{H}_{1}\left(z\right)=\frac{2}{\pi}(1-J_{0}% \left(z\right))+\frac{4}{\pi}\sum_{k=1}^{\infty}\frac{J_{2k}\left(z\right)}{4k% ^{2}-1}}} StruveH(1, z)=(2)/(Pi)*(1 - BesselJ(0, z))+(4)/(Pi)*sum((BesselJ(2*k, z))/(4*(k)^(2)- 1), k = 1..infinity) StruveH[1, z]=Divide[2,Pi]*(1 - BesselJ[0, z])+Divide[4,Pi]*Sum[Divide[BesselJ[2*k, z],4*(k)^(2)- 1], {k, 1, Infinity}] Failure Successful Skip -
11.4.E22 2 Ο€ ⁒ ( 1 - J 0 ⁑ ( z ) ) + 4 Ο€ ⁒ βˆ‘ k = 1 ∞ J 2 ⁒ k ⁑ ( z ) 4 ⁒ k 2 - 1 = 4 ⁒ βˆ‘ k = 0 ∞ J 2 ⁒ k + 1 2 ⁑ ( 1 2 ⁒ z ) ⁒ J 2 ⁒ k + 3 2 ⁑ ( 1 2 ⁒ z ) 2 πœ‹ 1 Bessel-J 0 𝑧 4 πœ‹ superscript subscript π‘˜ 1 Bessel-J 2 π‘˜ 𝑧 4 superscript π‘˜ 2 1 4 superscript subscript π‘˜ 0 Bessel-J 2 π‘˜ 1 2 1 2 𝑧 Bessel-J 2 π‘˜ 3 2 1 2 𝑧 {\displaystyle{\displaystyle\frac{2}{\pi}(1-J_{0}\left(z\right))+\frac{4}{\pi}% \sum_{k=1}^{\infty}\frac{J_{2k}\left(z\right)}{4k^{2}-1}=4\sum_{k=0}^{\infty}J% _{2k+\frac{1}{2}}\left(\tfrac{1}{2}z\right)J_{2k+\frac{3}{2}}\left(\tfrac{1}{2% }z\right)}} (2)/(Pi)*(1 - BesselJ(0, z))+(4)/(Pi)*sum((BesselJ(2*k, z))/(4*(k)^(2)- 1), k = 1..infinity)= 4*sum(BesselJ(2*k +(1)/(2), (1)/(2)*z)*BesselJ(2*k +(3)/(2), (1)/(2)*z), k = 0..infinity) Divide[2,Pi]*(1 - BesselJ[0, z])+Divide[4,Pi]*Sum[Divide[BesselJ[2*k, z],4*(k)^(2)- 1], {k, 1, Infinity}]= 4*Sum[BesselJ[2*k +Divide[1,2], Divide[1,2]*z]*BesselJ[2*k +Divide[3,2], Divide[1,2]*z], {k, 0, Infinity}] Failure Failure Skip Error
11.4.E23 𝐇 Ξ½ - 1 ⁑ ( z ) + 𝐇 Ξ½ + 1 ⁑ ( z ) = 2 ⁒ Ξ½ z ⁒ 𝐇 Ξ½ ⁑ ( z ) + ( 1 2 ⁒ z ) Ξ½ Ο€ ⁒ Ξ“ ⁑ ( Ξ½ + 3 2 ) Struve-H 𝜈 1 𝑧 Struve-H 𝜈 1 𝑧 2 𝜈 𝑧 Struve-H 𝜈 𝑧 superscript 1 2 𝑧 𝜈 πœ‹ Euler-Gamma 𝜈 3 2 {\displaystyle{\displaystyle\mathbf{H}_{\nu-1}\left(z\right)+\mathbf{H}_{\nu+1% }\left(z\right)=\frac{2\nu}{z}\mathbf{H}_{\nu}\left(z\right)+\frac{(\tfrac{1}{% 2}z)^{\nu}}{\sqrt{\pi}\Gamma\left(\nu+\tfrac{3}{2}\right)}}} StruveH(nu - 1, z)+ StruveH(nu + 1, z)=(2*nu)/(z)*StruveH(nu, z)+(((1)/(2)*z)^(nu))/(sqrt(Pi)*GAMMA(nu +(3)/(2))) StruveH[\[Nu]- 1, z]+ StruveH[\[Nu]+ 1, z]=Divide[2*\[Nu],z]*StruveH[\[Nu], z]+Divide[(Divide[1,2]*z)^(\[Nu]),Sqrt[Pi]*Gamma[\[Nu]+Divide[3,2]]] Failure Successful Successful -
11.4.E24 𝐇 Ξ½ - 1 ⁑ ( z ) - 𝐇 Ξ½ + 1 ⁑ ( z ) = 2 ⁒ 𝐇 Ξ½ β€² ⁑ ( z ) - ( 1 2 ⁒ z ) Ξ½ Ο€ ⁒ Ξ“ ⁑ ( Ξ½ + 3 2 ) Struve-H 𝜈 1 𝑧 Struve-H 𝜈 1 𝑧 2 diffop Struve-H 𝜈 1 𝑧 superscript 1 2 𝑧 𝜈 πœ‹ Euler-Gamma 𝜈 3 2 {\displaystyle{\displaystyle\mathbf{H}_{\nu-1}\left(z\right)-\mathbf{H}_{\nu+1% }\left(z\right)=2\!\mathbf{H}_{\nu}'\left(z\right)-\frac{(\tfrac{1}{2}z)^{\nu}% }{\sqrt{\pi}\Gamma\left(\nu+\tfrac{3}{2}\right)}}} StruveH(nu - 1, z)- StruveH(nu + 1, z)= 2*subs( temp=z, diff( StruveH(nu, temp), temp$(1) ) )-(((1)/(2)*z)^(nu))/(sqrt(Pi)*GAMMA(nu +(3)/(2))) StruveH[\[Nu]- 1, z]- StruveH[\[Nu]+ 1, z]= 2*(D[StruveH[\[Nu], temp], {temp, 1}]/.temp-> z)-Divide[(Divide[1,2]*z)^(\[Nu]),Sqrt[Pi]*Gamma[\[Nu]+Divide[3,2]]] Failure Successful Successful -
11.4.E25 𝐋 Ξ½ - 1 ⁑ ( z ) - 𝐋 Ξ½ + 1 ⁑ ( z ) = 2 ⁒ Ξ½ z ⁒ 𝐋 Ξ½ ⁑ ( z ) + ( 1 2 ⁒ z ) Ξ½ Ο€ ⁒ Ξ“ ⁑ ( Ξ½ + 3 2 ) modified-Struve-L 𝜈 1 𝑧 modified-Struve-L 𝜈 1 𝑧 2 𝜈 𝑧 modified-Struve-L 𝜈 𝑧 superscript 1 2 𝑧 𝜈 πœ‹ Euler-Gamma 𝜈 3 2 {\displaystyle{\displaystyle\mathbf{L}_{\nu-1}\left(z\right)-\mathbf{L}_{\nu+1% }\left(z\right)=\frac{2\nu}{z}\mathbf{L}_{\nu}\left(z\right)+\frac{(\tfrac{1}{% 2}z)^{\nu}}{\sqrt{\pi}\Gamma\left(\nu+\tfrac{3}{2}\right)}}} StruveL(nu - 1, z)- StruveL(nu + 1, z)=(2*nu)/(z)*StruveL(nu, z)+(((1)/(2)*z)^(nu))/(sqrt(Pi)*GAMMA(nu +(3)/(2))) StruveL[\[Nu]- 1, z]- StruveL[\[Nu]+ 1, z]=Divide[2*\[Nu],z]*StruveL[\[Nu], z]+Divide[(Divide[1,2]*z)^(\[Nu]),Sqrt[Pi]*Gamma[\[Nu]+Divide[3,2]]] Failure Successful Successful -
11.4.E26 𝐋 Ξ½ - 1 ⁑ ( z ) + 𝐋 Ξ½ + 1 ⁑ ( z ) = 2 ⁒ 𝐋 Ξ½ β€² ⁑ ( z ) - ( 1 2 ⁒ z ) Ξ½ Ο€ ⁒ Ξ“ ⁑ ( Ξ½ + 3 2 ) modified-Struve-L 𝜈 1 𝑧 modified-Struve-L 𝜈 1 𝑧 2 diffop modified-Struve-L 𝜈 1 𝑧 superscript 1 2 𝑧 𝜈 πœ‹ Euler-Gamma 𝜈 3 2 {\displaystyle{\displaystyle\mathbf{L}_{\nu-1}\left(z\right)+\mathbf{L}_{\nu+1% }\left(z\right)=2\!\mathbf{L}_{\nu}'\left(z\right)-\frac{(\tfrac{1}{2}z)^{\nu}% }{\sqrt{\pi}\Gamma\left(\nu+\tfrac{3}{2}\right)}}} StruveL(nu - 1, z)+ StruveL(nu + 1, z)= 2*subs( temp=z, diff( StruveL(nu, temp), temp$(1) ) )-(((1)/(2)*z)^(nu))/(sqrt(Pi)*GAMMA(nu +(3)/(2))) StruveL[\[Nu]- 1, z]+ StruveL[\[Nu]+ 1, z]= 2*(D[StruveL[\[Nu], temp], {temp, 1}]/.temp-> z)-Divide[(Divide[1,2]*z)^(\[Nu]),Sqrt[Pi]*Gamma[\[Nu]+Divide[3,2]]] Failure Successful Successful -
11.4.E27 d d z ⁑ ( z Ξ½ ⁒ 𝐇 Ξ½ ⁑ ( z ) ) = z Ξ½ ⁒ 𝐇 Ξ½ - 1 ⁑ ( z ) derivative 𝑧 superscript 𝑧 𝜈 Struve-H 𝜈 𝑧 superscript 𝑧 𝜈 Struve-H 𝜈 1 𝑧 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\left(z^{\nu}\mathbf% {H}_{\nu}\left(z\right)\right)=z^{\nu}\mathbf{H}_{\nu-1}\left(z\right)}} diff((z)^(nu)* StruveH(nu, z), z)= (z)^(nu)* StruveH(nu - 1, z) D[(z)^(\[Nu])* StruveH[\[Nu], z], z]= (z)^(\[Nu])* StruveH[\[Nu]- 1, z] Failure Successful Successful -
11.4.E28 d d z ⁑ ( z - Ξ½ ⁒ 𝐇 Ξ½ ⁑ ( z ) ) = 2 - Ξ½ Ο€ ⁒ Ξ“ ⁑ ( Ξ½ + 3 2 ) - z - Ξ½ ⁒ 𝐇 Ξ½ + 1 ⁑ ( z ) derivative 𝑧 superscript 𝑧 𝜈 Struve-H 𝜈 𝑧 superscript 2 𝜈 πœ‹ Euler-Gamma 𝜈 3 2 superscript 𝑧 𝜈 Struve-H 𝜈 1 𝑧 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\left(z^{-\nu}% \mathbf{H}_{\nu}\left(z\right)\right)=\frac{2^{-\nu}}{\sqrt{\pi}\Gamma\left(% \nu+\tfrac{3}{2}\right)}-z^{-\nu}\mathbf{H}_{\nu+1}\left(z\right)}} diff((z)^(- nu)* StruveH(nu, z), z)=((2)^(- nu))/(sqrt(Pi)*GAMMA(nu +(3)/(2)))- (z)^(- nu)* StruveH(nu + 1, z) D[(z)^(- \[Nu])* StruveH[\[Nu], z], z]=Divide[(2)^(- \[Nu]),Sqrt[Pi]*Gamma[\[Nu]+Divide[3,2]]]- (z)^(- \[Nu])* StruveH[\[Nu]+ 1, z] Successful Successful - -
