Bessel Functions - 10.47 Definitions and Basic Properties

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
10.47.E1 z 2 ⁢ d 2 w d z 2 + 2 ⁢ z ⁢ d w d z + ( z 2 - n ⁢ ( n + 1 ) ) ⁢ w = 0 superscript 𝑧 2 derivative 𝑤 𝑧 2 2 𝑧 derivative 𝑤 𝑧 superscript 𝑧 2 𝑛 𝑛 1 𝑤 0 {\displaystyle{\displaystyle z^{2}\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+% 2z\frac{\mathrm{d}w}{\mathrm{d}z}+\left(z^{2}-n(n+1)\right)w=0}}
z^{2}\deriv[2]{w}{z}+2z\deriv{w}{z}+\left(z^{2}-n(n+1)\right)w = 0		 

(z)^(2)* diff(w, [z$(2)])+ 2*z*diff(w, z)+((z)^(2)- n*(n + 1))*w = 0

(z)^(2)* D[w, {z, 2}]+ 2*z*D[w, z]+((z)^(2)- n*(n + 1))*w == 0
Failure Failure
Failed [210 / 210]
Result: -1.732050808+.3733632160e-9*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, n = 1}


Result: -5.196152424-2.000000000*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, n = 2}

... skip entries to safe data
Failed [210 / 210]
Result: Complex[-1.7320508075688772, 1.1102230246251565*^-16]
Test Values: {Rule[n, 1], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-5.196152422706633, -1.9999999999999996]
Test Values: {Rule[n, 2], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
10.47.E2 z 2 ⁢ d 2 w d z 2 + 2 ⁢ z ⁢ d w d z - ( z 2 + n ⁢ ( n + 1 ) ) ⁢ w = 0 superscript 𝑧 2 derivative 𝑤 𝑧 2 2 𝑧 derivative 𝑤 𝑧 superscript 𝑧 2 𝑛 𝑛 1 𝑤 0 {\displaystyle{\displaystyle z^{2}\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+% 2z\frac{\mathrm{d}w}{\mathrm{d}z}-\left(z^{2}+n(n+1)\right)w=0}}
z^{2}\deriv[2]{w}{z}+2z\deriv{w}{z}-\left(z^{2}+n(n+1)\right)w = 0		 

(z)^(2)* diff(w, [z$(2)])+ 2*z*diff(w, z)-((z)^(2)+ n*(n + 1))*w = 0

(z)^(2)* D[w, {z, 2}]+ 2*z*D[w, z]-((z)^(2)+ n*(n + 1))*w == 0
Failure Failure
Failed [210 / 210]
Result: -1.732050808-2.000000000*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, n = 1}


Result: -5.196152424-4.000000000*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, n = 2}

... skip entries to safe data
Failed [210 / 210]
Result: Complex[-1.7320508075688776, -1.9999999999999998]
Test Values: {Rule[n, 1], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-5.196152422706632, -3.9999999999999996]
Test Values: {Rule[n, 2], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
10.47.E3 𝗃 n ⁡ ( z ) = 1 2 ⁢ π / z ⁢ J n + 1 2 ⁡ ( z ) spherical-Bessel-J 𝑛 𝑧 1 2 𝜋 𝑧 Bessel-J 𝑛 1 2 𝑧 {\displaystyle{\displaystyle\mathsf{j}_{n}\left(z\right)=\sqrt{\tfrac{1}{2}\pi% /z}J_{n+\frac{1}{2}}\left(z\right)}}
\sphBesselJ{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\BesselJ{n+\frac{1}{2}}@{z}		 ℜ ⁡ ( ( n + 1 2 ) + k + 1 ) > 0 𝑛 1 2 𝑘 1 0 {\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0}} 
Error

