Bessel Functions - 10.16 Relations to Other Functions

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
10.16#Ex1 J 1 2 ⁑ ( z ) = Y - 1 2 ⁑ ( z ) Bessel-J 1 2 𝑧 Bessel-Y-Weber 1 2 𝑧 {\displaystyle{\displaystyle J_{\frac{1}{2}}\left(z\right)=Y_{-\frac{1}{2}}% \left(z\right)}}
\BesselJ{\frac{1}{2}}@{z} = \BesselY{-\frac{1}{2}}@{z}		 β„œ ⁑ ( ( 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( - 1 2 ) ) + k + 1 ) > 0 formulae-sequence 1 2 π‘˜ 1 0 formulae-sequence 1 2 π‘˜ 1 0 1 2 π‘˜ 1 0 {\displaystyle{\displaystyle\Re((\frac{1}{2})+k+1)>0,\Re((-\frac{1}{2})+k+1)>0% ,\Re((-(-\frac{1}{2}))+k+1)>0}} 
BesselJ((1)/(2), z) = BesselY(-(1)/(2), z)

BesselJ[Divide[1,2], z] == BesselY[-Divide[1,2], z]
Successful Successful Skip - symbolical successful subtest Successful [Tested: 7]
10.16#Ex1 Y - 1 2 ⁑ ( z ) = ( 2 Ο€ ⁒ z ) 1 2 ⁒ sin ⁑ z Bessel-Y-Weber 1 2 𝑧 superscript 2 πœ‹ 𝑧 1 2 𝑧 {\displaystyle{\displaystyle Y_{-\frac{1}{2}}\left(z\right)=\left(\frac{2}{\pi z% }\right)^{\frac{1}{2}}\sin z}}
\BesselY{-\frac{1}{2}}@{z} = \left(\frac{2}{\pi z}\right)^{\frac{1}{2}}\sin@@{z}		 β„œ ⁑ ( ( 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( - 1 2 ) ) + k + 1 ) > 0 formulae-sequence 1 2 π‘˜ 1 0 formulae-sequence 1 2 π‘˜ 1 0 1 2 π‘˜ 1 0 {\displaystyle{\displaystyle\Re((\frac{1}{2})+k+1)>0,\Re((-\frac{1}{2})+k+1)>0% ,\Re((-(-\frac{1}{2}))+k+1)>0}} 
BesselY(-(1)/(2), z) = ((2)/(Pi*z))^((1)/(2))* sin(z)

BesselY[-Divide[1,2], z] == (Divide[2,Pi*z])^(Divide[1,2])* Sin[z]
Failure Failure Successful [Tested: 7] Successful [Tested: 7]
10.16#Ex2 J - 1 2 ⁑ ( z ) = - Y 1 2 ⁑ ( z ) Bessel-J 1 2 𝑧 Bessel-Y-Weber 1 2 𝑧 {\displaystyle{\displaystyle J_{-\frac{1}{2}}\left(z\right)=-Y_{\frac{1}{2}}% \left(z\right)}}
\BesselJ{-\frac{1}{2}}@{z} = -\BesselY{\frac{1}{2}}@{z}		 β„œ ⁑ ( ( - 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( 1 2 ) ) + k + 1 ) > 0 formulae-sequence 1 2 π‘˜ 1 0 formulae-sequence 1 2 π‘˜ 1 0 1 2 π‘˜ 1 0 {\displaystyle{\displaystyle\Re((-\frac{1}{2})+k+1)>0,\Re((\frac{1}{2})+k+1)>0% ,\Re((-(\frac{1}{2}))+k+1)>0}} 
BesselJ(-(1)/(2), z) = - BesselY((1)/(2), z)

BesselJ[-Divide[1,2], z] == - BesselY[Divide[1,2], z]
Successful Successful Skip - symbolical successful subtest Successful [Tested: 7]
10.16#Ex2 - Y 1 2 ⁑ ( z ) = ( 2 Ο€ ⁒ z ) 1 2 ⁒ cos ⁑ z Bessel-Y-Weber 1 2 𝑧 superscript 2 πœ‹ 𝑧 1 2 𝑧 {\displaystyle{\displaystyle-Y_{\frac{1}{2}}\left(z\right)=\left(\frac{2}{\pi z% }\right)^{\frac{1}{2}}\cos z}}
-\BesselY{\frac{1}{2}}@{z} = \left(\frac{2}{\pi z}\right)^{\frac{1}{2}}\cos@@{z}		 β„œ ⁑ ( ( - 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( 1 2 ) ) + k + 1 ) > 0 formulae-sequence 1 2 π‘˜ 1 0 formulae-sequence 1 2 π‘˜ 1 0 1 2 π‘˜ 1 0 {\displaystyle{\displaystyle\Re((-\frac{1}{2})+k+1)>0,\Re((\frac{1}{2})+k+1)>0% ,\Re((-(\frac{1}{2}))+k+1)>0}} 
- BesselY((1)/(2), z) = ((2)/(Pi*z))^((1)/(2))* cos(z)

