Struve and Related Functions - 11.4 Basic Properties

From testwiki
Revision as of 11:28, 28 June 2021 by Admin (talk | contribs) (Admin moved page Main Page to Verifying DLMF with Maple and Mathematica)
(diff) ← Older revision | Latest revision (diff) | Newer revision β†’ (diff)
Jump to navigation Jump to search


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
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}}}
\StruveK{n+\frac{1}{2}}@{z} = \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}		 β„œ ⁑ ( ( n + 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( n + 1 2 ) ) + k + 1 ) > 0 , β„œ ⁑ ( n + ( n + 1 2 ) + 3 2 ) > 0 formulae-sequence 𝑛 1 2 π‘˜ 1 0 formulae-sequence 𝑛 1 2 π‘˜ 1 0 𝑛 𝑛 1 2 3 2 0 {\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-(n+\frac{1}{2}))+% k+1)>0,\Re(n+(n+\frac{1}{2})+\tfrac{3}{2})>0}} 
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}, GenerateConditions->None]
Error Failure -
Failed [6 / 21]
Result: Plus[0.9229158558166265, Times[-0.4886025119029198, DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[4, []], Times[Plus[-18, Times[-8, ]], [Plus[1, ]]], Times[Plus[30, Times[22, ], Times[4, Power[, 2]], Power[1.5, 2]], [Plus[2, ]]], Times[-1, Plus[3, ], Power[1.5, 2], [Plus[3, ]]]], 0], Equal[[1], Plus[1, Times[2, Power[1.5, -2]]]], Equal[[2], Plus[Rational[1, 2], Times[12, Power[1.5, -4]], Times[2, Power[1.5, -2]]]], Equal[[3], Plus[Rational[1, 6], Times[120, Power[1.5, -6]], Times[12, Power[1.5, -4]], Power[1.5, -2]]], Equal[[4], Plus[Rational[1, 24], Times[1680, Power[1.5, -8]], Times[120, Power[1.5, -6]], Times[6, Power[1.5, -4]], Times[Rational[1, 3], Power[1.5, -2]]]]}]][1.0]]], {Rule[n, 1], Rule[z, 1.5]}


Result: Plus[1.3775876377262881, Times[-0.36645188392718997, DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[4, []], Times[Plus[-18, Times[-8, ]], [Plus[1, ]]], Times[Plus[30, Times[22, ], Times[4, Power[, 2]], Power[1.5, 2]], [Plus[2, ]]], Times[-1, Plus[3, ], Power[1.5, 2], [Plus[3, ]]]], 0], Equal[[1], Plus[1, Times[2, Power[1.5, -2]]]], Equal[[2], Plus[Rational[1, 2], Times[12, Power[1.5, -4]], Times[2, Power[1.5, -2]]]], Equal[[3], Plus[Rational[1, 6], Times[120, Power[1.5, -6]], Times[12, Power[1.5, -4]], Power[1.5, -2]]], Equal[[4], Plus[Rational[1, 24], Times[1680, Power[1.5, -8]], Times[120, Power[1.5, -6]], Times[6, Power[1.5, -4]], Times[Rational[1, 3], Power[1.5, -2]]]]}]][2.0]]], {Rule[n, 2], Rule[z, 1.5]}

... skip entries to safe data
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}}}
\modStruveL{n+\frac{1}{2}}@{z} = \modBesselI{-n-\frac{1}{2}}@{z}-\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}		 β„œ ⁑ ( ( - n - 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( n + ( n + 1 2 ) + 3 2 ) > 0 formulae-sequence 𝑛 1 2 π‘˜ 1 0 𝑛 𝑛 1 2 3 2 0 {\displaystyle{\displaystyle\Re((-n-\frac{1}{2})+k+1)>0,\Re(n+(n+\frac{1}{2})+% \tfrac{3}{2})>0}} 
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}, GenerateConditions->None]
Failure Failure Successful [Tested: 21]
Failed [6 / 21]
Result: Plus[-0.05428916798921324, Times[0.4886025119029198, DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[4, []], Times[Plus[-18, Times[-8, ]], [Plus[1, ]]], Times[Plus[30, Times[22, ], Times[4, Power[, 2]], Times[-1, Power[1.5, 2]]], [Plus[2, ]]], Times[Plus[3, ], Power[1.5, 2], [Plus[3, ]]]], 0], Equal[[1], Plus[1, Times[-2, Power[1.5, -2]]]], Equal[[2], Plus[Rational[1, 2], Times[12, Power[1.5, -4]], Times[-2, Power[1.5, -2]]]], Equal[[3], Plus[Rational[1, 6], Times[-120, Power[1.5, -6]], Times[12, Power[1.5, -4]], Times[-1, Power[1.5, -2]]]], Equal[[4], Plus[Rational[1, 24], Times[1680, Power[1.5, -8]], Times[-120, Power[1.5, -6]], Times[6, Power[1.5, -4]], Times[Rational[-1, 3], Power[1.5, -2]]]]}]][1.0]]], {Rule[n, 1], Rule[z, 1.5]}