11.4.E29 d d z ⁑ ( z Ξ½ ⁒ 𝐋 Ξ½ ⁑ ( z ) ) = z Ξ½ ⁒ 𝐋 Ξ½ - 1 ⁑ ( z ) derivative 𝑧 superscript 𝑧 𝜈 modified-Struve-L 𝜈 𝑧 superscript 𝑧 𝜈 modified-Struve-L 𝜈 1 𝑧 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\left(z^{\nu}\mathbf% {L}_{\nu}\left(z\right)\right)=z^{\nu}\mathbf{L}_{\nu-1}\left(z\right)}} diff((z)^(nu)* StruveL(nu, z), z)= (z)^(nu)* StruveL(nu - 1, z) D[(z)^(\[Nu])* StruveL[\[Nu], z], z]= (z)^(\[Nu])* StruveL[\[Nu]- 1, z] Failure Successful Successful -
11.4.E30 d d z ⁑ ( z - Ξ½ ⁒ 𝐋 Ξ½ ⁑ ( z ) ) = 2 - Ξ½ Ο€ ⁒ Ξ“ ⁑ ( Ξ½ + 3 2 ) + z - Ξ½ ⁒ 𝐋 Ξ½ + 1 ⁑ ( z ) derivative 𝑧 superscript 𝑧 𝜈 modified-Struve-L 𝜈 𝑧 superscript 2 𝜈 πœ‹ Euler-Gamma 𝜈 3 2 superscript 𝑧 𝜈 modified-Struve-L 𝜈 1 𝑧 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\left(z^{-\nu}% \mathbf{L}_{\nu}\left(z\right)\right)=\frac{2^{-\nu}}{\sqrt{\pi}\Gamma\left(% \nu+\tfrac{3}{2}\right)}+z^{-\nu}\mathbf{L}_{\nu+1}\left(z\right)}} diff((z)^(- nu)* StruveL(nu, z), z)=((2)^(- nu))/(sqrt(Pi)*GAMMA(nu +(3)/(2)))+ (z)^(- nu)* StruveL(nu + 1, z) D[(z)^(- \[Nu])* StruveL[\[Nu], z], z]=Divide[(2)^(- \[Nu]),Sqrt[Pi]*Gamma[\[Nu]+Divide[3,2]]]+ (z)^(- \[Nu])* StruveL[\[Nu]+ 1, z] Successful Successful - -
11.4#Ex1 𝐇 0 β€² ⁑ ( z ) = 2 Ο€ - 𝐇 1 ⁑ ( z ) diffop Struve-H 0 1 𝑧 2 πœ‹ Struve-H 1 𝑧 {\displaystyle{\displaystyle\mathbf{H}_{0}'\left(z\right)=\frac{2}{\pi}-% \mathbf{H}_{1}\left(z\right)}} subs( temp=z, diff( StruveH(0, temp), temp$(1) ) )=(2)/(Pi)- StruveH(1, z) (D[StruveH[0, temp], {temp, 1}]/.temp-> z)=Divide[2,Pi]- StruveH[1, z] Successful Successful - -
11.4#Ex2 d d z ⁑ ( z ⁒ 𝐇 1 ⁑ ( z ) ) = z ⁒ 𝐇 0 ⁑ ( z ) derivative 𝑧 𝑧 Struve-H 1 𝑧 𝑧 Struve-H 0 𝑧 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}(z\mathbf{H}_{1}% \left(z\right))=z\mathbf{H}_{0}\left(z\right)}} diff(z*StruveH(1, z), z)= z*StruveH(0, z) D[z*StruveH[1, z], z]= z*StruveH[0, z] Successful Successful - -
11.4#Ex3 𝐋 0 β€² ⁑ ( z ) = 2 Ο€ + 𝐋 1 ⁑ ( z ) diffop modified-Struve-L 0 1 𝑧 2 πœ‹ modified-Struve-L 1 𝑧 {\displaystyle{\displaystyle\mathbf{L}_{0}'\left(z\right)=\frac{2}{\pi}+% \mathbf{L}_{1}\left(z\right)}} subs( temp=z, diff( StruveL(0, temp), temp$(1) ) )=(2)/(Pi)+ StruveL(1, z) (D[StruveL[0, temp], {temp, 1}]/.temp-> z)=Divide[2,Pi]+ StruveL[1, z] Successful Successful - -
11.4#Ex4 d d z ⁑ ( z ⁒ 𝐋 1 ⁑ ( z ) ) = z ⁒ 𝐋 0 ⁑ ( z ) derivative 𝑧 𝑧 modified-Struve-L 1 𝑧 𝑧 modified-Struve-L 0 𝑧 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}(z\mathbf{L}_{1}% \left(z\right))=z\mathbf{L}_{0}\left(z\right)}} diff(z*StruveL(1, z), z)= z*StruveL(0, z) D[z*StruveL[1, z], z]= z*StruveL[0, z] Successful Successful - -
11.5.E1 𝐇 Ξ½ ⁑ ( z ) = 2 ⁒ ( 1 2 ⁒ z ) Ξ½ Ο€ ⁒ Ξ“ ⁑ ( Ξ½ + 1 2 ) ⁒ ∫ 0 1 ( 1 - t 2 ) Ξ½ - 1 2 ⁒ sin ⁑ ( z ⁒ t ) ⁒ d t Struve-H 𝜈 𝑧 2 superscript 1 2 𝑧 𝜈 πœ‹ Euler-Gamma 𝜈 1 2 superscript subscript 0 1 superscript 1 superscript 𝑑 2 𝜈 1 2 𝑧 𝑑 𝑑 {\displaystyle{\displaystyle\mathbf{H}_{\nu}\left(z\right)=\frac{2(\tfrac{1}{2% }z)^{\nu}}{\sqrt{\pi}\Gamma\left(\nu+\tfrac{1}{2}\right)}\int_{0}^{1}(1-t^{2})% ^{\nu-\frac{1}{2}}\sin\left(zt\right)\mathrm{d}t}} StruveH(nu, z)=(2*((1)/(2)*z)^(nu))/(sqrt(Pi)*GAMMA(nu +(1)/(2)))*int((1 - (t)^(2))^(nu -(1)/(2))* sin(z*t), t = 0..1) StruveH[\[Nu], z]=Divide[2*(Divide[1,2]*z)^(\[Nu]),Sqrt[Pi]*Gamma[\[Nu]+Divide[1,2]]]*Integrate[(1 - (t)^(2))^(\[Nu]-Divide[1,2])* Sin[z*t], {t, 0, 1}] Successful Successful - -
11.5.E1 2 ⁒ ( 1 2 ⁒ z ) Ξ½ Ο€ ⁒ Ξ“ ⁑ ( Ξ½ + 1 2 ) ⁒ ∫ 0 1 ( 1 - t 2 ) Ξ½ - 1 2 ⁒ sin ⁑ ( z ⁒ t ) ⁒ d t = 2 ⁒ ( 1 2 ⁒ z ) Ξ½ Ο€ ⁒ Ξ“ ⁑ ( Ξ½ + 1 2 ) ⁒ ∫ 0 Ο€ / 2 sin ⁑ ( z ⁒ cos ⁑ ΞΈ ) ⁒ ( sin ⁑ ΞΈ ) 2 ⁒ Ξ½ ⁒ d ΞΈ 2 superscript 1 2 𝑧 𝜈 πœ‹ Euler-Gamma 𝜈 1 2 superscript subscript 0 1 superscript 1 superscript 𝑑 2 𝜈 1 2 𝑧 𝑑 𝑑 2 superscript 1 2 𝑧 𝜈 πœ‹ Euler-Gamma 𝜈 1 2 superscript subscript 0 πœ‹ 2 𝑧 πœƒ superscript πœƒ 2 𝜈 πœƒ {\displaystyle{\displaystyle\frac{2(\tfrac{1}{2}z)^{\nu}}{\sqrt{\pi}\Gamma% \left(\nu+\tfrac{1}{2}\right)}\int_{0}^{1}(1-t^{2})^{\nu-\frac{1}{2}}\sin\left% (zt\right)\mathrm{d}t=\frac{2(\tfrac{1}{2}z)^{\nu}}{\sqrt{\pi}\Gamma\left(\nu+% \tfrac{1}{2}\right)}\int_{0}^{\pi/2}\sin\left(z\cos\theta\right)(\sin\theta)^{% 2\nu}\mathrm{d}\theta}} (2*((1)/(2)*z)^(nu))/(sqrt(Pi)*GAMMA(nu +(1)/(2)))*int((1 - (t)^(2))^(nu -(1)/(2))* sin(z*t), t = 0..1)=(2*((1)/(2)*z)^(nu))/(sqrt(Pi)*GAMMA(nu +(1)/(2)))*int(sin(z*cos(theta))*(sin(theta))^(2*nu), theta = 0..Pi/ 2) Divide[2*(Divide[1,2]*z)^(\[Nu]),Sqrt[Pi]*Gamma[\[Nu]+Divide[1,2]]]*Integrate[(1 - (t)^(2))^(\[Nu]-Divide[1,2])* Sin[z*t], {t, 0, 1}]=Divide[2*(Divide[1,2]*z)^(\[Nu]),Sqrt[Pi]*Gamma[\[Nu]+Divide[1,2]]]*Integrate[Sin[z*Cos[\[Theta]]]*(Sin[\[Theta]])^(2*\[Nu]), {\[Theta], 0, Pi/ 2}] Successful Successful - -
11.5.E2 𝐊 Ξ½ ⁑ ( z ) = 2 ⁒ ( 1 2 ⁒ z ) Ξ½ Ο€ ⁒ Ξ“ ⁑ ( Ξ½ + 1 2 ) ⁒ ∫ 0 ∞ e - z ⁒ t ⁒ ( 1 + t 2 ) Ξ½ - 1 2 ⁒ d t associated-Struve-K 𝜈 𝑧 2 superscript 1 2 𝑧 𝜈 πœ‹ Euler-Gamma 𝜈 1 2 superscript subscript 0 superscript 𝑒 𝑧 𝑑 superscript 1 superscript 𝑑 2 𝜈 1 2 𝑑 {\displaystyle{\displaystyle\mathbf{K}_{\nu}\left(z\right)=\frac{2(\tfrac{1}{2% }z)^{\nu}}{\sqrt{\pi}\Gamma\left(\nu+\tfrac{1}{2}\right)}\int_{0}^{\infty}e^{-% zt}(1+t^{2})^{\nu-\frac{1}{2}}\mathrm{d}t}} StruveH(nu, z) - BesselY(nu, z)=(2*((1)/(2)*z)^(nu))/(sqrt(Pi)*GAMMA(nu +(1)/(2)))*int(exp(- z*t)*(1 + (t)^(2))^(nu -(1)/(2)), t = 0..infinity) StruveH[\[Nu], z] - BesselY[\[Nu], z]=Divide[2*(Divide[1,2]*z)^(\[Nu]),Sqrt[Pi]*Gamma[\[Nu]+Divide[1,2]]]*Integrate[Exp[- z*t]*(1 + (t)^(2))^(\[Nu]-Divide[1,2]), {t, 0, Infinity}] Successful Failure - Error
11.5.E3 𝐊 0 ⁑ ( z ) = 2 Ο€ ⁒ ∫ 0 ∞ e - z ⁒ sinh ⁑ t ⁒ d t associated-Struve-K 0 𝑧 2 πœ‹ superscript subscript 0 superscript 𝑒 𝑧 𝑑 𝑑 {\displaystyle{\displaystyle\mathbf{K}_{0}\left(z\right)=\frac{2}{\pi}\int_{0}% ^{\infty}e^{-z\sinh t}\mathrm{d}t}} StruveH(0, z) - BesselY(0, z)=(2)/(Pi)*int(exp(- z*sinh(t)), t = 0..infinity) StruveH[0, z] - BesselY[0, z]=Divide[2,Pi]*Integrate[Exp[- z*Sinh[t]], {t, 0, Infinity}] Successful Failure - Error