- BesselY[Divide[1,2], z] == (Divide[2,Pi*z])^(Divide[1,2])* Cos[z]
Failure Failure Successful [Tested: 7] Successful [Tested: 7]
10.16#Ex3 H 1 2 ( 1 ) ⁑ ( z ) = - i ⁒ H - 1 2 ( 1 ) ⁑ ( z ) Hankel-H-1-Bessel-third-kind 1 2 𝑧 𝑖 Hankel-H-1-Bessel-third-kind 1 2 𝑧 {\displaystyle{\displaystyle{H^{(1)}_{\frac{1}{2}}}\left(z\right)=-i{H^{(1)}_{% -\frac{1}{2}}}\left(z\right)}}
\HankelH{1}{\frac{1}{2}}@{z} = -i\HankelH{1}{-\frac{1}{2}}@{z}		 

HankelH1((1)/(2), z) = - I*HankelH1(-(1)/(2), z)

HankelH1[Divide[1,2], z] == - I*HankelH1[-Divide[1,2], z]
Successful Successful Skip - symbolical successful subtest Successful [Tested: 7]
10.16#Ex3 - i ⁒ H - 1 2 ( 1 ) ⁑ ( z ) = - i ⁒ ( 2 Ο€ ⁒ z ) 1 2 ⁒ e i ⁒ z 𝑖 Hankel-H-1-Bessel-third-kind 1 2 𝑧 𝑖 superscript 2 πœ‹ 𝑧 1 2 superscript 𝑒 𝑖 𝑧 {\displaystyle{\displaystyle-i{H^{(1)}_{-\frac{1}{2}}}\left(z\right)=-i\left(% \frac{2}{\pi z}\right)^{\frac{1}{2}}e^{iz}}}
-i\HankelH{1}{-\frac{1}{2}}@{z} = -i\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}e^{iz}		 

- I*HankelH1(-(1)/(2), z) = - I*((2)/(Pi*z))^((1)/(2))* exp(I*z)

- I*HankelH1[-Divide[1,2], z] == - I*(Divide[2,Pi*z])^(Divide[1,2])* Exp[I*z]
Failure Failure Successful [Tested: 7] Successful [Tested: 7]
10.16#Ex4 H 1 2 ( 2 ) ⁑ ( z ) = i ⁒ H - 1 2 ( 2 ) ⁑ ( z ) Hankel-H-2-Bessel-third-kind 1 2 𝑧 𝑖 Hankel-H-2-Bessel-third-kind 1 2 𝑧 {\displaystyle{\displaystyle{H^{(2)}_{\frac{1}{2}}}\left(z\right)=i{H^{(2)}_{-% \frac{1}{2}}}\left(z\right)}}
\HankelH{2}{\frac{1}{2}}@{z} = i\HankelH{2}{-\frac{1}{2}}@{z}		 

HankelH2((1)/(2), z) = I*HankelH2(-(1)/(2), z)

HankelH2[Divide[1,2], z] == I*HankelH2[-Divide[1,2], z]
Successful Successful Skip - symbolical successful subtest Successful [Tested: 7]
10.16#Ex4 i ⁒ H - 1 2 ( 2 ) ⁑ ( z ) = i ⁒ ( 2 Ο€ ⁒ z ) 1 2 ⁒ e - i ⁒ z 𝑖 Hankel-H-2-Bessel-third-kind 1 2 𝑧 𝑖 superscript 2 πœ‹ 𝑧 1 2 superscript 𝑒 𝑖 𝑧 {\displaystyle{\displaystyle i{H^{(2)}_{-\frac{1}{2}}}\left(z\right)=i\left(% \frac{2}{\pi z}\right)^{\frac{1}{2}}e^{-iz}}}
i\HankelH{2}{-\frac{1}{2}}@{z} = i\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}e^{-iz}		 

I*HankelH2(-(1)/(2), z) = I*((2)/(Pi*z))^((1)/(2))* exp(- I*z)

I*HankelH2[-Divide[1,2], z] == I*(Divide[2,Pi*z])^(Divide[1,2])* Exp[- I*z]
Failure Failure Successful [Tested: 7] Successful [Tested: 7]
10.16#Ex5 J 1 4 ⁑ ( z ) = - 2 - 1 4 ⁒ Ο€ - 1 2 ⁒ z - 1 4 ⁒ ( W ⁑ ( 0 , 2 ⁒ z 1 2 ) - W ⁑ ( 0 , - 2 ⁒ z 1 2 ) ) Bessel-J 1 4 𝑧 superscript 2 1 4 superscript πœ‹ 1 2 superscript 𝑧 1 4 parabolic-W 0 2 superscript 𝑧 1 2 parabolic-W 0 2 superscript 𝑧 1 2 {\displaystyle{\displaystyle J_{\frac{1}{4}}\left(z\right)=-2^{-\frac{1}{4}}% \pi^{-\frac{1}{2}}z^{-\frac{1}{4}}\left(W\left(0,2z^{\frac{1}{2}}\right)-W% \left(0,-2z^{\frac{1}{2}}\right)\right)}}
\BesselJ{\frac{1}{4}}@{z} = -2^{-\frac{1}{4}}\pi^{-\frac{1}{2}}z^{-\frac{1}{4}}\left(\paraW@{0}{2z^{\frac{1}{2}}}-\paraW@{0}{-2z^{\frac{1}{2}}}\right)		 β„œ ⁑ ( ( 1 4 ) + k + 1 ) > 0 1 4 π‘˜ 1 0 {\displaystyle{\displaystyle\Re((\frac{1}{4})+k+1)>0}} 
Error