Result: Plus[-0.726117621855728, Times[0.36645188392718997, DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[4, []], Times[Plus[-18, Times[-8, ]], [Plus[1, ]]], Times[Plus[30, Times[22, ], Times[4, Power[, 2]], Times[-1, Power[1.5, 2]]], [Plus[2, ]]], Times[Plus[3, ], Power[1.5, 2], [Plus[3, ]]]], 0], Equal[[1], Plus[1, Times[-2, Power[1.5, -2]]]], Equal[[2], Plus[Rational[1, 2], Times[12, Power[1.5, -4]], Times[-2, Power[1.5, -2]]]], Equal[[3], Plus[Rational[1, 6], Times[-120, Power[1.5, -6]], Times[12, Power[1.5, -4]], Times[-1, Power[1.5, -2]]]], Equal[[4], Plus[Rational[1, 24], Times[1680, Power[1.5, -8]], Times[-120, Power[1.5, -6]], Times[6, Power[1.5, -4]], Times[Rational[-1, 3], Power[1.5, -2]]]]}]][2.0]]], {Rule[n, 2], Rule[z, 1.5]}

... skip entries to safe data
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-\frac{1}{2}}@{z} = (-1)^{n}\BesselJ{n+\frac{1}{2}}@{z}		 β„œ ⁑ ( ( n + 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( n + ( - n - 1 2 ) + 3 2 ) > 0 formulae-sequence 𝑛 1 2 π‘˜ 1 0 𝑛 𝑛 1 2 3 2 0 {\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re(n+(-n-\frac{1}{2})+% \tfrac{3}{2})>0}} 
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 [Tested: 21] Successful [Tested: 21]
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)}}
\modStruveL{-n-\frac{1}{2}}@{z} = \modBesselI{n+\frac{1}{2}}@{z}		 β„œ ⁑ ( ( n + 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( n + ( - n - 1 2 ) + 3 2 ) > 0 formulae-sequence 𝑛 1 2 π‘˜ 1 0 𝑛 𝑛 1 2 3 2 0 {\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re(n+(-n-\frac{1}{2})+% \tfrac{3}{2})>0}} 
StruveL(- n -(1)/(2), z) = BesselI(n +(1)/(2), z)

StruveL[- n -Divide[1,2], z] == BesselI[n +Divide[1,2], z]
Failure Failure Error Successful [Tested: 21]
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{\frac{1}{2}}@{z} = \left(\frac{2}{\pi z}\right)^{\frac{1}{2}}(1-\cos@@{z})		 β„œ ⁑ ( n + ( 1 2 ) + 3 2 ) > 0 𝑛 1 2 3 2 0 {\displaystyle{\displaystyle\Re(n+(\frac{1}{2})+\tfrac{3}{2})>0}} 
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 [Tested: 7] Successful [Tested: 7]
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{-\frac{1}{2}}@{z} = \left(\frac{2}{\pi z}\right)^{\frac{1}{2}}\sin@@{z}		 β„œ ⁑ ( n + ( - 1 2 ) + 3 2 ) > 0 𝑛 1 2 3 2 0 {\displaystyle{\displaystyle\Re(n+(-\frac{1}{2})+\tfrac{3}{2})>0}} 
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 [Tested: 7] Successful [Tested: 7]
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)}}
\modStruveL{\frac{1}{2}}@{z} = \left(\frac{2}{\pi z}\right)^{\frac{1}{2}}(\cosh@@{z}-1)		 β„œ ⁑ ( n + ( 1 2 ) + 3 2 ) > 0 𝑛 1 2 3 2 0 {\displaystyle{\displaystyle\Re(n+(\frac{1}{2})+\tfrac{3}{2})>0}} 
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 [Tested: 7] Successful [Tested: 7]
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}}
\modStruveL{-\frac{1}{2}}@{z} = \left(\frac{2}{\pi z}\right)^{\frac{1}{2}}\sinh@@{z}		 β„œ ⁑ ( n + ( - 1 2 ) + 3 2 ) > 0 𝑛 1 2 3 2 0 {\displaystyle{\displaystyle\Re(n+(-\frac{1}{2})+\tfrac{3}{2})>0}} 
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 Error Successful [Tested: 7]
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{\frac{3}{2}}@{z} = \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)		 β„œ ⁑ ( n + ( 3 2 ) + 3 2 ) > 0 𝑛 3 2 3 2 0 {\displaystyle{\displaystyle\Re(n+(\frac{3}{2})+\tfrac{3}{2})>0}} 
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 [Tested: 7] Successful [Tested: 7]
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{-\frac{3}{2}}@{z} = \left(\frac{2}{\pi z}\right)^{\frac{1}{2}}\left(\cos@@{z}-\frac{\sin@@{z}}{z}\right)		 β„œ ⁑ ( n + ( - 3 2 ) + 3 2 ) > 0 𝑛 3 2 3 2 0 {\displaystyle{\displaystyle\Re(n+(-\frac{3}{2})+\tfrac{3}{2})>0}} 
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 [Tested: 7] Successful [Tested: 7]
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)}}
\modStruveL{\frac{3}{2}}@{z} = -\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)		 β„œ ⁑ ( n + ( 3 2 ) + 3 2 ) > 0 𝑛 3 2 3 2 0 {\displaystyle{\displaystyle\Re(n+(\frac{3}{2})+\tfrac{3}{2})>0}} 
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 [Tested: 7] Successful [Tested: 7]
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)}}
\modStruveL{-\frac{3}{2}}@{z} = \left(\frac{2}{\pi z}\right)^{\frac{1}{2}}\left(\cosh@@{z}-\frac{\sinh@@{z}}{z}\right)		 β„œ ⁑ ( n + ( - 3 2 ) + 3 2 ) > 0 𝑛 3 2 3 2 0 {\displaystyle{\displaystyle\Re(n+(-\frac{3}{2})+\tfrac{3}{2})>0}} 
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 Error Successful [Tested: 7]
11.4.E13 𝐇 Ξ½ ⁑ ( x ) β‰₯ 0 Struve-H 𝜈 π‘₯ 0 {\displaystyle{\displaystyle\mathbf{H}_{\nu}\left(x\right)\geq 0}}
\StruveH{\nu}@{x} \geq 0		 x > 0 , Ξ½ β‰₯ 1 2 , β„œ ⁑ ( n + Ξ½ + 3 2 ) > 0 formulae-sequence π‘₯ 0 formulae-sequence 𝜈 1 2 𝑛 𝜈 3 2 0 {\displaystyle{\displaystyle x>0,\nu\geq\tfrac{1}{2},\Re(n+\nu+\tfrac{3}{2})>0}} 
StruveH(nu, x) >= 0