11.5.E4 𝐌 Ξ½ ⁑ ( z ) = - 2 ⁒ ( 1 2 ⁒ z ) Ξ½ Ο€ ⁒ Ξ“ ⁑ ( Ξ½ + 1 2 ) ⁒ ∫ 0 1 e - z ⁒ t ⁒ ( 1 - t 2 ) Ξ½ - 1 2 ⁒ d t associated-Struve-M 𝜈 𝑧 2 superscript 1 2 𝑧 𝜈 πœ‹ Euler-Gamma 𝜈 1 2 superscript subscript 0 1 superscript 𝑒 𝑧 𝑑 superscript 1 superscript 𝑑 2 𝜈 1 2 𝑑 {\displaystyle{\displaystyle\mathbf{M}_{\nu}\left(z\right)=-\frac{2(\tfrac{1}{% 2}z)^{\nu}}{\sqrt{\pi}\Gamma\left(\nu+\tfrac{1}{2}\right)}\int_{0}^{1}e^{-zt}(% 1-t^{2})^{\nu-\frac{1}{2}}\mathrm{d}t}} StruveL(nu, z) - BesselI(nu, z)= -(2*((1)/(2)*z)^(nu))/(sqrt(Pi)*GAMMA(nu +(1)/(2)))*int(exp(- z*t)*(1 - (t)^(2))^(nu -(1)/(2)), t = 0..1) StruveL[\[Nu], z] - BesselI[\[Nu], z]= -Divide[2*(Divide[1,2]*z)^(\[Nu]),Sqrt[Pi]*Gamma[\[Nu]+Divide[1,2]]]*Integrate[Exp[- z*t]*(1 - (t)^(2))^(\[Nu]-Divide[1,2]), {t, 0, 1}] Successful Successful - -
11.5.E5 𝐌 0 ⁑ ( z ) = - 2 Ο€ ⁒ ∫ 0 Ο€ / 2 e - z ⁒ cos ⁑ ΞΈ ⁒ d ΞΈ associated-Struve-M 0 𝑧 2 πœ‹ superscript subscript 0 πœ‹ 2 superscript 𝑒 𝑧 πœƒ πœƒ {\displaystyle{\displaystyle\mathbf{M}_{0}\left(z\right)=-\frac{2}{\pi}\int_{0% }^{\pi/2}e^{-z\cos\theta}\mathrm{d}\theta}} StruveL(0, z) - BesselI(0, z)= -(2)/(Pi)*int(exp(- z*cos(theta)), theta = 0..Pi/ 2) StruveL[0, z] - BesselI[0, z]= -Divide[2,Pi]*Integrate[Exp[- z*Cos[\[Theta]]], {\[Theta], 0, Pi/ 2}] Successful Successful - -
11.5.E6 𝐋 Ξ½ ⁑ ( z ) = 2 ⁒ ( 1 2 ⁒ z ) Ξ½ Ο€ ⁒ Ξ“ ⁑ ( Ξ½ + 1 2 ) ⁒ ∫ 0 Ο€ / 2 sinh ⁑ ( z ⁒ cos ⁑ ΞΈ ) ⁒ ( sin ⁑ ΞΈ ) 2 ⁒ Ξ½ ⁒ d ΞΈ modified-Struve-L 𝜈 𝑧 2 superscript 1 2 𝑧 𝜈 πœ‹ Euler-Gamma 𝜈 1 2 superscript subscript 0 πœ‹ 2 𝑧 πœƒ superscript πœƒ 2 𝜈 πœƒ {\displaystyle{\displaystyle\mathbf{L}_{\nu}\left(z\right)=\frac{2(\tfrac{1}{2% }z)^{\nu}}{\sqrt{\pi}\Gamma\left(\nu+\tfrac{1}{2}\right)}\int_{0}^{\pi/2}\sinh% \left(z\cos\theta\right)(\sin\theta)^{2\nu}\mathrm{d}\theta}} StruveL(nu, z)=(2*((1)/(2)*z)^(nu))/(sqrt(Pi)*GAMMA(nu +(1)/(2)))*int(sinh(z*cos(theta))*(sin(theta))^(2*nu), theta = 0..Pi/ 2) StruveL[\[Nu], z]=Divide[2*(Divide[1,2]*z)^(\[Nu]),Sqrt[Pi]*Gamma[\[Nu]+Divide[1,2]]]*Integrate[Sinh[z*Cos[\[Theta]]]*(Sin[\[Theta]])^(2*\[Nu]), {\[Theta], 0, Pi/ 2}] Successful Failure - Error
11.5.E7 I - Ξ½ ⁑ ( x ) - 𝐋 Ξ½ ⁑ ( x ) = 2 ⁒ ( 1 2 ⁒ x ) Ξ½ Ο€ ⁒ Ξ“ ⁑ ( Ξ½ + 1 2 ) ⁒ ∫ 0 ∞ ( 1 + t 2 ) Ξ½ - 1 2 ⁒ sin ⁑ ( x ⁒ t ) ⁒ d t modified-Bessel-first-kind 𝜈 π‘₯ modified-Struve-L 𝜈 π‘₯ 2 superscript 1 2 π‘₯ 𝜈 πœ‹ Euler-Gamma 𝜈 1 2 superscript subscript 0 superscript 1 superscript 𝑑 2 𝜈 1 2 π‘₯ 𝑑 𝑑 {\displaystyle{\displaystyle I_{-\nu}\left(x\right)-\mathbf{L}_{\nu}\left(x% \right)=\frac{2(\tfrac{1}{2}x)^{\nu}}{\sqrt{\pi}\Gamma\left(\nu+\tfrac{1}{2}% \right)}\int_{0}^{\infty}(1+t^{2})^{\nu-\frac{1}{2}}\sin\left(xt\right)\mathrm% {d}t}} BesselI(- nu, x)- StruveL(nu, x)=(2*((1)/(2)*x)^(nu))/(sqrt(Pi)*GAMMA(nu +(1)/(2)))*int((1 + (t)^(2))^(nu -(1)/(2))* sin(x*t), t = 0..infinity) BesselI[- \[Nu], x]- StruveL[\[Nu], x]=Divide[2*(Divide[1,2]*x)^(\[Nu]),Sqrt[Pi]*Gamma[\[Nu]+Divide[1,2]]]*Integrate[(1 + (t)^(2))^(\[Nu]-Divide[1,2])* Sin[x*t], {t, 0, Infinity}] Failure Failure Skip Error
11.5.E8 ( 1 2 ⁒ x ) - Ξ½ - 1 ⁒ 𝐇 Ξ½ ⁑ ( x ) = - 1 2 ⁒ Ο€ ⁒ i ⁒ ∫ - i ⁒ ∞ i ⁒ ∞ Ο€ ⁒ csc ⁑ ( Ο€ ⁒ s ) Ξ“ ⁑ ( 3 2 + s ) ⁒ Ξ“ ⁑ ( 3 2 + Ξ½ + s ) ⁒ ( 1 4 ⁒ x 2 ) s ⁒ d s superscript 1 2 π‘₯ 𝜈 1 Struve-H 𝜈 π‘₯ 1 2 πœ‹ 𝑖 superscript subscript 𝑖 𝑖 πœ‹ πœ‹ 𝑠 Euler-Gamma 3 2 𝑠 Euler-Gamma 3 2 𝜈 𝑠 superscript 1 4 superscript π‘₯ 2 𝑠 𝑠 {\displaystyle{\displaystyle(\tfrac{1}{2}x)^{-\nu-1}\mathbf{H}_{\nu}\left(x% \right)=-\frac{1}{2\pi i}\int_{-i\infty}^{i\infty}\frac{\pi\csc\left(\pi s% \right)}{\Gamma\left(\tfrac{3}{2}+s\right)\Gamma\left(\tfrac{3}{2}+\nu+s\right% )}(\tfrac{1}{4}x^{2})^{s}\mathrm{d}s}} ((1)/(2)*x)^(- nu - 1)* StruveH(nu, x)= -(1)/(2*Pi*I)*int((Pi*csc(Pi*s))/(GAMMA((3)/(2)+ s)*GAMMA((3)/(2)+ nu + s))*((1)/(4)*(x)^(2))^(s), s = - I*infinity..I*infinity) (Divide[1,2]*x)^(- \[Nu]- 1)* StruveH[\[Nu], x]= -Divide[1,2*Pi*I]*Integrate[Divide[Pi*Csc[Pi*s],Gamma[Divide[3,2]+ s]*Gamma[Divide[3,2]+ \[Nu]+ s]]*(Divide[1,4]*(x)^(2))^(s), {s, - I*Infinity, I*Infinity}] Failure Failure Skip Error
11.5.E9 ( 1 2 ⁒ z ) - Ξ½ - 1 ⁒ 𝐋 Ξ½ ⁑ ( z ) = 1 2 ⁒ Ο€ ⁒ i ⁒ ∫ ∞ ( 0 + ) Ο€ ⁒ csc ⁑ ( Ο€ ⁒ s ) Ξ“ ⁑ ( 3 2 + s ) ⁒ Ξ“ ⁑ ( 3 2 + Ξ½ + s ) ⁒ ( - 1 4 ⁒ z 2 ) s ⁒ d s superscript 1 2 𝑧 𝜈 1 modified-Struve-L 𝜈 𝑧 1 2 πœ‹ 𝑖 superscript subscript limit-from 0 πœ‹ πœ‹ 𝑠 Euler-Gamma 3 2 𝑠 Euler-Gamma 3 2 𝜈 𝑠 superscript 1 4 superscript 𝑧 2 𝑠 𝑠 {\displaystyle{\displaystyle(\tfrac{1}{2}z)^{-\nu-1}\mathbf{L}_{\nu}\left(z% \right)=\frac{1}{2\pi i}\int_{\infty}^{(0+)}\frac{\pi\csc\left(\pi s\right)}{% \Gamma\left(\tfrac{3}{2}+s\right)\Gamma\left(\tfrac{3}{2}+\nu+s\right)}(-% \tfrac{1}{4}z^{2})^{s}\mathrm{d}s}} ((1)/(2)*z)^(- nu - 1)* StruveL(nu, z)=(1)/(2*Pi*I)*int((Pi*csc(Pi*s))/(GAMMA((3)/(2)+ s)*GAMMA((3)/(2)+ nu + s))*(-(1)/(4)*(z)^(2))^(s), s = infinity..(0 +)) (Divide[1,2]*z)^(- \[Nu]- 1)* StruveL[\[Nu], z]=Divide[1,2*Pi*I]*Integrate[Divide[Pi*Csc[Pi*s],Gamma[Divide[3,2]+ s]*Gamma[Divide[3,2]+ \[Nu]+ s]]*(-Divide[1,4]*(z)^(2))^(s), {s, Infinity, (0 +)}] Error Failure - Error
11.7.E1 ∫ z Ξ½ ⁒ 𝐇 Ξ½ - 1 ⁑ ( z ) ⁒ d z = z Ξ½ ⁒ 𝐇 Ξ½ ⁑ ( z ) superscript 𝑧 𝜈 Struve-H 𝜈 1 𝑧 𝑧 superscript 𝑧 𝜈 Struve-H 𝜈 𝑧 {\displaystyle{\displaystyle\int z^{\nu}\mathbf{H}_{\nu-1}\left(z\right)% \mathrm{d}z=z^{\nu}\mathbf{H}_{\nu}\left(z\right)}} int((z)^(nu)* StruveH(nu - 1, z), z)= (z)^(nu)* StruveH(nu, z) Integrate[(z)^(\[Nu])* StruveH[\[Nu]- 1, z], z]= (z)^(\[Nu])* StruveH[\[Nu], z] Successful Successful - -
11.7.E2 ∫ z - Ξ½ ⁒ 𝐇 Ξ½ + 1 ⁑ ( z ) ⁒ d z = - z - Ξ½ ⁒ 𝐇 Ξ½ ⁑ ( z ) + 2 - Ξ½ ⁒ z Ο€ ⁒ Ξ“ ⁑ ( Ξ½ + 3 2 ) superscript 𝑧 𝜈 Struve-H 𝜈 1 𝑧 𝑧 superscript 𝑧 𝜈 Struve-H 𝜈 𝑧 superscript 2 𝜈 𝑧 πœ‹ Euler-Gamma 𝜈 3 2 {\displaystyle{\displaystyle\int z^{-\nu}\mathbf{H}_{\nu+1}\left(z\right)% \mathrm{d}z=-z^{-\nu}\mathbf{H}_{\nu}\left(z\right)+\frac{2^{-\nu}z}{\sqrt{\pi% }\Gamma\left(\nu+\tfrac{3}{2}\right)}}} int((z)^(- nu)* StruveH(nu + 1, z), z)= - (z)^(- nu)* StruveH(nu, z)+((2)^(- nu)* z)/(sqrt(Pi)*GAMMA(nu +(3)/(2))) Integrate[(z)^(- \[Nu])* StruveH[\[Nu]+ 1, z], z]= - (z)^(- \[Nu])* StruveH[\[Nu], z]+Divide[(2)^(- \[Nu])* z,Sqrt[Pi]*Gamma[\[Nu]+Divide[3,2]]] Successful Successful - -