BesselJ[Divide[1,4], z] == - (2)^(-Divide[1,4])* (Pi)^(-Divide[1,2])* (z)^(-Divide[1,4])*(Sqrt[(Sqrt[1+Exp[2*Pi*(0)]]-Exp[Pi*(0)])/2] * Exp[Divide[Pi*(0),4]] * ( Exp[I*(Pi/8 + Arg[GAMMA[1/2 + I*(0)]]/2)] * ParabolicCylinderD[- 1/2 - I*(0), 2*(z)^(Divide[1,2]) * Exp[-Divide[Pi*I,4]]] + Exp[-I*(Pi/8 + Arg[GAMMA[1/2 + I*(0)]]/2)] * ParabolicCylinderD[- 1/2 + I*(0), 2*(z)^(Divide[1,2]) * Exp[Divide[Pi*I,4]]] )- Sqrt[(Sqrt[1+Exp[2*Pi*(0)]]-Exp[Pi*(0)])/2] * Exp[Divide[Pi*(0),4]] * ( Exp[I*(Pi/8 + Arg[GAMMA[1/2 + I*(0)]]/2)] * ParabolicCylinderD[- 1/2 - I*(0), - 2*(z)^(Divide[1,2]) * Exp[-Divide[Pi*I,4]]] + Exp[-I*(Pi/8 + Arg[GAMMA[1/2 + I*(0)]]/2)] * ParabolicCylinderD[- 1/2 + I*(0), - 2*(z)^(Divide[1,2]) * Exp[Divide[Pi*I,4]]] ))
Missing Macro Error Failure -
Failed [7 / 7]
Result: Plus[Complex[0.8427727646508262, -0.04212015747529019], Times[Complex[0.4703662267003617, -0.06192488852586185], Plus[Times[0.4550898605622274, Plus[Times[Complex[0.3150667711363517, -1.1318933470332309], Power[2.718281828459045, Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]], Times[Complex[0.1941072423227021, 0.35884759380625464], Power[2.718281828459045, Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]], Times[-0.4550898605622274, Plus[Times[Complex[1.684848183162187, 0.4798071226199044], Power[2.718281828459045, Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]], Times[Complex[1.8058077119758371, -1.0109338182195815], Power[2.718281828459045, Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]]]]]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Complex[0.7942814592773979, 0.6544287188687908], Times[Complex[0.41086410074312574, -0.23721249916439713], Plus[Times[0.4550898605622274, Plus[Times[Complex[1.9382359752879499, -0.7976721648462198], Power[2.718281828459045, Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]], Times[Complex[0.22978077998995444, -0.1584303699393873], Power[2.718281828459045, Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]], Times[-0.4550898605622274, Plus[Times[Complex[0.8690225748967872, 1.5500248253586082], Power[2.718281828459045, Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]], Times[Complex[2.5774777701947826, 0.910783030451775], Power[2.718281828459045, Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]]]]]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.16#Ex6 J - 1 4 ⁑ ( z ) = 2 - 1 4 ⁒ Ο€ - 1 2 ⁒ z - 1 4 ⁒ ( W ⁑ ( 0 , 2 ⁒ z 1 2 ) + W ⁑ ( 0 , - 2 ⁒ z 1 2 ) ) Bessel-J 1 4 𝑧 superscript 2 1 4 superscript πœ‹ 1 2 superscript 𝑧 1 4 parabolic-W 0 2 superscript 𝑧 1 2 parabolic-W 0 2 superscript 𝑧 1 2 {\displaystyle{\displaystyle J_{-\frac{1}{4}}\left(z\right)=2^{-\frac{1}{4}}% \pi^{-\frac{1}{2}}z^{-\frac{1}{4}}\left(W\left(0,2z^{\frac{1}{2}}\right)+W% \left(0,-2z^{\frac{1}{2}}\right)\right)}}
\BesselJ{-\frac{1}{4}}@{z} = 2^{-\frac{1}{4}}\pi^{-\frac{1}{2}}z^{-\frac{1}{4}}\left(\paraW@{0}{2z^{\frac{1}{2}}}+\paraW@{0}{-2z^{\frac{1}{2}}}\right)		 β„œ ⁑ ( ( - 1 4 ) + k + 1 ) > 0 1 4 π‘˜ 1 0 {\displaystyle{\displaystyle\Re((-\frac{1}{4})+k+1)>0}} 
Error