StruveH[\[Nu], x] >= 0
Failure Failure Successful [Tested: 9] Successful [Tested: 9]
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} = \frac{2(\tfrac{1}{2}z)^{\nu+1}}{\sqrt{\pi}\EulerGamma@{\nu+\tfrac{3}{2}}}(1+\vartheta)		 Ξ½ β‰  - 3 2 , β„œ ⁑ ( Ξ½ + 3 2 ) > 0 , β„œ ⁑ ( n + Ξ½ + 3 2 ) > 0 formulae-sequence 𝜈 3 2 formulae-sequence 𝜈 3 2 0 𝑛 𝜈 3 2 0 {\displaystyle{\displaystyle\nu\neq-\tfrac{3}{2},\Re(\nu+\tfrac{3}{2})>0,\Re(n% +\nu+\tfrac{3}{2})>0}} 
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
Failed [300 / 300]
Result: -.1471445522-.1672488986*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, vartheta = 1/2*3^(1/2)+1/2*I}


Result: .1483631977-.1537807385*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, vartheta = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.14714455195987888, -0.16724889870966364]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ο‘, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.8437410873580948, -0.4272690725617171]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ο‘, 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
11.4.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)}}
|\vartheta| < \frac{2}{3}\exp@{\frac{\tfrac{1}{4}|z|^{2}}{|\nu_{0}+\tfrac{3}{2}|}-1}		 

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
Failed [300 / 300]
Result: 1. < .2719639306
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, vartheta = 1/2*3^(1/2)+1/2*I, nu[0] = 1/2*3^(1/2)+1/2*I}


Result: 1. < .2962703575
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, vartheta = 1/2*3^(1/2)+1/2*I, nu[0] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: False
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ο‘, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[Ξ½, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: False
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ο‘, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[Ξ½, 0], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
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}@{ze^{m\pi i}} = e^{m\pi i(\nu+1)}\StruveH{\nu}@{z}		 β„œ ⁑ ( n + Ξ½ + 3 2 ) > 0 𝑛 𝜈 3 2 0 {\displaystyle{\displaystyle\Re(n+\nu+\tfrac{3}{2})>0}} 
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
Failed [36 / 70]
Result: .7482205956+.6031447740*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, m = 3}


Result: -.4043537260-.2594960110*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2), m = 3}