11.7.E3 ∫ z Ξ½ ⁒ 𝐋 Ξ½ - 1 ⁑ ( z ) ⁒ d z = z Ξ½ ⁒ 𝐋 Ξ½ ⁑ ( z ) superscript 𝑧 𝜈 modified-Struve-L 𝜈 1 𝑧 𝑧 superscript 𝑧 𝜈 modified-Struve-L 𝜈 𝑧 {\displaystyle{\displaystyle\int z^{\nu}\mathbf{L}_{\nu-1}\left(z\right)% \mathrm{d}z=z^{\nu}\mathbf{L}_{\nu}\left(z\right)}} int((z)^(nu)* StruveL(nu - 1, z), z)= (z)^(nu)* StruveL(nu, z) Integrate[(z)^(\[Nu])* StruveL[\[Nu]- 1, z], z]= (z)^(\[Nu])* StruveL[\[Nu], z] Failure Successful Skip -
11.7.E4 ∫ z - Ξ½ ⁒ 𝐋 Ξ½ + 1 ⁑ ( z ) ⁒ d z = z - Ξ½ ⁒ 𝐋 Ξ½ ⁑ ( z ) - 2 - Ξ½ ⁒ z Ο€ ⁒ Ξ“ ⁑ ( Ξ½ + 3 2 ) superscript 𝑧 𝜈 modified-Struve-L 𝜈 1 𝑧 𝑧 superscript 𝑧 𝜈 modified-Struve-L 𝜈 𝑧 superscript 2 𝜈 𝑧 πœ‹ Euler-Gamma 𝜈 3 2 {\displaystyle{\displaystyle\int z^{-\nu}\mathbf{L}_{\nu+1}\left(z\right)% \mathrm{d}z=z^{-\nu}\mathbf{L}_{\nu}\left(z\right)-\frac{2^{-\nu}z}{\sqrt{\pi}% \Gamma\left(\nu+\tfrac{3}{2}\right)}}} int((z)^(- nu)* StruveL(nu + 1, z), z)= (z)^(- nu)* StruveL(nu, z)-((2)^(- nu)* z)/(sqrt(Pi)*GAMMA(nu +(3)/(2))) Integrate[(z)^(- \[Nu])* StruveL[\[Nu]+ 1, z], z]= (z)^(- \[Nu])* StruveL[\[Nu], z]-Divide[(2)^(- \[Nu])* z,Sqrt[Pi]*Gamma[\[Nu]+Divide[3,2]]] Successful Successful - -
11.7.E5 f Ξ½ ⁒ ( z ) = ∫ 0 z t Ξ½ ⁒ 𝐇 Ξ½ ⁑ ( t ) ⁒ d t subscript 𝑓 𝜈 𝑧 superscript subscript 0 𝑧 superscript 𝑑 𝜈 Struve-H 𝜈 𝑑 𝑑 {\displaystyle{\displaystyle f_{\nu}(z)=\int_{0}^{z}t^{\nu}\mathbf{H}_{\nu}% \left(t\right)\mathrm{d}t}} f[nu]*(z)= int((t)^(nu)* StruveH(nu, t), t = 0..z) Subscript[f, \[Nu]]*(z)= Integrate[(t)^(\[Nu])* StruveH[\[Nu], t], {t, 0, z}] Failure Failure Skip Error
11.7.E6 f Ξ½ + 1 ⁒ ( z ) = ( 2 ⁒ Ξ½ + 1 ) ⁒ f Ξ½ ⁒ ( z ) - z Ξ½ + 1 ⁒ 𝐇 Ξ½ ⁑ ( z ) + ( 1 2 ⁒ z 2 ) Ξ½ + 1 ( Ξ½ + 1 ) ⁒ Ο€ ⁒ Ξ“ ⁑ ( Ξ½ + 3 2 ) subscript 𝑓 𝜈 1 𝑧 2 𝜈 1 subscript 𝑓 𝜈 𝑧 superscript 𝑧 𝜈 1 Struve-H 𝜈 𝑧 superscript 1 2 superscript 𝑧 2 𝜈 1 𝜈 1 πœ‹ Euler-Gamma 𝜈 3 2 {\displaystyle{\displaystyle f_{\nu+1}(z)=(2\nu+1)f_{\nu}(z)-z^{\nu+1}\mathbf{% H}_{\nu}\left(z\right)+\frac{(\tfrac{1}{2}z^{2})^{\nu+1}}{(\nu+1)\sqrt{\pi}% \Gamma\left(\nu+\tfrac{3}{2}\right)}}} f[nu + 1]*(z)=(2*nu + 1)* f[nu]*(z)- (z)^(nu + 1)* StruveH(nu, z)+(((1)/(2)*(z)^(2))^(nu + 1))/((nu + 1)*sqrt(Pi)*GAMMA(nu +(3)/(2))) Subscript[f, \[Nu]+ 1]*(z)=(2*\[Nu]+ 1)* Subscript[f, \[Nu]]*(z)- (z)^(\[Nu]+ 1)* StruveH[\[Nu], z]+Divide[(Divide[1,2]*(z)^(2))^(\[Nu]+ 1),(\[Nu]+ 1)*Sqrt[Pi]*Gamma[\[Nu]+Divide[3,2]]] Failure Failure
Fail
.588297823+3.112391650*I <- {z = 2^(1/2)+I*2^(1/2), f[nu] = 2^(1/2)+I*2^(1/2), f[nu+1] = 2^(1/2)+I*2^(1/2), nu = -1/2}
4.588297821-.8876083477*I <- {z = 2^(1/2)+I*2^(1/2), f[nu] = 2^(1/2)+I*2^(1/2), f[nu+1] = 2^(1/2)-I*2^(1/2), nu = -1/2}
.588297823-4.887608346*I <- {z = 2^(1/2)+I*2^(1/2), f[nu] = 2^(1/2)+I*2^(1/2), f[nu+1] = -2^(1/2)-I*2^(1/2), nu = -1/2}
-3.411702175-.8876083477*I <- {z = 2^(1/2)+I*2^(1/2), f[nu] = 2^(1/2)+I*2^(1/2), f[nu+1] = -2^(1/2)+I*2^(1/2), nu = -1/2}
... skip entries to safe data
 Error
11.7.E7 ∫ 0 Ο€ / 2 𝐇 Ξ½ ⁑ ( z ⁒ sin ⁑ ΞΈ ) ⁒ ( sin ⁑ ΞΈ ) Ξ½ + 1 ( cos ⁑ ΞΈ ) 2 ⁒ Ξ½ ⁒ d ΞΈ = 2 - Ξ½ Ο€ ⁒ Ξ“ ⁑ ( 1 2 - Ξ½ ) ⁒ z Ξ½ - 1 ⁒ ( 1 - cos ⁑ z ) superscript subscript 0 πœ‹ 2 Struve-H 𝜈 𝑧 πœƒ superscript πœƒ 𝜈 1 superscript πœƒ 2 𝜈 πœƒ superscript 2 𝜈 πœ‹ Euler-Gamma 1 2 𝜈 superscript 𝑧 𝜈 1 1 𝑧 {\displaystyle{\displaystyle\int_{0}^{\pi/2}\mathbf{H}_{\nu}\left(z\sin\theta% \right)\frac{(\sin\theta)^{\nu+1}}{(\cos\theta)^{2\nu}}\mathrm{d}\theta=\frac{% 2^{-\nu}}{\sqrt{\pi}}\Gamma\left(\tfrac{1}{2}-\nu\right)z^{\nu-1}(1-\cos z)}} int(StruveH(nu, z*sin(theta))*((sin(theta))^(nu + 1))/((cos(theta))^(2*nu)), theta = 0..Pi/ 2)=((2)^(- nu))/(sqrt(Pi))*GAMMA((1)/(2)- nu)*(z)^(nu - 1)*(1 - cos(z)) Integrate[StruveH[\[Nu], z*Sin[\[Theta]]]*Divide[(Sin[\[Theta]])^(\[Nu]+ 1),(Cos[\[Theta]])^(2*\[Nu])], {\[Theta], 0, Pi/ 2}]=Divide[(2)^(- \[Nu]),Sqrt[Pi]]*Gamma[Divide[1,2]- \[Nu]]*(z)^(\[Nu]- 1)*(1 - Cos[z]) Successful Failure - Error
11.7#Ex1 ∫ 0 ∞ 𝐇 0 ⁑ ( t ) ⁒ d t t = 1 2 ⁒ Ο€ superscript subscript 0 Struve-H 0 𝑑 𝑑 𝑑 1 2 πœ‹ {\displaystyle{\displaystyle\int_{0}^{\infty}\mathbf{H}_{0}\left(t\right)\,% \frac{\mathrm{d}t}{t}=\tfrac{1}{2}\pi}} int(StruveH(0, t)*(1)/(t), t = 0..infinity)=(1)/(2)*Pi Integrate[StruveH[0, t]*Divide[1,t], {t, 0, Infinity}]=Divide[1,2]*Pi Successful Successful - -
11.7#Ex2 ∫ 0 ∞ 𝐇 1 ⁑ ( t ) ⁒ d t t 2 = 1 4 ⁒ Ο€ superscript subscript 0 Struve-H 1 𝑑 𝑑 superscript 𝑑 2 1 4 πœ‹ {\displaystyle{\displaystyle\int_{0}^{\infty}\mathbf{H}_{1}\left(t\right)\,% \frac{\mathrm{d}t}{t^{2}}=\tfrac{1}{4}\pi}} int(StruveH(1, t)*(1)/((t)^(2)), t = 0..infinity)=(1)/(4)*Pi Integrate[StruveH[1, t]*Divide[1,(t)^(2)], {t, 0, Infinity}]=Divide[1,4]*Pi Successful Successful - -
11.7.E9 ∫ 0 ∞ 𝐇 Ξ½ ⁑ ( t ) ⁒ d t = - cot ⁑ ( 1 2 ⁒ Ο€ ⁒ Ξ½ ) superscript subscript 0 Struve-H 𝜈 𝑑 𝑑 1 2 πœ‹ 𝜈 {\displaystyle{\displaystyle\int_{0}^{\infty}\mathbf{H}_{\nu}\left(t\right)% \mathrm{d}t=-\cot\left(\tfrac{1}{2}\pi\nu\right)}} int(StruveH(nu, t), t = 0..infinity)= - cot((1)/(2)*Pi*nu) Integrate[StruveH[\[Nu], t], {t, 0, Infinity}]= - Cot[Divide[1,2]*Pi*\[Nu]] Successful Failure - Error
11.7.E10 ∫ 0 ∞ t - Ξ½ - 1 ⁒ 𝐇 Ξ½ ⁑ ( t ) ⁒ d t = Ο€ 2 Ξ½ + 1 ⁒ Ξ“ ⁑ ( Ξ½ + 1 ) superscript subscript 0 superscript 𝑑 𝜈 1 Struve-H 𝜈 𝑑 𝑑 πœ‹ superscript 2 𝜈 1 Euler-Gamma 𝜈 1 {\displaystyle{\displaystyle\int_{0}^{\infty}t^{-\nu-1}\mathbf{H}_{\nu}\left(t% \right)\mathrm{d}t=\frac{\pi}{2^{\nu+1}\Gamma\left(\nu+1\right)}}} int((t)^(- nu - 1)* StruveH(nu, t), t = 0..infinity)=(Pi)/((2)^(nu + 1)* GAMMA(nu + 1)) Integrate[(t)^(- \[Nu]- 1)* StruveH[\[Nu], t], {t, 0, Infinity}]=Divide[Pi,(2)^(\[Nu]+ 1)* Gamma[\[Nu]+ 1]] Successful Failure - Error