BesselJ[-Divide[1,4], z] == (2)^(-Divide[1,4])* (Pi)^(-Divide[1,2])* (z)^(-Divide[1,4])*(Sqrt[(Sqrt[1+Exp[2*Pi*(0)]]-Exp[Pi*(0)])/2] * Exp[Divide[Pi*(0),4]] * ( Exp[I*(Pi/8 + Arg[GAMMA[1/2 + I*(0)]]/2)] * ParabolicCylinderD[- 1/2 - I*(0), 2*(z)^(Divide[1,2]) * Exp[-Divide[Pi*I,4]]] + Exp[-I*(Pi/8 + Arg[GAMMA[1/2 + I*(0)]]/2)] * ParabolicCylinderD[- 1/2 + I*(0), 2*(z)^(Divide[1,2]) * Exp[Divide[Pi*I,4]]] )+ Sqrt[(Sqrt[1+Exp[2*Pi*(0)]]-Exp[Pi*(0)])/2] * Exp[Divide[Pi*(0),4]] * ( Exp[I*(Pi/8 + Arg[GAMMA[1/2 + I*(0)]]/2)] * ParabolicCylinderD[- 1/2 - I*(0), - 2*(z)^(Divide[1,2]) * Exp[-Divide[Pi*I,4]]] + Exp[-I*(Pi/8 + Arg[GAMMA[1/2 + I*(0)]]/2)] * ParabolicCylinderD[- 1/2 + I*(0), - 2*(z)^(Divide[1,2]) * Exp[Divide[Pi*I,4]]] ))
Missing Macro Error Aborted -
Failed [7 / 7]
Result: Plus[Complex[0.7570692040611657, -0.36205959587261455], Times[Complex[-0.4703662267003617, 0.06192488852586186], Plus[Times[0.4550898605622274, Plus[Times[Complex[0.3150667711363517, -1.1318933470332309], Power[2.718281828459045, Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]], Times[Complex[0.1941072423227021, 0.35884759380625464], Power[2.718281828459045, Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]], Times[0.4550898605622274, Plus[Times[Complex[1.684848183162187, 0.4798071226199044], Power[2.718281828459045, Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]], Times[Complex[1.8058077119758371, -1.0109338182195815], Power[2.718281828459045, Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]]]]]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Complex[1.1199640481676587, -0.30003362129733535], Times[Complex[-0.41086410074312574, 0.2372124991643971], Plus[Times[0.4550898605622274, Plus[Times[Complex[1.9382359752879499, -0.7976721648462198], Power[2.718281828459045, Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]], Times[Complex[0.22978077998995444, -0.1584303699393873], Power[2.718281828459045, Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]], Times[0.4550898605622274, Plus[Times[Complex[0.8690225748967872, 1.5500248253586082], Power[2.718281828459045, Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]], Times[Complex[2.5774777701947826, 0.910783030451775], Power[2.718281828459045, Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]]]]]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.16#Ex7 J 3 4 ⁑ ( z ) = - 2 - 1 4 ⁒ Ο€ - 1 2 ⁒ z - 3 4 ⁒ ( W β€² ⁑ ( 0 , 2 ⁒ z 1 2 ) - W β€² ⁑ ( 0 , - 2 ⁒ z 1 2 ) ) Bessel-J 3 4 𝑧 superscript 2 1 4 superscript πœ‹ 1 2 superscript 𝑧 3 4 diffop parabolic-W 1 0 2 superscript 𝑧 1 2 diffop parabolic-W 1 0 2 superscript 𝑧 1 2 {\displaystyle{\displaystyle J_{\frac{3}{4}}\left(z\right)=-2^{-\frac{1}{4}}% \pi^{-\frac{1}{2}}z^{-\frac{3}{4}}\left(W'\left(0,2z^{\frac{1}{2}}\right)-W'% \left(0,-2z^{\frac{1}{2}}\right)\right)}}
\BesselJ{\frac{3}{4}}@{z} = -2^{-\frac{1}{4}}\pi^{-\frac{1}{2}}z^{-\frac{3}{4}}\left(\paraW'@{0}{2z^{\frac{1}{2}}}-\paraW'@{0}{-2z^{\frac{1}{2}}}\right)		 β„œ ⁑ ( ( 3 4 ) + k + 1 ) > 0 3 4 π‘˜ 1 0 {\displaystyle{\displaystyle\Re((\frac{3}{4})+k+1)>0}} 
Error