... skip entries to safe data
Failed [48 / 70]
Result: Complex[0.7482205967366697, 0.6031447730973842]
Test Values: {Rule[m, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.8264714651575658, -11.333535783044978]
Test Values: {Rule[m, 3], 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
11.4.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)}}
\modStruveL{\nu}@{ze^{m\pi i}} = e^{m\pi i(\nu+1)}\modStruveL{\nu}@{z}		 β„œ ⁑ ( n + Ξ½ + 3 2 ) > 0 𝑛 𝜈 3 2 0 {\displaystyle{\displaystyle\Re(n+\nu+\tfrac{3}{2})>0}} 
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
Failed [36 / 70]
Result: .7484016339+.7418531852*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, m = 3}


Result: -.3910618545-.1976660760*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2), m = 3}

... skip entries to safe data
Failed [48 / 70]
Result: Complex[0.7484016356562583, 0.741853184386289]
Test Values: {Rule[m, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.1393494415684403, -14.42209495054837]
Test Values: {Rule[m, 3], 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
11.4.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} = \frac{4}{\pi^{1/2}\EulerGamma@{\nu+\tfrac{1}{2}}}\*\sum_{k=0}^{\infty}\frac{(2k+\nu+1)\EulerGamma@{k+\nu+1}}{k!(2k+1)(2k+2\nu+1)}\BesselJ{2k+\nu+1}@{z}		 β„œ ⁑ ( ( 2 ⁒ k + Ξ½ + 1 ) + k + 1 ) > 0 , β„œ ⁑ ( Ξ½ + 1 2 ) > 0 , β„œ ⁑ ( k + Ξ½ + 1 ) > 0 , β„œ ⁑ ( n + Ξ½ + 3 2 ) > 0 formulae-sequence 2 π‘˜ 𝜈 1 π‘˜ 1 0 formulae-sequence 𝜈 1 2 0 formulae-sequence π‘˜ 𝜈 1 0 𝑛 𝜈 3 2 0 {\displaystyle{\displaystyle\Re((2k+\nu+1)+k+1)>0,\Re(\nu+\tfrac{1}{2})>0,\Re(% k+\nu+1)>0,\Re(n+\nu+\tfrac{3}{2})>0}} 
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}, GenerateConditions->None]
Failure Failure Successful [Tested: 7]
Failed [35 / 35]
Result: Plus[Complex[0.19324594490102928, 0.050519652606000824], Times[Complex[-2.8810800784728325, -0.07996643500485433], NSum[Times[Power[Plus[1, Times[2, k]], -1], Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Times[2, k]], Power[Plus[1, Times[2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[2, k]], -1], BesselJ[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Times[2, k]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Power[Factorial[k], -1], Gamma[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]], k]]]
Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Complex[-0.11400577441337441, 0.7764453237975459], Times[Complex[-3.5865453830779916, 1.1372180444285063], NSum[Times[Power[Plus[1, Times[2, k]], -1], Plus[1, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]], Times[2, k]], Power[Plus[1, Times[2, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]], Times[2, k]], -1], BesselJ[Plus[1, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]], Times[2, k]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Power[Factorial[k], -1], Gamma[Plus[1, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]], k]]]
Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {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
11.4.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} = \left(\frac{z}{2\pi}\right)^{1/2}\sum_{k=0}^{\infty}\frac{(\tfrac{1}{2}z)^{k}}{k!(k+\tfrac{1}{2})}\BesselJ{k+\nu+\frac{1}{2}}@{z}		 β„œ ⁑ ( ( k + Ξ½ + 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( n + Ξ½ + 3 2 ) > 0 formulae-sequence π‘˜ 𝜈 1 2 π‘˜ 1 0 𝑛 𝜈 3 2 0 {\displaystyle{\displaystyle\Re((k+\nu+\frac{1}{2})+k+1)>0,\Re(n+\nu+\tfrac{3}% {2})>0}} 
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}, GenerateConditions->None]
Failure Failure Skipped - Because timed out
Failed [70 / 70]
Result: Plus[Complex[0.19324594490102928, 0.050519652606000824], Times[Complex[-0.38534865183839906, -0.10325386006452089], NSum[Times[Power[2, Times[-1, k]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], k], Power[Plus[Rational[1, 2], k], -1], BesselJ[Plus[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Power[Factorial[k], -1]]
Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Complex[0.7460861755377195, -0.054406581179451755], Times[Complex[-0.38534865183839906, -0.10325386006452089], NSum[Times[Power[2, Times[-1, k]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], k], Power[Plus[Rational[1, 2], k], -1], BesselJ[Plus[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]], k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Power[Factorial[k], -1]]
Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {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
11.4.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} = \frac{(\tfrac{1}{2}z)^{\nu+\frac{1}{2}}}{\EulerGamma@{\nu+\tfrac{1}{2}}}\sum_{k=0}^{\infty}\frac{(\tfrac{1}{2}z)^{k}}{k!(k+\nu+\tfrac{1}{2})}\BesselJ{k+\frac{1}{2}}@{z}		 β„œ ⁑ ( ( k + 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( Ξ½ + 1 2 ) > 0 , β„œ ⁑ ( n + Ξ½ + 3 2 ) > 0 formulae-sequence π‘˜ 1 2 π‘˜ 1 0 formulae-sequence 𝜈 1 2 0 𝑛 𝜈 3 2 0 {\displaystyle{\displaystyle\Re((k+\frac{1}{2})+k+1)>0,\Re(\nu+\tfrac{1}{2})>0% ,\Re(n+\nu+\tfrac{3}{2})>0}} 
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}, GenerateConditions->None]
Failure Failure Successful [Tested: 35]
Failed [35 / 35]
Result: Plus[Complex[0.19324594490102928, 0.050519652606000824], Times[Complex[-0.35177626861232025, -0.14724813153619726], NSum[Times[Power[2, Times[-1, k]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], k], Power[Plus[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], k], -1], BesselJ[Plus[Rational[1, 2], k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Power[Factorial[k], -1]]
Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Complex[-0.11400577441337441, 0.7764453237975459], Times[Complex[-0.8980289919269182, -0.9563358827585198], NSum[Times[Power[2, Times[-1, k]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], k], Power[Plus[Rational[1, 2], Power[E, Times[Complex[0, Rational[-1, 3]], Pi]], k], -1], BesselJ[Plus[Rational[1, 2], k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Power[Factorial[k], -1]]
Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {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
11.4.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} = \frac{4}{\pi}\sum_{k=0}^{\infty}\frac{\BesselJ{2k+1}@{z}}{2k+1}		 β„œ ⁑ ( ( 2 ⁒ k + 1 ) + k + 1 ) > 0 , β„œ ⁑ ( n + 0 + 3 2 ) > 0 formulae-sequence 2 π‘˜ 1 π‘˜ 1 0 𝑛 0 3 2 0 {\displaystyle{\displaystyle\Re((2k+1)+k+1)>0,\Re(n+0+\tfrac{3}{2})>0}} 
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}, GenerateConditions->None]
Failure Successful Successful [Tested: 7] Successful [Tested: 7]
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)}}
\frac{4}{\pi}\sum_{k=0}^{\infty}\frac{\BesselJ{2k+1}@{z}}{2k+1} = 2\sum_{k=0}^{\infty}(-1)^{k}\BesselJ{k+\frac{1}{2}}^{2}@{\tfrac{1}{2}z}		 β„œ ⁑ ( ( 2 ⁒ k + 1 ) + k + 1 ) > 0 , β„œ ⁑ ( n + 0 + 3 2 ) > 0 , β„œ ⁑ ( ( k + 1 2 ) + k + 1 ) > 0 formulae-sequence 2 π‘˜ 1 π‘˜ 1 0 formulae-sequence 𝑛 0 3 2 0 π‘˜ 1 2 π‘˜ 1 0 {\displaystyle{\displaystyle\Re((2k+1)+k+1)>0,\Re(n+0+\tfrac{3}{2})>0,\Re((k+% \frac{1}{2})+k+1)>0}} 
(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}, GenerateConditions->None] == 2*Sum[(- 1)^(k)* (BesselJ[k +Divide[1,2], Divide[1,2]*z])^(2), {k, 0, Infinity}, GenerateConditions->None]
Failure Failure Successful [Tested: 7]
Failed [7 / 7]
Result: Plus[Complex[0.5489285468594604, 0.24901722825393072], Times[-2.0, NSum[Times[Power[-1, k], Power[BesselJ[Plus[Rational[1, 2], k], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], 2]]
Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Complex[-0.39043053959878776, 0.5488285427518664], Times[-2.0, NSum[Times[Power[-1, k], Power[BesselJ[Plus[Rational[1, 2], k], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]], 2]]
Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
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} = \frac{2}{\pi}(1-\BesselJ{0}@{z})+\frac{4}{\pi}\sum_{k=1}^{\infty}\frac{\BesselJ{2k}@{z}}{4k^{2}-1}		 β„œ ⁑ ( 0 + k + 1 ) > 0 , β„œ ⁑ ( ( 2 ⁒ k ) + k + 1 ) > 0 , β„œ ⁑ ( n + 1 + 3 2 ) > 0 formulae-sequence 0 π‘˜ 1 0 formulae-sequence 2 π‘˜ π‘˜ 1 0 𝑛 1 3 2 0 {\displaystyle{\displaystyle\Re(0+k+1)>0,\Re((2k)+k+1)>0,\Re(n+1+\tfrac{3}{2})% >0}} 
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}, GenerateConditions->None]
Failure Successful Successful [Tested: 7] Successful [Tested: 7]
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)}}
\frac{2}{\pi}(1-\BesselJ{0}@{z})+\frac{4}{\pi}\sum_{k=1}^{\infty}\frac{\BesselJ{2k}@{z}}{4k^{2}-1} = 4\sum_{k=0}^{\infty}\BesselJ{2k+\frac{1}{2}}@{\tfrac{1}{2}z}\BesselJ{2k+\frac{3}{2}}@{\tfrac{1}{2}z}		 β„œ ⁑ ( 0 + k + 1 ) > 0 , β„œ ⁑ ( ( 2 ⁒ k ) + k + 1 ) > 0 , β„œ ⁑ ( n + 1 + 3 2 ) > 0 , β„œ ⁑ ( ( 2 ⁒ k + 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( 2 ⁒ k + 3 2 ) + k + 1 ) > 0 formulae-sequence 0 π‘˜ 1 0 formulae-sequence 2 π‘˜ π‘˜ 1 0 formulae-sequence 𝑛 1 3 2 0 formulae-sequence 2 π‘˜ 1 2 π‘˜ 1 0 2 π‘˜ 3 2 π‘˜ 1 0 {\displaystyle{\displaystyle\Re(0+k+1)>0,\Re((2k)+k+1)>0,\Re(n+1+\tfrac{3}{2})% >0,\Re((2k+\frac{1}{2})+k+1)>0,\Re((2k+\frac{3}{2})+k+1)>0}} 
(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}, GenerateConditions->None] == 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}, GenerateConditions->None]
Failure Failure Successful [Tested: 7]
Failed [7 / 7]
Result: Plus[Complex[0.11277588530299563, 0.1715300454702578], Times[-4.0, NSum[Times[BesselJ[Plus[Rational[1, 2], Times[2, k]], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], BesselJ[Plus[Rational[3, 2], Times[2, k]], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]
Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Plus[Complex[-0.09862236423565694, -0.19602243923212043], Times[-4.0, NSum[Times[BesselJ[Plus[Rational[1, 2], Times[2, k]], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]], BesselJ[Plus[Rational[3, 2], Times[2, k]], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]]
Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
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} = \frac{2\nu}{z}\StruveH{\nu}@{z}+\frac{(\tfrac{1}{2}z)^{\nu}}{\sqrt{\pi}\EulerGamma@{\nu+\tfrac{3}{2}}}		 β„œ ⁑ ( Ξ½ + 3 2 ) > 0 , β„œ ⁑ ( n + ( Ξ½ - 1 ) + 3 2 ) > 0 , β„œ ⁑ ( n + ( Ξ½ + 1 ) + 3 2 ) > 0 , β„œ ⁑ ( n + Ξ½ + 3 2 ) > 0 formulae-sequence 𝜈 3 2 0 formulae-sequence 𝑛 𝜈 1 3 2 0 formulae-sequence 𝑛 𝜈 1 3 2 0 𝑛 𝜈 3 2 0 {\displaystyle{\displaystyle\Re(\nu+\tfrac{3}{2})>0,\Re(n+(\nu-1)+\tfrac{3}{2}% )>0,\Re(n+(\nu+1)+\tfrac{3}{2})>0,\Re(n+\nu+\tfrac{3}{2})>0}} 
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 [Tested: 56] Successful [Tested: 56]
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\StruveH{\nu}'@{z}-\frac{(\tfrac{1}{2}z)^{\nu}}{\sqrt{\pi}\EulerGamma@{\nu+\tfrac{3}{2}}}		 β„œ ⁑ ( Ξ½ + 3 2 ) > 0 , β„œ ⁑ ( n + ( Ξ½ - 1 ) + 3 2 ) > 0 , β„œ ⁑ ( n + ( Ξ½ + 1 ) + 3 2 ) > 0 , β„œ ⁑ ( n + Ξ½ + 3 2 ) > 0 formulae-sequence 𝜈 3 2 0 formulae-sequence 𝑛 𝜈 1 3 2 0 formulae-sequence 𝑛 𝜈 1 3 2 0 𝑛 𝜈 3 2 0 {\displaystyle{\displaystyle\Re(\nu+\tfrac{3}{2})>0,\Re(n+(\nu-1)+\tfrac{3}{2}% )>0,\Re(n+(\nu+1)+\tfrac{3}{2})>0,\Re(n+\nu+\tfrac{3}{2})>0}} 
StruveH(nu - 1, z)- StruveH(nu + 1, z) = 2*diff( StruveH(nu, z), z$(1) )-(((1)/(2)*z)^(nu))/(sqrt(Pi)*GAMMA(nu +(3)/(2)))
 