11.7.E11 ∫ 0 ∞ t ΞΌ - Ξ½ - 1 ⁒ 𝐇 Ξ½ ⁑ ( t ) ⁒ d t = Ξ“ ⁑ ( 1 2 ⁒ ΞΌ ) ⁒ 2 ΞΌ - Ξ½ - 1 ⁒ tan ⁑ ( 1 2 ⁒ Ο€ ⁒ ΞΌ ) Ξ“ ⁑ ( Ξ½ - 1 2 ⁒ ΞΌ + 1 ) superscript subscript 0 superscript 𝑑 πœ‡ 𝜈 1 Struve-H 𝜈 𝑑 𝑑 Euler-Gamma 1 2 πœ‡ superscript 2 πœ‡ 𝜈 1 1 2 πœ‹ πœ‡ Euler-Gamma 𝜈 1 2 πœ‡ 1 {\displaystyle{\displaystyle\int_{0}^{\infty}t^{\mu-\nu-1}\mathbf{H}_{\nu}% \left(t\right)\mathrm{d}t=\frac{\Gamma\left(\tfrac{1}{2}\mu\right)2^{\mu-\nu-1% }\tan\left(\tfrac{1}{2}\pi\mu\right)}{\Gamma\left(\nu-\tfrac{1}{2}\mu+1\right)% }}} int((t)^(mu - nu - 1)* StruveH(nu, t), t = 0..infinity)=(GAMMA((1)/(2)*mu)*(2)^(mu - nu - 1)* tan((1)/(2)*Pi*mu))/(GAMMA(nu -(1)/(2)*mu + 1)) Integrate[(t)^(\[Mu]- \[Nu]- 1)* StruveH[\[Nu], t], {t, 0, Infinity}]=Divide[Gamma[Divide[1,2]*\[Mu]]*(2)^(\[Mu]- \[Nu]- 1)* Tan[Divide[1,2]*Pi*\[Mu]],Gamma[\[Nu]-Divide[1,2]*\[Mu]+ 1]] Successful Failure - Error
11.7.E12 ∫ 0 ∞ t - ΞΌ - Ξ½ ⁒ 𝐇 ΞΌ ⁑ ( t ) ⁒ 𝐇 Ξ½ ⁑ ( t ) ⁒ d t = Ο€ ⁒ Ξ“ ⁑ ( ΞΌ + Ξ½ ) 2 ΞΌ + Ξ½ ⁒ Ξ“ ⁑ ( ΞΌ + Ξ½ + 1 2 ) ⁒ Ξ“ ⁑ ( ΞΌ + 1 2 ) ⁒ Ξ“ ⁑ ( Ξ½ + 1 2 ) superscript subscript 0 superscript 𝑑 πœ‡ 𝜈 Struve-H πœ‡ 𝑑 Struve-H 𝜈 𝑑 𝑑 πœ‹ Euler-Gamma πœ‡ 𝜈 superscript 2 πœ‡ 𝜈 Euler-Gamma πœ‡ 𝜈 1 2 Euler-Gamma πœ‡ 1 2 Euler-Gamma 𝜈 1 2 {\displaystyle{\displaystyle\int_{0}^{\infty}t^{-\mu-\nu}\mathbf{H}_{\mu}\left% (t\right)\mathbf{H}_{\nu}\left(t\right)\mathrm{d}t=\frac{\sqrt{\pi}\Gamma\left% (\mu+\nu\right)}{2^{\mu+\nu}\Gamma\left(\mu+\nu+\tfrac{1}{2}\right)\Gamma\left% (\mu+\tfrac{1}{2}\right)\Gamma\left(\nu+\tfrac{1}{2}\right)}}} int((t)^(- mu - nu)* StruveH(mu, t)*StruveH(nu, t), t = 0..infinity)=(sqrt(Pi)*GAMMA(mu + nu))/((2)^(mu + nu)* GAMMA(mu + nu +(1)/(2))*GAMMA(mu +(1)/(2))*GAMMA(nu +(1)/(2))) Integrate[(t)^(- \[Mu]- \[Nu])* StruveH[\[Mu], t]*StruveH[\[Nu], t], {t, 0, Infinity}]=Divide[Sqrt[Pi]*Gamma[\[Mu]+ \[Nu]],(2)^(\[Mu]+ \[Nu])* Gamma[\[Mu]+ \[Nu]+Divide[1,2]]*Gamma[\[Mu]+Divide[1,2]]*Gamma[\[Nu]+Divide[1,2]]] Error Failure - Error
11.7.E13 ∫ 0 ∞ e - a ⁒ t ⁒ 𝐇 0 ⁑ ( t ) ⁒ d t = 2 Ο€ ⁒ 1 + a 2 ⁒ ln ⁑ ( 1 + 1 + a 2 a ) superscript subscript 0 superscript 𝑒 π‘Ž 𝑑 Struve-H 0 𝑑 𝑑 2 πœ‹ 1 superscript π‘Ž 2 1 1 superscript π‘Ž 2 π‘Ž {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-at}\mathbf{H}_{0}\left(t% \right)\mathrm{d}t=\frac{2}{\pi\sqrt{1+a^{2}}}\ln\left(\frac{1+\sqrt{1+a^{2}}}% {a}\right)}} int(exp(- a*t)*StruveH(0, t), t = 0..infinity)=(2)/(Pi*sqrt(1 + (a)^(2)))*ln((1 +sqrt(1 + (a)^(2)))/(a)) Integrate[Exp[- a*t]*StruveH[0, t], {t, 0, Infinity}]=Divide[2,Pi*Sqrt[1 + (a)^(2)]]*Log[Divide[1 +Sqrt[1 + (a)^(2)],a]] Failure Failure Skip Error
11.7.E14 ∫ 0 ∞ e - a ⁒ t ⁒ 𝐇 1 ⁑ ( t ) ⁒ d t = 2 Ο€ ⁒ a - 2 ⁒ a Ο€ ⁒ 1 + a 2 ⁒ ln ⁑ ( 1 + 1 + a 2 a ) superscript subscript 0 superscript 𝑒 π‘Ž 𝑑 Struve-H 1 𝑑 𝑑 2 πœ‹ π‘Ž 2 π‘Ž πœ‹ 1 superscript π‘Ž 2 1 1 superscript π‘Ž 2 π‘Ž {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-at}\mathbf{H}_{1}\left(t% \right)\mathrm{d}t=\frac{2}{\pi a}-\frac{2a}{\pi\sqrt{1+a^{2}}}\ln\left(\frac{% 1+\sqrt{1+a^{2}}}{a}\right)}} int(exp(- a*t)*StruveH(1, t), t = 0..infinity)=(2)/(Pi*a)-(2*a)/(Pi*sqrt(1 + (a)^(2)))*ln((1 +sqrt(1 + (a)^(2)))/(a)) Integrate[Exp[- a*t]*StruveH[1, t], {t, 0, Infinity}]=Divide[2,Pi*a]-Divide[2*a,Pi*Sqrt[1 + (a)^(2)]]*Log[Divide[1 +Sqrt[1 + (a)^(2)],a]] Failure Failure Skip Error
11.7.E15 ∫ 0 ∞ e - a ⁒ t ⁒ 𝐋 0 ⁑ ( t ) ⁒ d t = 2 Ο€ ⁒ a 2 - 1 ⁒ arcsin ⁑ ( 1 a ) superscript subscript 0 superscript 𝑒 π‘Ž 𝑑 modified-Struve-L 0 𝑑 𝑑 2 πœ‹ superscript π‘Ž 2 1 1 π‘Ž {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-at}\mathbf{L}_{0}\left(t% \right)\mathrm{d}t=\frac{2}{\pi\sqrt{a^{2}\!-\!1}}\operatorname{arcsin}\left(% \frac{1}{a}\right)}} int(exp(- a*t)*StruveL(0, t), t = 0..infinity)=(2)/(Pi*sqrt((a)^(2)- 1))*arcsin((1)/(a)) Integrate[Exp[- a*t]*StruveL[0, t], {t, 0, Infinity}]=Divide[2,Pi*Sqrt[(a)^(2)- 1]]*ArcSin[Divide[1,a]] Failure Failure Skip Error
11.7#Ex4 = 2 ⁒ a Ο€ ⁒ a 2 - 1 ⁒ arctan ⁑ ( 1 a 2 - 1 ) - 2 Ο€ ⁒ a absent 2 π‘Ž πœ‹ superscript π‘Ž 2 1 1 superscript π‘Ž 2 1 2 πœ‹ π‘Ž {\displaystyle{\displaystyle=\frac{2a}{\pi\sqrt{a^{2}\!-\!1}}\operatorname{% arctan}\left(\frac{1}{\sqrt{a^{2}\!-\!1}}\right)-\frac{2}{\pi a}}} =(2*a)/(Pi*sqrt((a)^(2)- 1))*arctan((1)/(sqrt((a)^(2)- 1)))-(2)/(Pi*a) =Divide[2*a,Pi*Sqrt[(a)^(2)- 1]]*ArcTan[Divide[1,Sqrt[(a)^(2)- 1]]]-Divide[2,Pi*a] Error Failure - Error
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}}} 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
Fail
-.1150287995-.4235009013*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
-.380733971+.363721811*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
-37.31009604-.1600773484*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
2.846320383+2.784324879*I <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Error
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 = LommelS1(mu, nu, z)+ A*BesselJ(nu, z)+ B*BesselY(nu, z) Error Failure Error Skip -
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)}}} LommelS1(mu, nu, z)= (z)^(mu + 1)* sum((- 1)^(k)*((z)^(2*k))/(a[k + 1]*(mu , nu)), k = 0..infinity) Error Failure Error Skip -
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)}} 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 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)}} LommelS1(mu, - nu, z)= LommelS1(mu, nu, z) Error Successful 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)}} LommelS2(mu, - nu, z)= LommelS2(mu, nu, z) Error Successful 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)}} 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 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)}} 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 Error Skip -