BesselJ[Divide[3,4], z] == - (2)^(-Divide[1,4])* (Pi)^(-Divide[1,2])* (z)^(-Divide[3,4])*((D[Sqrt[(Sqrt[1+Exp[2*Pi*(0)]]-Exp[Pi*(0)])/2] * Exp[Divide[Pi*(0),4]] * ( Exp[I*(Pi/8 + Arg[GAMMA[1/2 + I*(0)]]/2)] * ParabolicCylinderD[- 1/2 - I*(0), temp * Exp[-Divide[Pi*I,4]]] + Exp[-I*(Pi/8 + Arg[GAMMA[1/2 + I*(0)]]/2)] * ParabolicCylinderD[- 1/2 + I*(0), temp * Exp[Divide[Pi*I,4]]] ), {temp, 1}]/.temp-> 2*(z)^(Divide[1,2]))- (D[Sqrt[(Sqrt[1+Exp[2*Pi*(0)]]-Exp[Pi*(0)])/2] * Exp[Divide[Pi*(0),4]] * ( Exp[I*(Pi/8 + Arg[GAMMA[1/2 + I*(0)]]/2)] * ParabolicCylinderD[- 1/2 - I*(0), temp * Exp[-Divide[Pi*I,4]]] + Exp[-I*(Pi/8 + Arg[GAMMA[1/2 + I*(0)]]/2)] * ParabolicCylinderD[- 1/2 + I*(0), temp * Exp[Divide[Pi*I,4]]] ), {temp, 1}]/.temp-> - 2*(z)^(Divide[1,2])))
Missing Macro Error Failure -
Failed [7 / 7]
Result: Plus[Complex[0.5824093961234496, 0.15854248220296385], Times[Complex[0.43831154566767444, -0.18155458676026498], Plus[Times[0.4550898605622274, Plus[Times[Complex[-1.0141669743850696, 0.548925751618472], Power[2.718281828459045, Plus[Complex[0.0, 0.7853981633974483], Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]], Times[Complex[-0.3595065696883391, -0.29725176260213915], Power[2.718281828459045, Plus[Complex[0.0, -0.7853981633974483], Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]]], Times[-0.4550898605622274, Plus[Times[Complex[0.48667094453227255, 0.3574086420945919], Power[2.718281828459045, Plus[Complex[0.0, 0.7853981633974483], Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]], Times[Complex[-0.16798946016445826, 1.2035861563152026], Power[2.718281828459045, Plus[Complex[0.0, -0.7853981633974483], Times[Complex[0.0, 1.0], Plus[0.39<syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.0836786417162193, 0.6909849218136797], Times[Complex[0.0, -0.4744249983287943], Plus[Times[-0.4550898605622274, Plus[Times[Complex[-1.52733809531001, -0.015580244977093649], Power[2.718281828459045, Plus[Complex[0.0, 0.7853981633974483], Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]], Times[Complex[-1.3790215645615536, -1.2403191305633965], Power[2.718281828459045, Plus[Complex[0.0, -0.7853981633974483], Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]]], Times[0.4550898605622274, Plus[Times[Complex[-0.154282678975249, -1.0920025998149403], Power[2.718281828459045, Plus[Complex[0.0, 0.7853981633974483], Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]], Times[Complex[-0.302599209723706, 0.13273628577136276], Power[2.718281828459045, Plus[Complex[0.0, -0.7853981633974483], Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]]]]]]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.16#Ex8 J - 3 4 ⁑ ( z ) = - 2 - 1 4 ⁒ Ο€ - 1 2 ⁒ z - 3 4 ⁒ ( W β€² ⁑ ( 0 , 2 ⁒ z 1 2 ) + W β€² ⁑ ( 0 , - 2 ⁒ z 1 2 ) ) Bessel-J 3 4 𝑧 superscript 2 1 4 superscript πœ‹ 1 2 superscript 𝑧 3 4 diffop parabolic-W 1 0 2 superscript 𝑧 1 2 diffop parabolic-W 1 0 2 superscript 𝑧 1 2 {\displaystyle{\displaystyle J_{-\frac{3}{4}}\left(z\right)=-2^{-\frac{1}{4}}% \pi^{-\frac{1}{2}}z^{-\frac{3}{4}}\left(W'\left(0,2z^{\frac{1}{2}}\right)+W'% \left(0,-2z^{\frac{1}{2}}\right)\right)}}
\BesselJ{-\frac{3}{4}}@{z} = -2^{-\frac{1}{4}}\pi^{-\frac{1}{2}}z^{-\frac{3}{4}}\left(\paraW'@{0}{2z^{\frac{1}{2}}}+\paraW'@{0}{-2z^{\frac{1}{2}}}\right)		 β„œ ⁑ ( ( - 3 4 ) + k + 1 ) > 0 3 4 π‘˜ 1 0 {\displaystyle{\displaystyle\Re((-\frac{3}{4})+k+1)>0}} 
Error