StruveH[\[Nu]- 1, z]- StruveH[\[Nu]+ 1, z] == 2*D[StruveH[\[Nu], z], {z, 1}]-Divide[(Divide[1,2]*z)^\[Nu],Sqrt[Pi]*Gamma[\[Nu]+Divide[3,2]]]
 Failure Successful Successful [Tested: 56] Successful [Tested: 56]
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)}}}
\modStruveL{\nu-1}@{z}-\modStruveL{\nu+1}@{z} = \frac{2\nu}{z}\modStruveL{\nu}@{z}+\frac{(\tfrac{1}{2}z)^{\nu}}{\sqrt{\pi}\EulerGamma@{\nu+\tfrac{3}{2}}}		 β„œ ⁑ ( Ξ½ + 3 2 ) > 0 , β„œ ⁑ ( n + ( Ξ½ - 1 ) + 3 2 ) > 0 , β„œ ⁑ ( n + ( Ξ½ + 1 ) + 3 2 ) > 0 , β„œ ⁑ ( n + Ξ½ + 3 2 ) > 0 formulae-sequence 𝜈 3 2 0 formulae-sequence 𝑛 𝜈 1 3 2 0 formulae-sequence 𝑛 𝜈 1 3 2 0 𝑛 𝜈 3 2 0 {\displaystyle{\displaystyle\Re(\nu+\tfrac{3}{2})>0,\Re(n+(\nu-1)+\tfrac{3}{2}% )>0,\Re(n+(\nu+1)+\tfrac{3}{2})>0,\Re(n+\nu+\tfrac{3}{2})>0}} 
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 [Tested: 56] Successful [Tested: 56]
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)}}}
\modStruveL{\nu-1}@{z}+\modStruveL{\nu+1}@{z} = 2\modStruveL{\nu}'@{z}-\frac{(\tfrac{1}{2}z)^{\nu}}{\sqrt{\pi}\EulerGamma@{\nu+\tfrac{3}{2}}}		 β„œ ⁑ ( Ξ½ + 3 2 ) > 0 , β„œ ⁑ ( n + ( Ξ½ - 1 ) + 3 2 ) > 0 , β„œ ⁑ ( n + ( Ξ½ + 1 ) + 3 2 ) > 0 , β„œ ⁑ ( n + Ξ½ + 3 2 ) > 0 formulae-sequence 𝜈 3 2 0 formulae-sequence 𝑛 𝜈 1 3 2 0 formulae-sequence 𝑛 𝜈 1 3 2 0 𝑛 𝜈 3 2 0 {\displaystyle{\displaystyle\Re(\nu+\tfrac{3}{2})>0,\Re(n+(\nu-1)+\tfrac{3}{2}% )>0,\Re(n+(\nu+1)+\tfrac{3}{2})>0,\Re(n+\nu+\tfrac{3}{2})>0}} 
StruveL(nu - 1, z)+ StruveL(nu + 1, z) = 2*diff( StruveL(nu, z), z$(1) )-(((1)/(2)*z)^(nu))/(sqrt(Pi)*GAMMA(nu +(3)/(2)))
 