11.10.E1 𝐉 Ξ½ ⁑ ( z ) = 1 Ο€ ⁒ ∫ 0 Ο€ cos ⁑ ( Ξ½ ⁒ ΞΈ - z ⁒ sin ⁑ ΞΈ ) ⁒ d ΞΈ Anger-J 𝜈 𝑧 1 πœ‹ superscript subscript 0 πœ‹ 𝜈 πœƒ 𝑧 πœƒ πœƒ {\displaystyle{\displaystyle\mathbf{J}_{\nu}\left(z\right)=\frac{1}{\pi}\int_{% 0}^{\pi}\cos\left(\nu\theta-z\sin\theta\right)\mathrm{d}\theta}} AngerJ(nu, z)=(1)/(Pi)*int(cos(nu*theta - z*sin(theta)), theta = 0..Pi) AngerJ[\[Nu], z]=Divide[1,Pi]*Integrate[Cos[\[Nu]*\[Theta]- z*Sin[\[Theta]]], {\[Theta], 0, Pi}] Failure Failure Skip Error
11.10.E2 𝐄 Ξ½ ⁑ ( z ) = 1 Ο€ ⁒ ∫ 0 Ο€ sin ⁑ ( Ξ½ ⁒ ΞΈ - z ⁒ sin ⁑ ΞΈ ) ⁒ d ΞΈ Weber-E 𝜈 𝑧 1 πœ‹ superscript subscript 0 πœ‹ 𝜈 πœƒ 𝑧 πœƒ πœƒ {\displaystyle{\displaystyle\mathbf{E}_{\nu}\left(z\right)=\frac{1}{\pi}\int_{% 0}^{\pi}\sin\left(\nu\theta-z\sin\theta\right)\mathrm{d}\theta}} WeberE(nu, z)=(1)/(Pi)*int(sin(nu*theta - z*sin(theta)), theta = 0..Pi) WeberE[\[Nu], z]=Divide[1,Pi]*Integrate[Sin[\[Nu]*\[Theta]- z*Sin[\[Theta]]], {\[Theta], 0, Pi}] Failure Failure Skip Error
11.10.E3 1 Ο€ ⁒ ∫ 0 2 ⁒ Ο€ cos ⁑ ( Ξ½ ⁒ ΞΈ - z ⁒ sin ⁑ ΞΈ ) ⁒ d ΞΈ = ( 1 + cos ⁑ ( 2 ⁒ Ο€ ⁒ Ξ½ ) ) ⁒ 𝐉 Ξ½ ⁑ ( z ) + sin ⁑ ( 2 ⁒ Ο€ ⁒ Ξ½ ) ⁒ 𝐄 Ξ½ ⁑ ( z ) 1 πœ‹ superscript subscript 0 2 πœ‹ 𝜈 πœƒ 𝑧 πœƒ πœƒ 1 2 πœ‹ 𝜈 Anger-J 𝜈 𝑧 2 πœ‹ 𝜈 Weber-E 𝜈 𝑧 {\displaystyle{\displaystyle\frac{1}{\pi}\int_{0}^{2\pi}\cos\left(\nu\theta-z% \sin\theta\right)\mathrm{d}\theta=(1+\cos\left(2\pi\nu\right))\,\mathbf{J}_{% \nu}\left(z\right)+\sin\left(2\pi\nu\right)\mathbf{E}_{\nu}\left(z\right)}} (1)/(Pi)*int(cos(nu*theta - z*sin(theta)), theta = 0..2*Pi)=(1 + cos(2*Pi*nu))* AngerJ(nu, z)+ sin(2*Pi*nu)*WeberE(nu, z) Divide[1,Pi]*Integrate[Cos[\[Nu]*\[Theta]- z*Sin[\[Theta]]], {\[Theta], 0, 2*Pi}]=(1 + Cos[2*Pi*\[Nu]])* AngerJ[\[Nu], z]+ Sin[2*Pi*\[Nu]]*WeberE[\[Nu], z] Failure Failure Skip Error
11.10.E8 𝐉 Ξ½ ⁑ ( z ) = cos ⁑ ( 1 2 ⁒ Ο€ ⁒ Ξ½ ) ⁒ S 1 ⁒ ( Ξ½ , z ) + sin ⁑ ( 1 2 ⁒ Ο€ ⁒ Ξ½ ) ⁒ S 2 ⁒ ( Ξ½ , z ) Anger-J 𝜈 𝑧 1 2 πœ‹ 𝜈 subscript 𝑆 1 𝜈 𝑧 1 2 πœ‹ 𝜈 subscript 𝑆 2 𝜈 𝑧 {\displaystyle{\displaystyle\mathbf{J}_{\nu}\left(z\right)=\cos\left(\tfrac{1}% {2}\pi\nu\right)\,S_{1}(\nu,z)+\sin\left(\tfrac{1}{2}\pi\nu\right)\,S_{2}(\nu,% z)}} AngerJ(nu, z)= cos((1)/(2)*Pi*nu)*S[1]*(nu , z)+ sin((1)/(2)*Pi*nu)*S[2]*(nu , z) AngerJ[\[Nu], z]= Cos[Divide[1,2]*Pi*\[Nu]]*Subscript[S, 1]*(\[Nu], z)+ Sin[Divide[1,2]*Pi*\[Nu]]*Subscript[S, 2]*(\[Nu], z) Failure Failure Error Error
11.10.E9 𝐄 Ξ½ ⁑ ( z ) = sin ⁑ ( 1 2 ⁒ Ο€ ⁒ Ξ½ ) ⁒ S 1 ⁒ ( Ξ½ , z ) - cos ⁑ ( 1 2 ⁒ Ο€ ⁒ Ξ½ ) ⁒ S 2 ⁒ ( Ξ½ , z ) Weber-E 𝜈 𝑧 1 2 πœ‹ 𝜈 subscript 𝑆 1 𝜈 𝑧 1 2 πœ‹ 𝜈 subscript 𝑆 2 𝜈 𝑧 {\displaystyle{\displaystyle\mathbf{E}_{\nu}\left(z\right)=\sin\left(\tfrac{1}% {2}\pi\nu\right)\,S_{1}(\nu,z)-\cos\left(\tfrac{1}{2}\pi\nu\right)\,S_{2}(\nu,% z)}} WeberE(nu, z)= sin((1)/(2)*Pi*nu)*S[1]*(nu , z)- cos((1)/(2)*Pi*nu)*S[2]*(nu , z) WeberE[\[Nu], z]= Sin[Divide[1,2]*Pi*\[Nu]]*Subscript[S, 1]*(\[Nu], z)- Cos[Divide[1,2]*Pi*\[Nu]]*Subscript[S, 2]*(\[Nu], z) Failure Failure Error Error
11.10.E10 S 1 ⁒ ( Ξ½ , z ) = βˆ‘ k = 0 ∞ ( - 1 ) k ⁒ ( 1 2 ⁒ z ) 2 ⁒ k Ξ“ ⁑ ( k + 1 2 ⁒ Ξ½ + 1 ) ⁒ Ξ“ ⁑ ( k - 1 2 ⁒ Ξ½ + 1 ) subscript 𝑆 1 𝜈 𝑧 superscript subscript π‘˜ 0 superscript 1 π‘˜ superscript 1 2 𝑧 2 π‘˜ Euler-Gamma π‘˜ 1 2 𝜈 1 Euler-Gamma π‘˜ 1 2 𝜈 1 {\displaystyle{\displaystyle S_{1}(\nu,z)=\sum_{k=0}^{\infty}\frac{(-1)^{k}(% \tfrac{1}{2}z)^{2k}}{\Gamma\left(k\!+\!\tfrac{1}{2}\nu+1\right)\Gamma\left(k\!% -\!\tfrac{1}{2}\nu\!+\!1\right)}}} S[1]*(nu , z)= sum(((- 1)^(k)*((1)/(2)*z)^(2*k))/(GAMMA(k +(1)/(2)*nu + 1)*GAMMA(k -(1)/(2)*nu + 1)), k = 0..infinity) Subscript[S, 1]*(\[Nu], z)= Sum[Divide[(- 1)^(k)*(Divide[1,2]*z)^(2*k),Gamma[k +Divide[1,2]*\[Nu]+ 1]*Gamma[k -Divide[1,2]*\[Nu]+ 1]], {k, 0, Infinity}] Failure Failure Skip Error
11.10.E11 S 2 ⁒ ( Ξ½ , z ) = βˆ‘ k = 0 ∞ ( - 1 ) k ⁒ ( 1 2 ⁒ z ) 2 ⁒ k + 1 Ξ“ ⁑ ( k + 1 2 ⁒ Ξ½ + 3 2 ) ⁒ Ξ“ ⁑ ( k - 1 2 ⁒ Ξ½ + 3 2 ) subscript 𝑆 2 𝜈 𝑧 superscript subscript π‘˜ 0 superscript 1 π‘˜ superscript 1 2 𝑧 2 π‘˜ 1 Euler-Gamma π‘˜ 1 2 𝜈 3 2 Euler-Gamma π‘˜ 1 2 𝜈 3 2 {\displaystyle{\displaystyle S_{2}(\nu,z)=\sum_{k=0}^{\infty}\frac{(-1)^{k}(% \tfrac{1}{2}z)^{2k+1}}{\Gamma\left(k\!+\!\tfrac{1}{2}\nu\!+\!\tfrac{3}{2}% \right)\Gamma\left(k\!-\!\tfrac{1}{2}\nu\!+\!\tfrac{3}{2}\right)}}} S[2]*(nu , z)= sum(((- 1)^(k)*((1)/(2)*z)^(2*k + 1))/(GAMMA(k +(1)/(2)*nu +(3)/(2))*GAMMA(k -(1)/(2)*nu +(3)/(2))), k = 0..infinity) Subscript[S, 2]*(\[Nu], z)= Sum[Divide[(- 1)^(k)*(Divide[1,2]*z)^(2*k + 1),Gamma[k +Divide[1,2]*\[Nu]+Divide[3,2]]*Gamma[k -Divide[1,2]*\[Nu]+Divide[3,2]]], {k, 0, Infinity}] Failure Failure Skip Error
11.10#Ex1 𝐉 Ξ½ ⁑ ( - z ) = 𝐉 - Ξ½ ⁑ ( z ) Anger-J 𝜈 𝑧 Anger-J 𝜈 𝑧 {\displaystyle{\displaystyle\mathbf{J}_{\nu}\left(-z\right)=\mathbf{J}_{-\nu}% \left(z\right)}} AngerJ(nu, - z)= AngerJ(- nu, z) AngerJ[\[Nu], - z]= AngerJ[- \[Nu], z] Successful Successful - -
11.10#Ex2 𝐄 Ξ½ ⁑ ( - z ) = - 𝐄 - Ξ½ ⁑ ( z ) Weber-E 𝜈 𝑧 Weber-E 𝜈 𝑧 {\displaystyle{\displaystyle\mathbf{E}_{\nu}\left(-z\right)=-\mathbf{E}_{-\nu}% \left(z\right)}} WeberE(nu, - z)= - WeberE(- nu, z) WeberE[\[Nu], - z]= - WeberE[- \[Nu], z] Successful Successful - -
11.10.E13 sin ⁑ ( Ο€ ⁒ Ξ½ ) ⁒ 𝐉 Ξ½ ⁑ ( z ) = cos ⁑ ( Ο€ ⁒ Ξ½ ) ⁒ 𝐄 Ξ½ ⁑ ( z ) - 𝐄 - Ξ½ ⁑ ( z ) πœ‹ 𝜈 Anger-J 𝜈 𝑧 πœ‹ 𝜈 Weber-E 𝜈 𝑧 Weber-E 𝜈 𝑧 {\displaystyle{\displaystyle\sin\left(\pi\nu\right)\,\mathbf{J}_{\nu}\left(z% \right)=\cos\left(\pi\nu\right)\,\mathbf{E}_{\nu}\left(z\right)-\mathbf{E}_{-% \nu}\left(z\right)}} sin(Pi*nu)*AngerJ(nu, z)= cos(Pi*nu)*WeberE(nu, z)- WeberE(- nu, z) Sin[Pi*\[Nu]]*AngerJ[\[Nu], z]= Cos[Pi*\[Nu]]*WeberE[\[Nu], z]- WeberE[- \[Nu], z] Successful Failure - Error