BesselJ[-Divide[3,4], z] == - (2)^(-Divide[1,4])* (Pi)^(-Divide[1,2])* (z)^(-Divide[3,4])*((D[Sqrt[(Sqrt[1+Exp[2*Pi*(0)]]-Exp[Pi*(0)])/2] * Exp[Divide[Pi*(0),4]] * ( Exp[I*(Pi/8 + Arg[GAMMA[1/2 + I*(0)]]/2)] * ParabolicCylinderD[- 1/2 - I*(0), temp * Exp[-Divide[Pi*I,4]]] + Exp[-I*(Pi/8 + Arg[GAMMA[1/2 + I*(0)]]/2)] * ParabolicCylinderD[- 1/2 + I*(0), temp * Exp[Divide[Pi*I,4]]] ), {temp, 1}]/.temp-> 2*(z)^(Divide[1,2]))+ (D[Sqrt[(Sqrt[1+Exp[2*Pi*(0)]]-Exp[Pi*(0)])/2] * Exp[Divide[Pi*(0),4]] * ( Exp[I*(Pi/8 + Arg[GAMMA[1/2 + I*(0)]]/2)] * ParabolicCylinderD[- 1/2 - I*(0), temp * Exp[-Divide[Pi*I,4]]] + Exp[-I*(Pi/8 + Arg[GAMMA[1/2 + I*(0)]]/2)] * ParabolicCylinderD[- 1/2 + I*(0), temp * Exp[Divide[Pi*I,4]]] ), {temp, 1}]/.temp-> - 2*(z)^(Divide[1,2])))
Missing Macro Error Failure -
Failed [7 / 7]
Result: Plus[Complex[0.05605283808026881, -0.4145839244466886], Times[Complex[0.43831154566767444, -0.18155458676026498], Plus[Times[0.4550898605622274, Plus[Times[Complex[-1.0141669743850696, 0.548925751618472], Power[2.718281828459045, Plus[Complex[0.0, 0.7853981633974483], Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]], Times[Complex[-0.3595065696883391, -0.29725176260213915], Power[2.718281828459045, Plus[Complex[0.0, -0.7853981633974483], Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]]], Times[0.4550898605622274, Plus[Times[Complex[0.48667094453227255, 0.3574086420945919], Power[2.718281828459045, Plus[Complex[0.0, 0.7853981633974483], Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]], Times[Complex[-0.16798946016445826, 1.2035861563152026], Power[2.718281828459045, Plus[Complex[0.0, -0.7853981633974483], Times[Complex[0.0, 1.0], Plus[0.39<syntaxhighlight lang=mathematica>Result: Plus[Complex[0.44186162583484034, -0.6708696264637843], Times[Complex[0.0, -0.4744249983287943], Plus[Times[0.4550898605622274, Plus[Times[Complex[-1.52733809531001, -0.015580244977093649], Power[2.718281828459045, Plus[Complex[0.0, 0.7853981633974483], Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]], Times[Complex[-1.3790215645615536, -1.2403191305633965], Power[2.718281828459045, Plus[Complex[0.0, -0.7853981633974483], Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]]], Times[0.4550898605622274, Plus[Times[Complex[-0.154282678975249, -1.0920025998149403], Power[2.718281828459045, Plus[Complex[0.0, 0.7853981633974483], Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]], Times[Complex[-0.302599209723706, 0.13273628577136276], Power[2.718281828459045, Plus[Complex[0.0, -0.7853981633974483], Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]]]]]]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.16.E5 J Ξ½ ⁑ ( z ) = ( 1 2 ⁒ z ) Ξ½ ⁒ e - i ⁒ z Ξ“ ⁑ ( Ξ½ + 1 ) ⁒ M ⁑ ( Ξ½ + 1 2 , 2 ⁒ Ξ½ + 1 , + 2 ⁒ i ⁒ z ) Bessel-J 𝜈 𝑧 superscript 1 2 𝑧 𝜈 superscript 𝑒 𝑖 𝑧 Euler-Gamma 𝜈 1 Kummer-confluent-hypergeometric-M 𝜈 1 2 2 𝜈 1 2 𝑖 𝑧 {\displaystyle{\displaystyle J_{\nu}\left(z\right)=\frac{(\tfrac{1}{2}z)^{\nu}% e^{-iz}}{\Gamma\left(\nu+1\right)}M\left(\nu+\tfrac{1}{2},2\nu+1,+2iz\right)}}
\BesselJ{\nu}@{z} = \frac{(\tfrac{1}{2}z)^{\nu}e^{- iz}}{\EulerGamma@{\nu+1}}\KummerconfhyperM@{\nu+\tfrac{1}{2}}{2\nu+1}{+ 2iz}		 β„œ ⁑ ( Ξ½ + k + 1 ) > 0 , β„œ ⁑ ( Ξ½ + 1 ) > 0 formulae-sequence 𝜈 π‘˜ 1 0 𝜈 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re(\nu+1)>0}} 
BesselJ(nu, z) = (((1)/(2)*z)^(nu)* exp(- I*z))/(GAMMA(nu + 1))*KummerM(nu +(1)/(2), 2*nu + 1, + 2*I*z)