StruveL[\[Nu]- 1, z]+ StruveL[\[Nu]+ 1, z] == 2*D[StruveL[\[Nu], z], {z, 1}]-Divide[(Divide[1,2]*z)^\[Nu],Sqrt[Pi]*Gamma[\[Nu]+Divide[3,2]]]
 Failure Successful Successful [Tested: 56] Successful [Tested: 56]
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)}}
\deriv{}{z}\left(z^{\nu}\StruveH{\nu}@{z}\right) = z^{\nu}\StruveH{\nu-1}@{z}		 β„œ ⁑ ( n + Ξ½ + 3 2 ) > 0 , β„œ ⁑ ( n + ( Ξ½ - 1 ) + 3 2 ) > 0 formulae-sequence 𝑛 𝜈 3 2 0 𝑛 𝜈 1 3 2 0 {\displaystyle{\displaystyle\Re(n+\nu+\tfrac{3}{2})>0,\Re(n+(\nu-1)+\tfrac{3}{% 2})>0}} 
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 [Tested: 70] Successful [Tested: 70]
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)}}
\deriv{}{z}\left(z^{-\nu}\StruveH{\nu}@{z}\right) = \frac{2^{-\nu}}{\sqrt{\pi}\EulerGamma@{\nu+\tfrac{3}{2}}}-z^{-\nu}\StruveH{\nu+1}@{z}		 β„œ ⁑ ( Ξ½ + 3 2 ) > 0 , β„œ ⁑ ( n + Ξ½ + 3 2 ) > 0 , β„œ ⁑ ( n + ( Ξ½ + 1 ) + 3 2 ) > 0 formulae-sequence 𝜈 3 2 0 formulae-sequence 𝑛 𝜈 3 2 0 𝑛 𝜈 1 3 2 0 {\displaystyle{\displaystyle\Re(\nu+\tfrac{3}{2})>0,\Re(n+\nu+\tfrac{3}{2})>0,% \Re(n+(\nu+1)+\tfrac{3}{2})>0}} 
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 - Successful [Tested: 56]
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)}}
\deriv{}{z}\left(z^{\nu}\modStruveL{\nu}@{z}\right) = z^{\nu}\modStruveL{\nu-1}@{z}		 β„œ ⁑ ( n + Ξ½ + 3 2 ) > 0 , β„œ ⁑ ( n + ( Ξ½ - 1 ) + 3 2 ) > 0 formulae-sequence 𝑛 𝜈 3 2 0 𝑛 𝜈 1 3 2 0 {\displaystyle{\displaystyle\Re(n+\nu+\tfrac{3}{2})>0,\Re(n+(\nu-1)+\tfrac{3}{% 2})>0}} 
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 [Tested: 70] Successful [Tested: 70]
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)}}
\deriv{}{z}\left(z^{-\nu}\modStruveL{\nu}@{z}\right) = \frac{2^{-\nu}}{\sqrt{\pi}\EulerGamma@{\nu+\tfrac{3}{2}}}+z^{-\nu}\modStruveL{\nu+1}@{z}		 β„œ ⁑ ( Ξ½ + 3 2 ) > 0 , β„œ ⁑ ( n + Ξ½ + 3 2 ) > 0 , β„œ ⁑ ( n + ( Ξ½ + 1 ) + 3 2 ) > 0 formulae-sequence 𝜈 3 2 0 formulae-sequence 𝑛 𝜈 3 2 0 𝑛 𝜈 1 3 2 0 {\displaystyle{\displaystyle\Re(\nu+\tfrac{3}{2})>0,\Re(n+\nu+\tfrac{3}{2})>0,% \Re(n+(\nu+1)+\tfrac{3}{2})>0}} 
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 - Successful [Tested: 56]
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)}}
\StruveH{0}'@{z} = \frac{2}{\pi}-\StruveH{1}@{z}		 β„œ ⁑ ( n + 0 + 3 2 ) > 0 , β„œ ⁑ ( n + 1 + 3 2 ) > 0 formulae-sequence 𝑛 0 3 2 0 𝑛 1 3 2 0 {\displaystyle{\displaystyle\Re(n+0+\tfrac{3}{2})>0,\Re(n+1+\tfrac{3}{2})>0}} 
diff( StruveH(0, z), z$(1) ) = (2)/(Pi)- StruveH(1, z)
 