11.10.E14 sin ⁑ ( Ο€ ⁒ Ξ½ ) ⁒ 𝐄 Ξ½ ⁑ ( z ) = 𝐉 - Ξ½ ⁑ ( z ) - cos ⁑ ( Ο€ ⁒ Ξ½ ) ⁒ 𝐉 Ξ½ ⁑ ( z ) πœ‹ 𝜈 Weber-E 𝜈 𝑧 Anger-J 𝜈 𝑧 πœ‹ 𝜈 Anger-J 𝜈 𝑧 {\displaystyle{\displaystyle\sin\left(\pi\nu\right)\,\mathbf{E}_{\nu}\left(z% \right)=\mathbf{J}_{-\nu}\left(z\right)-\cos\left(\pi\nu\right)\,\mathbf{J}_{% \nu}\left(z\right)}} sin(Pi*nu)*WeberE(nu, z)= AngerJ(- nu, z)- cos(Pi*nu)*AngerJ(nu, z) Sin[Pi*\[Nu]]*WeberE[\[Nu], z]= AngerJ[- \[Nu], z]- Cos[Pi*\[Nu]]*AngerJ[\[Nu], z] Successful Failure - Error
11.10.E17 𝐉 Ξ½ ⁑ ( z ) = sin ⁑ ( Ο€ ⁒ Ξ½ ) Ο€ ⁒ ( s 0 , Ξ½ ⁑ ( z ) - Ξ½ ⁒ s - 1 , Ξ½ ⁑ ( z ) ) Anger-J 𝜈 𝑧 πœ‹ 𝜈 πœ‹ Lommel-s 0 𝜈 𝑧 𝜈 Lommel-s 1 𝜈 𝑧 {\displaystyle{\displaystyle\mathbf{J}_{\nu}\left(z\right)=\frac{\sin\left(\pi% \nu\right)}{\pi}(s_{{0},{\nu}}\left(z\right)-\nu s_{{-1},{\nu}}\left(z\right))}} AngerJ(nu, z)=(sin(Pi*nu))/(Pi)*(LommelS1(0, nu, z)- nu*LommelS1(- 1, nu, z)) Error Successful Error - -
11.10.E18 𝐄 Ξ½ ⁑ ( z ) = - 1 Ο€ ⁒ ( 1 + cos ⁑ ( Ο€ ⁒ Ξ½ ) ) ⁒ s 0 , Ξ½ ⁑ ( z ) - Ξ½ Ο€ ⁒ ( 1 - cos ⁑ ( Ο€ ⁒ Ξ½ ) ) ⁒ s - 1 , Ξ½ ⁑ ( z ) Weber-E 𝜈 𝑧 1 πœ‹ 1 πœ‹ 𝜈 Lommel-s 0 𝜈 𝑧 𝜈 πœ‹ 1 πœ‹ 𝜈 Lommel-s 1 𝜈 𝑧 {\displaystyle{\displaystyle\mathbf{E}_{\nu}\left(z\right)=-\frac{1}{\pi}(1+% \cos\left(\pi\nu\right))s_{{0},{\nu}}\left(z\right)\\ -\frac{\nu}{\pi}(1-\cos\left(\pi\nu\right))s_{{-1},{\nu}}\left(z\right)}} WeberE(nu, z)= -(1)/(Pi)*(1 + cos(Pi*nu))* LommelS1(0, nu, z)-(nu)/(Pi)*(1 - cos(Pi*nu))* LommelS1(- 1, nu, z) Error Successful Error - -
11.10.E19 𝐄 1 2 ⁑ ( z ) = ( 1 2 ⁒ Ο€ ⁒ z ) - 1 2 ⁒ ( A + ⁒ ( Ο‡ ) ⁒ cos ⁑ z - A - ⁒ ( Ο‡ ) ⁒ sin ⁑ z ) Weber-E 1 2 𝑧 superscript 1 2 πœ‹ 𝑧 1 2 subscript 𝐴 πœ’ 𝑧 subscript 𝐴 πœ’ 𝑧 {\displaystyle{\displaystyle\mathbf{E}_{\frac{1}{2}}\left(z\right)\\ =(\tfrac{1}{2}\pi z)^{-\frac{1}{2}}(A_{+}(\chi)\cos z-A_{-}(\chi)\sin z)}} WeberE((1)/(2), z)=((1)/(2)*Pi*z)^(-(1)/(2))*(A[+]*(chi)*cos(z)- A[-]*(chi)*sin(z)) WeberE[Divide[1,2], z]=(Divide[1,2]*Pi*z)^(-Divide[1,2])*(Subscript[A, +]*(\[Chi])*Cos[z]- Subscript[A, -]*(\[Chi])*Sin[z]) Error Error - -
11.10.E20 - 𝐄 - 1 2 ⁑ ( z ) = ( 1 2 ⁒ Ο€ ⁒ z ) - 1 2 ⁒ ( A + ⁒ ( Ο‡ ) ⁒ sin ⁑ z + A - ⁒ ( Ο‡ ) ⁒ cos ⁑ z ) Weber-E 1 2 𝑧 superscript 1 2 πœ‹ 𝑧 1 2 subscript 𝐴 πœ’ 𝑧 subscript 𝐴 πœ’ 𝑧 {\displaystyle{\displaystyle-\mathbf{E}_{-\frac{1}{2}}\left(z\right)\\ =(\tfrac{1}{2}\pi z)^{-\frac{1}{2}}(A_{+}(\chi)\sin z+A_{-}(\chi)\cos z)}} - WeberE(-(1)/(2), z)=((1)/(2)*Pi*z)^(-(1)/(2))*(A[+]*(chi)*sin(z)+ A[-]*(chi)*cos(z)) - WeberE[-Divide[1,2], z]=(Divide[1,2]*Pi*z)^(-Divide[1,2])*(Subscript[A, +]*(\[Chi])*Sin[z]+ Subscript[A, -]*(\[Chi])*Cos[z]) Error Error - -
11.10#Ex3 A + ⁒ ( Ο‡ ) = C ⁑ ( Ο‡ ) + S ⁑ ( Ο‡ ) subscript 𝐴 πœ’ Fresnel-cosine-integral πœ’ Fresnel-sine-integral πœ’ {\displaystyle{\displaystyle A_{+}(\chi)=C\left(\chi\right)+S\left(\chi\right)}} A[+]*(chi)= FresnelC(chi)+ FresnelS(chi) Subscript[A, +]*(\[Chi])= FresnelC[\[Chi]]+ FresnelS[\[Chi]] Error Failure - Error
11.10#Ex3 A - ⁒ ( Ο‡ ) = C ⁑ ( Ο‡ ) - S ⁑ ( Ο‡ ) subscript 𝐴 πœ’ Fresnel-cosine-integral πœ’ Fresnel-sine-integral πœ’ {\displaystyle{\displaystyle A_{-}(\chi)=C\left(\chi\right)-S\left(\chi\right)}} A[-]*(chi)= FresnelC(chi)- FresnelS(chi) Subscript[A, -]*(\[Chi])= FresnelC[\[Chi]]- FresnelS[\[Chi]] Error Failure - Error
11.10.E22 𝐄 n ⁑ ( z ) = - 𝐇 n ⁑ ( z ) + 1 Ο€ ⁒ βˆ‘ k = 0 m 1 Ξ“ ⁑ ( k + 1 2 ) Ξ“ ⁑ ( n + 1 2 - k ) ⁒ ( 1 2 ⁒ z ) n - 2 ⁒ k - 1 Weber-E 𝑛 𝑧 Struve-H 𝑛 𝑧 1 πœ‹ superscript subscript π‘˜ 0 subscript π‘š 1 Euler-Gamma π‘˜ 1 2 Euler-Gamma 𝑛 1 2 π‘˜ superscript 1 2 𝑧 𝑛 2 π‘˜ 1 {\displaystyle{\displaystyle\mathbf{E}_{n}\left(z\right)=-\mathbf{H}_{n}\left(% z\right)+\frac{1}{\pi}\sum_{k=0}^{m_{1}}\frac{\Gamma\left(k+\tfrac{1}{2}\right% )}{\Gamma\left(n\!+\!\tfrac{1}{2}\!-\!k\right)}(\tfrac{1}{2}z)^{n-2k-1}}} WeberE(n, z)= - StruveH(n, z)+(1)/(Pi)*sum((GAMMA(k +(1)/(2)))/(GAMMA(n +(1)/(2)- k))*((1)/(2)*z)^(n - 2*k - 1), k = 0..m[1]) WeberE[n, z]= - StruveH[n, z]+Divide[1,Pi]*Sum[Divide[Gamma[k +Divide[1,2]],Gamma[n +Divide[1,2]- k]]*(Divide[1,2]*z)^(n - 2*k - 1), {k, 0, Subscript[m, 1]}] Failure Failure Skip Error
11.10.E23 𝐄 - n ⁑ ( z ) = - 𝐇 - n ⁑ ( z ) + ( - 1 ) n + 1 Ο€ ⁒ βˆ‘ k = 0 m 2 Ξ“ ⁑ ( n - k - 1 2 ) Ξ“ ⁑ ( k + 3 2 ) ⁒ ( 1 2 ⁒ z ) - n + 2 ⁒ k + 1 Weber-E 𝑛 𝑧 Struve-H 𝑛 𝑧 superscript 1 𝑛 1 πœ‹ superscript subscript π‘˜ 0 subscript π‘š 2 Euler-Gamma 𝑛 π‘˜ 1 2 Euler-Gamma π‘˜ 3 2 superscript 1 2 𝑧 𝑛 2 π‘˜ 1 {\displaystyle{\displaystyle\mathbf{E}_{-n}\left(z\right)=-\mathbf{H}_{-n}% \left(z\right)+\frac{(-1)^{n+1}}{\pi}\sum_{k=0}^{m_{2}}\frac{\Gamma\left(n\!-% \!k\!-\!\tfrac{1}{2}\right)}{\Gamma\left(k+\tfrac{3}{2}\right)}(\tfrac{1}{2}z)% ^{-n+2k+1}}} WeberE(- n, z)= - StruveH(- n, z)+((- 1)^(n + 1))/(Pi)*sum((GAMMA(n - k -(1)/(2)))/(GAMMA(k +(3)/(2)))*((1)/(2)*z)^(- n + 2*k + 1), k = 0..m[2]) WeberE[- n, z]= - StruveH[- n, z]+Divide[(- 1)^(n + 1),Pi]*Sum[Divide[Gamma[n - k -Divide[1,2]],Gamma[k +Divide[3,2]]]*(Divide[1,2]*z)^(- n + 2*k + 1), {k, 0, Subscript[m, 2]}] Failure Failure Skip Error
11.10#Ex5 m 1 = ⌊ 1 2 ⁒ n - 1 2 βŒ‹ subscript π‘š 1 1 2 𝑛 1 2 {\displaystyle{\displaystyle m_{1}=\left\lfloor\tfrac{1}{2}n-\tfrac{1}{2}% \right\rfloor}} m[1]= floor((1)/(2)*n -(1)/(2)) Subscript[m, 1]= Floor[Divide[1,2]*n -Divide[1,2]] Failure Failure
Fail
1.414213562+1.414213562*I <- {m[1] = 2^(1/2)+I*2^(1/2), n = 1}
1.414213562+1.414213562*I <- {m[1] = 2^(1/2)+I*2^(1/2), n = 2}
.414213562+1.414213562*I <- {m[1] = 2^(1/2)+I*2^(1/2), n = 3}
1.414213562-1.414213562*I <- {m[1] = 2^(1/2)-I*2^(1/2), n = 1}
... skip entries to safe data
 Error
11.10#Ex6 m 2 = ⌈ 1 2 ⁒ n - 3 2 βŒ‰ subscript π‘š 2 1 2 𝑛 3 2 {\displaystyle{\displaystyle m_{2}=\left\lceil\tfrac{1}{2}n-\tfrac{3}{2}\right% \rceil}} m[2]= ceil((1)/(2)*n -(3)/(2)) Subscript[m, 2]= Ceiling[Divide[1,2]*n -Divide[3,2]] Failure Failure
Fail
2.414213562+1.414213562*I <- {m[2] = 2^(1/2)+I*2^(1/2), n = 1}
1.414213562+1.414213562*I <- {m[2] = 2^(1/2)+I*2^(1/2), n = 2}
1.414213562+1.414213562*I <- {m[2] = 2^(1/2)+I*2^(1/2), n = 3}