BesselJ[\[Nu], z] == Divide[(Divide[1,2]*z)^\[Nu]* Exp[- I*z],Gamma[\[Nu]+ 1]]*Hypergeometric1F1[\[Nu]+Divide[1,2], 2*\[Nu]+ 1, + 2*I*z]
Failure Successful
Failed [7 / 56]
Result: -.827986137e-1+.7317301038*I
Test Values: {nu = -1/2, z = 1/2*3^(1/2)+1/2*I}


Result: -.8060140108+.3257248263*I
Test Values: {nu = -1/2, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [7 / 56]
Result: Complex[-0.08279861346468581, 0.7317301035002939]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ξ½, -0.5]}


Result: Complex[-0.8060140105131326, 0.32572482654389856]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[Ξ½, -0.5]}

... skip entries to safe data
10.16.E5 J Ξ½ ⁑ ( z ) = ( 1 2 ⁒ z ) Ξ½ ⁒ e + i ⁒ z Ξ“ ⁑ ( Ξ½ + 1 ) ⁒ M ⁑ ( Ξ½ + 1 2 , 2 ⁒ Ξ½ + 1 , - 2 ⁒ i ⁒ z ) Bessel-J 𝜈 𝑧 superscript 1 2 𝑧 𝜈 superscript 𝑒 𝑖 𝑧 Euler-Gamma 𝜈 1 Kummer-confluent-hypergeometric-M 𝜈 1 2 2 𝜈 1 2 𝑖 𝑧 {\displaystyle{\displaystyle J_{\nu}\left(z\right)=\frac{(\tfrac{1}{2}z)^{\nu}% e^{+iz}}{\Gamma\left(\nu+1\right)}M\left(\nu+\tfrac{1}{2},2\nu+1,-2iz\right)}}
\BesselJ{\nu}@{z} = \frac{(\tfrac{1}{2}z)^{\nu}e^{+ iz}}{\EulerGamma@{\nu+1}}\KummerconfhyperM@{\nu+\tfrac{1}{2}}{2\nu+1}{- 2iz}		 β„œ ⁑ ( Ξ½ + k + 1 ) > 0 , β„œ ⁑ ( Ξ½ + 1 ) > 0 formulae-sequence 𝜈 π‘˜ 1 0 𝜈 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re(\nu+1)>0}} 
BesselJ(nu, z) = (((1)/(2)*z)^(nu)* exp(+ I*z))/(GAMMA(nu + 1))*KummerM(nu +(1)/(2), 2*nu + 1, - 2*I*z)

BesselJ[\[Nu], z] == Divide[(Divide[1,2]*z)^\[Nu]* Exp[+ I*z],Gamma[\[Nu]+ 1]]*Hypergeometric1F1[\[Nu]+Divide[1,2], 2*\[Nu]+ 1, - 2*I*z]
Failure Successful
Failed [7 / 56]
Result: .827986132e-1-.7317301035*I
Test Values: {nu = -1/2, z = 1/2*3^(1/2)+1/2*I}


Result: .8060140102-.3257248264*I
Test Values: {nu = -1/2, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [7 / 56]
Result: Complex[0.08279861346468548, -0.7317301035002935]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ξ½, -0.5]}


Result: Complex[0.8060140105131325, -0.325724826543898]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[Ξ½, -0.5]}

... skip entries to safe data
10.16.E7 J Ξ½ ⁑ ( z ) = e - ( 2 ⁒ Ξ½ + 1 ) ⁒ Ο€ ⁒ i / 4 2 2 ⁒ Ξ½ ⁒ Ξ“ ⁑ ( Ξ½ + 1 ) ⁒ ( 2 ⁒ z ) - 1 2 ⁒ M 0 , Ξ½ ⁑ ( + 2 ⁒ i ⁒ z ) Bessel-J 𝜈 𝑧 superscript 𝑒 2 𝜈 1 πœ‹ 𝑖 4 superscript 2 2 𝜈 Euler-Gamma 𝜈 1 superscript 2 𝑧 1 2 Whittaker-confluent-hypergeometric-M 0 𝜈 2 𝑖 𝑧 {\displaystyle{\displaystyle J_{\nu}\left(z\right)=\frac{e^{-(2\nu+1)\pi i/4}}% {2^{2\nu}\Gamma\left(\nu+1\right)}(2z)^{-\frac{1}{2}}M_{0,\nu}\left(+2iz\right% )}}
\BesselJ{\nu}@{z} = \frac{e^{-(2\nu+1)\pi i/4}}{2^{2\nu}\EulerGamma@{\nu+1}}(2z)^{-\frac{1}{2}}\WhittakerconfhyperM{0}{\nu}@{+ 2iz}		 β„œ ⁑ ( Ξ½ + k + 1 ) > 0 , β„œ ⁑ ( Ξ½ + 1 ) > 0 formulae-sequence 𝜈 π‘˜ 1 0 𝜈 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re(\nu+1)>0}} 
BesselJ(nu, z) = (exp(-(2*nu + 1)*Pi*I/4))/((2)^(2*nu)* GAMMA(nu + 1))*(2*z)^(-(1)/(2))* WhittakerM(0, nu, + 2*I*z)