D[StruveH[0, z], {z, 1}] == Divide[2,Pi]- StruveH[1, z]
 Successful Successful - Successful [Tested: 7]
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)}}
\deriv{}{z}(z\StruveH{1}@{z}) = z\StruveH{0}@{z}		 β„œ ⁑ ( n + 1 + 3 2 ) > 0 , β„œ ⁑ ( n + 0 + 3 2 ) > 0 formulae-sequence 𝑛 1 3 2 0 𝑛 0 3 2 0 {\displaystyle{\displaystyle\Re(n+1+\tfrac{3}{2})>0,\Re(n+0+\tfrac{3}{2})>0}} 
diff(z*StruveH(1, z), z) = z*StruveH(0, z)
 
D[z*StruveH[1, z], z] == z*StruveH[0, z]
 Successful Successful - Successful [Tested: 7]
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)}}
\modStruveL{0}'@{z} = \frac{2}{\pi}+\modStruveL{1}@{z}		 β„œ ⁑ ( n + 0 + 3 2 ) > 0 , β„œ ⁑ ( n + 1 + 3 2 ) > 0 formulae-sequence 𝑛 0 3 2 0 𝑛 1 3 2 0 {\displaystyle{\displaystyle\Re(n+0+\tfrac{3}{2})>0,\Re(n+1+\tfrac{3}{2})>0}} 
diff( StruveL(0, z), z$(1) ) = (2)/(Pi)+ StruveL(1, z)
 
D[StruveL[0, z], {z, 1}] == Divide[2,Pi]+ StruveL[1, z]
 Successful Successful - Successful [Tested: 7]
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)}}
\deriv{}{z}(z\modStruveL{1}@{z}) = z\modStruveL{0}@{z}		 β„œ ⁑ ( n + 1 + 3 2 ) > 0 , β„œ ⁑ ( n + 0 + 3 2 ) > 0 formulae-sequence 𝑛 1 3 2 0 𝑛 0 3 2 0 {\displaystyle{\displaystyle\Re(n+1+\tfrac{3}{2})>0,\Re(n+0+\tfrac{3}{2})>0}} 
diff(z*StruveL(1, z), z) = z*StruveL(0, z)
 
D[z*StruveL[1, z], z] == z*StruveL[0, z]
 Successful Successful - Successful [Tested: 7]
Retrieved from "https://lct.wmflabs.org/w/index.php?title=11.4&oldid=58233"