2.414213562-1.414213562*I <- {m[2] = 2^(1/2)-I*2^(1/2), n = 1}
... skip entries to safe data
 Error
11.10.E29 𝐉 n ⁑ ( z ) = J n ⁑ ( z ) Anger-J 𝑛 𝑧 Bessel-J 𝑛 𝑧 {\displaystyle{\displaystyle\mathbf{J}_{n}\left(z\right)=J_{n}\left(z\right)}} AngerJ(n, z)= BesselJ(n, z) AngerJ[n, z]= BesselJ[n, z] Failure Failure Successful Error
11.10.E32 𝐉 Ξ½ - 1 ⁑ ( z ) + 𝐉 Ξ½ + 1 ⁑ ( z ) = 2 ⁒ Ξ½ z ⁒ 𝐉 Ξ½ ⁑ ( z ) - 2 Ο€ ⁒ z ⁒ sin ⁑ ( Ο€ ⁒ Ξ½ ) Anger-J 𝜈 1 𝑧 Anger-J 𝜈 1 𝑧 2 𝜈 𝑧 Anger-J 𝜈 𝑧 2 πœ‹ 𝑧 πœ‹ 𝜈 {\displaystyle{\displaystyle\mathbf{J}_{\nu-1}\left(z\right)+\mathbf{J}_{\nu+1% }\left(z\right)=\frac{2\nu}{z}\mathbf{J}_{\nu}\left(z\right)-\frac{2}{\pi z}% \sin\left(\pi\nu\right)}} AngerJ(nu - 1, z)+ AngerJ(nu + 1, z)=(2*nu)/(z)*AngerJ(nu, z)-(2)/(Pi*z)*sin(Pi*nu) AngerJ[\[Nu]- 1, z]+ AngerJ[\[Nu]+ 1, z]=Divide[2*\[Nu],z]*AngerJ[\[Nu], z]-Divide[2,Pi*z]*Sin[Pi*\[Nu]] Failure Failure Successful Error
11.10.E33 𝐄 Ξ½ - 1 ⁑ ( z ) + 𝐄 Ξ½ + 1 ⁑ ( z ) = 2 ⁒ Ξ½ z ⁒ 𝐄 Ξ½ ⁑ ( z ) - 2 Ο€ ⁒ z ⁒ ( 1 - cos ⁑ ( Ο€ ⁒ Ξ½ ) ) Weber-E 𝜈 1 𝑧 Weber-E 𝜈 1 𝑧 2 𝜈 𝑧 Weber-E 𝜈 𝑧 2 πœ‹ 𝑧 1 πœ‹ 𝜈 {\displaystyle{\displaystyle\mathbf{E}_{\nu-1}\left(z\right)+\mathbf{E}_{\nu+1% }\left(z\right)=\frac{2\nu}{z}\mathbf{E}_{\nu}\left(z\right)-\frac{2}{\pi z}(1% -\cos\left(\pi\nu\right))}} WeberE(nu - 1, z)+ WeberE(nu + 1, z)=(2*nu)/(z)*WeberE(nu, z)-(2)/(Pi*z)*(1 - cos(Pi*nu)) WeberE[\[Nu]- 1, z]+ WeberE[\[Nu]+ 1, z]=Divide[2*\[Nu],z]*WeberE[\[Nu], z]-Divide[2,Pi*z]*(1 - Cos[Pi*\[Nu]]) Failure Failure Successful Error
11.10.E34 2 ⁒ 𝐉 Ξ½ β€² ⁑ ( z ) = 𝐉 Ξ½ - 1 ⁑ ( z ) - 𝐉 Ξ½ + 1 ⁑ ( z ) 2 diffop Anger-J 𝜈 1 𝑧 Anger-J 𝜈 1 𝑧 Anger-J 𝜈 1 𝑧 {\displaystyle{\displaystyle 2\mathbf{J}_{\nu}'\left(z\right)=\mathbf{J}_{\nu-% 1}\left(z\right)-\mathbf{J}_{\nu+1}\left(z\right)}} 2*subs( temp=z, diff( AngerJ(nu, temp), temp$(1) ) )= AngerJ(nu - 1, z)- AngerJ(nu + 1, z) 2*(D[AngerJ[\[Nu], temp], {temp, 1}]/.temp-> z)= AngerJ[\[Nu]- 1, z]- AngerJ[\[Nu]+ 1, z] Failure Successful Successful -
11.10.E35 2 ⁒ 𝐄 Ξ½ β€² ⁑ ( z ) = 𝐄 Ξ½ - 1 ⁑ ( z ) - 𝐄 Ξ½ + 1 ⁑ ( z ) 2 diffop Weber-E 𝜈 1 𝑧 Weber-E 𝜈 1 𝑧 Weber-E 𝜈 1 𝑧 {\displaystyle{\displaystyle 2\mathbf{E}_{\nu}'\left(z\right)=\mathbf{E}_{\nu-% 1}\left(z\right)-\mathbf{E}_{\nu+1}\left(z\right)}} 2*subs( temp=z, diff( WeberE(nu, temp), temp$(1) ) )= WeberE(nu - 1, z)- WeberE(nu + 1, z) 2*(D[WeberE[\[Nu], temp], {temp, 1}]/.temp-> z)= WeberE[\[Nu]- 1, z]- WeberE[\[Nu]+ 1, z] Failure Successful Successful -
11.10.E36 z ⁒ 𝐉 Ξ½ β€² ⁑ ( z ) + Ξ½ ⁒ 𝐉 Ξ½ ⁑ ( z ) = + z ⁒ 𝐉 Ξ½ - 1 ⁑ ( z ) + sin ⁑ ( Ο€ ⁒ Ξ½ ) Ο€ 𝑧 diffop Anger-J 𝜈 1 𝑧 𝜈 Anger-J 𝜈 𝑧 𝑧 Anger-J 𝜈 1 𝑧 πœ‹ 𝜈 πœ‹ {\displaystyle{\displaystyle z\mathbf{J}_{\nu}'\left(z\right)+\nu\mathbf{J}_{% \nu}\left(z\right)=+z\mathbf{J}_{\nu-1}\left(z\right)+\frac{\sin\left(\pi\nu% \right)}{\pi}}} z*subs( temp=z, diff( AngerJ(nu, temp), temp$(1) ) )+ nu*AngerJ(nu, z)= + z*AngerJ(nu - 1, z)+(sin(Pi*nu))/(Pi) z*(D[AngerJ[\[Nu], temp], {temp, 1}]/.temp-> z)+ \[Nu]*AngerJ[\[Nu], z]= + z*AngerJ[\[Nu]- 1, z]+Divide[Sin[Pi*\[Nu]],Pi] Failure Failure Successful Error
11.10.E36 z ⁒ 𝐉 Ξ½ β€² ⁑ ( z ) - Ξ½ ⁒ 𝐉 Ξ½ ⁑ ( z ) = - z ⁒ 𝐉 Ξ½ + 1 ⁑ ( z ) - sin ⁑ ( Ο€ ⁒ Ξ½ ) Ο€ 𝑧 diffop Anger-J 𝜈 1 𝑧 𝜈 Anger-J 𝜈 𝑧 𝑧 Anger-J 𝜈 1 𝑧 πœ‹ 𝜈 πœ‹ {\displaystyle{\displaystyle z\mathbf{J}_{\nu}'\left(z\right)-\nu\mathbf{J}_{% \nu}\left(z\right)=-z\mathbf{J}_{\nu+1}\left(z\right)-\frac{\sin\left(\pi\nu% \right)}{\pi}}} z*subs( temp=z, diff( AngerJ(nu, temp), temp$(1) ) )- nu*AngerJ(nu, z)= - z*AngerJ(nu + 1, z)-(sin(Pi*nu))/(Pi) z*(D[AngerJ[\[Nu], temp], {temp, 1}]/.temp-> z)- \[Nu]*AngerJ[\[Nu], z]= - z*AngerJ[\[Nu]+ 1, z]-Divide[Sin[Pi*\[Nu]],Pi] Successful Failure - Error
11.10.E37 z ⁒ 𝐄 Ξ½ β€² ⁑ ( z ) + Ξ½ ⁒ 𝐄 Ξ½ ⁑ ( z ) = + z ⁒ 𝐄 Ξ½ - 1 ⁑ ( z ) + ( 1 - cos ⁑ ( Ο€ ⁒ Ξ½ ) ) Ο€ 𝑧 diffop Weber-E 𝜈 1 𝑧 𝜈 Weber-E 𝜈 𝑧 𝑧 Weber-E 𝜈 1 𝑧 1 πœ‹ 𝜈 πœ‹ {\displaystyle{\displaystyle z\mathbf{E}_{\nu}'\left(z\right)+\nu\mathbf{E}_{% \nu}\left(z\right)=+z\mathbf{E}_{\nu-1}\left(z\right)+\frac{(1-\cos\left(\pi% \nu\right))}{\pi}}} z*subs( temp=z, diff( WeberE(nu, temp), temp$(1) ) )+ nu*WeberE(nu, z)= + z*WeberE(nu - 1, z)+(1 - cos(Pi*nu))/(Pi) z*(D[WeberE[\[Nu], temp], {temp, 1}]/.temp-> z)+ \[Nu]*WeberE[\[Nu], z]= + z*WeberE[\[Nu]- 1, z]+Divide[1 - Cos[Pi*\[Nu]],Pi] Failure Failure Successful Error
11.10.E37 z ⁒ 𝐄 Ξ½ β€² ⁑ ( z ) - Ξ½ ⁒ 𝐄 Ξ½ ⁑ ( z ) = - z ⁒ 𝐄 Ξ½ + 1 ⁑ ( z ) - ( 1 - cos ⁑ ( Ο€ ⁒ Ξ½ ) ) Ο€ 𝑧 diffop Weber-E 𝜈 1 𝑧 𝜈 Weber-E 𝜈 𝑧 𝑧 Weber-E 𝜈 1 𝑧 1 πœ‹ 𝜈 πœ‹ {\displaystyle{\displaystyle z\mathbf{E}_{\nu}'\left(z\right)-\nu\mathbf{E}_{% \nu}\left(z\right)=-z\mathbf{E}_{\nu+1}\left(z\right)-\frac{(1-\cos\left(\pi% \nu\right))}{\pi}}} z*subs( temp=z, diff( WeberE(nu, temp), temp$(1) ) )- nu*WeberE(nu, z)= - z*WeberE(nu + 1, z)-(1 - cos(Pi*nu))/(Pi) z*(D[WeberE[\[Nu], temp], {temp, 1}]/.temp-> z)- \[Nu]*WeberE[\[Nu], z]= - z*WeberE[\[Nu]+ 1, z]-Divide[1 - Cos[Pi*\[Nu]],Pi] Successful Failure - Error
11.11.E12 ΞΌ = 1 - Ξ» 2 - ln ⁑ ( 1 + 1 - Ξ» 2 Ξ» ) πœ‡ 1 superscript πœ† 2 1 1 superscript πœ† 2 πœ† {\displaystyle{\displaystyle\mu=\sqrt{1-\lambda^{2}}-\ln\left(\frac{1+\sqrt{1-% \lambda^{2}}}{\lambda}\right)}} mu =sqrt(1 - (lambda)^(2))- ln((1 +sqrt(1 - (lambda)^(2)))/(lambda)) \[Mu]=Sqrt[1 - (\[Lambda])^(2)]- Log[Divide[1 +Sqrt[1 - (\[Lambda])^(2)],\[Lambda]]] Failure Failure
Fail
.1801572199+1.430482725*I <- {lambda = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)+I*2^(1/2)}
.1801572199-1.397944399*I <- {lambda = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)-I*2^(1/2)}
-2.648269904-1.397944399*I <- {lambda = 2^(1/2)+I*2^(1/2), mu = -2^(1/2)-I*2^(1/2)}
-2.648269904+1.430482725*I <- {lambda = 2^(1/2)+I*2^(1/2), mu = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Error
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