BesselJ[\[Nu], z] == Divide[Exp[-(2*\[Nu]+ 1)*Pi*I/4],(2)^(2*\[Nu])* Gamma[\[Nu]+ 1]]*(2*z)^(-Divide[1,2])* WhittakerM[0, \[Nu], + 2*I*z]
Failure Failure
Failed [1 / 7]
Result: 1.448710179-.1398527410*I
Test Values: {z = -1/2+1/2*I*3^(1/2), nu = 1/4}

Failed [1 / 7]
Result: Complex[1.448710178146189, -0.13985274040860685]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[Ξ½, Rational[1, 4]]}

10.16.E7 J Ξ½ ⁑ ( z ) = e + ( 2 ⁒ Ξ½ + 1 ) ⁒ Ο€ ⁒ i / 4 2 2 ⁒ Ξ½ ⁒ Ξ“ ⁑ ( Ξ½ + 1 ) ⁒ ( 2 ⁒ z ) - 1 2 ⁒ M 0 , Ξ½ ⁑ ( - 2 ⁒ i ⁒ z ) Bessel-J 𝜈 𝑧 superscript 𝑒 2 𝜈 1 πœ‹ 𝑖 4 superscript 2 2 𝜈 Euler-Gamma 𝜈 1 superscript 2 𝑧 1 2 Whittaker-confluent-hypergeometric-M 0 𝜈 2 𝑖 𝑧 {\displaystyle{\displaystyle J_{\nu}\left(z\right)=\frac{e^{+(2\nu+1)\pi i/4}}% {2^{2\nu}\Gamma\left(\nu+1\right)}(2z)^{-\frac{1}{2}}M_{0,\nu}\left(-2iz\right% )}}
\BesselJ{\nu}@{z} = \frac{e^{+(2\nu+1)\pi i/4}}{2^{2\nu}\EulerGamma@{\nu+1}}(2z)^{-\frac{1}{2}}\WhittakerconfhyperM{0}{\nu}@{- 2iz}		 β„œ ⁑ ( Ξ½ + k + 1 ) > 0 , β„œ ⁑ ( Ξ½ + 1 ) > 0 formulae-sequence 𝜈 π‘˜ 1 0 𝜈 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re(\nu+1)>0}} 
BesselJ(nu, z) = (exp(+(2*nu + 1)*Pi*I/4))/((2)^(2*nu)* GAMMA(nu + 1))*(2*z)^(-(1)/(2))* WhittakerM(0, nu, - 2*I*z)

BesselJ[\[Nu], z] == Divide[Exp[+(2*\[Nu]+ 1)*Pi*I/4],(2)^(2*\[Nu])* Gamma[\[Nu]+ 1]]*(2*z)^(-Divide[1,2])* WhittakerM[0, \[Nu], - 2*I*z]
Failure Failure
Failed [1 / 7]
Result: 1.191860674-.595668984e-1*I
Test Values: {z = -1/2*3^(1/2)-1/2*I, nu = 1/4}

Failed [1 / 7]
Result: Complex[1.191860673767867, -0.059566897950845576]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]], Rule[Ξ½, Rational[1, 4]]}

10.16.E9 J Ξ½ ⁑ ( z ) = ( 1 2 ⁒ z ) Ξ½ Ξ“ ⁑ ( Ξ½ + 1 ) ⁒ F 1 0 ⁑ ( - ; Ξ½ + 1 ; - 1 4 ⁒ z 2 ) Bessel-J 𝜈 𝑧 superscript 1 2 𝑧 𝜈 Euler-Gamma 𝜈 1 Gauss-hypergeometric-pFq 0 1 𝜈 1 1 4 superscript 𝑧 2 {\displaystyle{\displaystyle J_{\nu}\left(z\right)=\frac{(\tfrac{1}{2}z)^{\nu}% }{\Gamma\left(\nu+1\right)}{{}_{0}F_{1}}\left(-;\nu+1;-\tfrac{1}{4}z^{2}\right% )}}
\BesselJ{\nu}@{z} = \frac{(\tfrac{1}{2}z)^{\nu}}{\EulerGamma@{\nu+1}}\genhyperF{0}{1}@{-}{\nu+1}{-\tfrac{1}{4}z^{2}}		 β„œ ⁑ ( Ξ½ + k + 1 ) > 0 , β„œ ⁑ ( Ξ½ + 1 ) > 0 formulae-sequence 𝜈 π‘˜ 1 0 𝜈 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re(\nu+1)>0}} 
BesselJ(nu, z) = (((1)/(2)*z)^(nu))/(GAMMA(nu + 1))*hypergeom([-], [nu + 1], -(1)/(4)*(z)^(2))

BesselJ[\[Nu], z] == Divide[(Divide[1,2]*z)^\[Nu],Gamma[\[Nu]+ 1]]*HypergeometricPFQ[{-}, {\[Nu]+ 1}, -Divide[1,4]*(z)^(2)]
Error Failure - Error
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