SphericalBesselJ[n, z] == Sqrt[Divide[1,2]*Pi/z]*BesselJ[n +Divide[1,2], z]
Missing Macro Error Failure Skip - symbolical successful subtest Successful [Tested: 21]
10.47.E3 1 2 ⁢ π / z ⁢ J n + 1 2 ⁡ ( z ) = ( - 1 ) n ⁢ 1 2 ⁢ π / z ⁢ Y - n - 1 2 ⁡ ( z ) 1 2 𝜋 𝑧 Bessel-J 𝑛 1 2 𝑧 superscript 1 𝑛 1 2 𝜋 𝑧 Bessel-Y-Weber 𝑛 1 2 𝑧 {\displaystyle{\displaystyle\sqrt{\tfrac{1}{2}\pi/z}J_{n+\frac{1}{2}}\left(z% \right)=(-1)^{n}\sqrt{\tfrac{1}{2}\pi/z}Y_{-n-\frac{1}{2}}\left(z\right)}}
\sqrt{\tfrac{1}{2}\pi/z}\BesselJ{n+\frac{1}{2}}@{z} = (-1)^{n}\sqrt{\tfrac{1}{2}\pi/z}\BesselY{-n-\frac{1}{2}}@{z}		 ℜ ⁡ ( ( n + 1 2 ) + k + 1 ) > 0 , ℜ ⁡ ( ( - n - 1 2 ) + k + 1 ) > 0 , ℜ ⁡ ( ( - ( - n - 1 2 ) ) + k + 1 ) > 0 formulae-sequence 𝑛 1 2 𝑘 1 0 formulae-sequence 𝑛 1 2 𝑘 1 0 𝑛 1 2 𝑘 1 0 {\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+% 1)>0,\Re((-(-n-\frac{1}{2}))+k+1)>0}} 
sqrt((1)/(2)*Pi/z)*BesselJ(n +(1)/(2), z) = (- 1)^(n)*sqrt((1)/(2)*Pi/z)*BesselY(- n -(1)/(2), z)

Sqrt[Divide[1,2]*Pi/z]*BesselJ[n +Divide[1,2], z] == (- 1)^(n)*Sqrt[Divide[1,2]*Pi/z]*BesselY[- n -Divide[1,2], z]
Failure Failure Successful [Tested: 21] Successful [Tested: 21]
10.47.E4 𝗒 n ⁡ ( z ) = 1 2 ⁢ π / z ⁢ Y n + 1 2 ⁡ ( z ) spherical-Bessel-Y 𝑛 𝑧 1 2 𝜋 𝑧 Bessel-Y-Weber 𝑛 1 2 𝑧 {\displaystyle{\displaystyle\mathsf{y}_{n}\left(z\right)=\sqrt{\tfrac{1}{2}\pi% /z}Y_{n+\frac{1}{2}}\left(z\right)}}
\sphBesselY{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\BesselY{n+\frac{1}{2}}@{z}		 ℜ ⁡ ( ( n + 1 2 ) + k + 1 ) > 0 , ℜ ⁡ ( ( - ( n + 1 2 ) ) + k + 1 ) > 0 formulae-sequence 𝑛 1 2 𝑘 1 0 𝑛 1 2 𝑘 1 0 {\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-(n+\frac{1}{2}))+% k+1)>0}} 
Error

SphericalBesselY[n, z] == Sqrt[Divide[1,2]*Pi/z]*BesselY[n +Divide[1,2], z]
Missing Macro Error Failure Skip - symbolical successful subtest Successful [Tested: 21]
10.47.E4 1 2 ⁢ π / z ⁢ Y n + 1 2 ⁡ ( z ) = ( - 1 ) n + 1 ⁢ 1 2 ⁢ π / z ⁢ J - n - 1 2 ⁡ ( z ) 1 2 𝜋 𝑧 Bessel-Y-Weber 𝑛 1 2 𝑧 superscript 1 𝑛 1 1 2 𝜋 𝑧 Bessel-J 𝑛 1 2 𝑧 {\displaystyle{\displaystyle\sqrt{\tfrac{1}{2}\pi/z}Y_{n+\frac{1}{2}}\left(z% \right)=(-1)^{n+1}\sqrt{\tfrac{1}{2}\pi/z}J_{-n-\frac{1}{2}}\left(z\right)}}
\sqrt{\tfrac{1}{2}\pi/z}\BesselY{n+\frac{1}{2}}@{z} = (-1)^{n+1}\sqrt{\tfrac{1}{2}\pi/z}\BesselJ{-n-\frac{1}{2}}@{z}		 ℜ ⁡ ( ( n + 1 2 ) + k + 1 ) > 0 , ℜ ⁡ ( ( - ( n + 1 2 ) ) + k + 1 ) > 0 , ℜ ⁡ ( ( - n - 1 2 ) + k + 1 ) > 0 formulae-sequence 𝑛 1 2 𝑘 1 0 formulae-sequence 𝑛 1 2 𝑘 1 0 𝑛 1 2 𝑘 1 0 {\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-(n+\frac{1}{2}))+% k+1)>0,\Re((-n-\frac{1}{2})+k+1)>0}} 
sqrt((1)/(2)*Pi/z)*BesselY(n +(1)/(2), z) = (- 1)^(n + 1)*sqrt((1)/(2)*Pi/z)*BesselJ(- n -(1)/(2), z)

Sqrt[Divide[1,2]*Pi/z]*BesselY[n +Divide[1,2], z] == (- 1)^(n + 1)*Sqrt[Divide[1,2]*Pi/z]*BesselJ[- n -Divide[1,2], z]
Failure Failure Successful [Tested: 21] Successful [Tested: 21]
10.47.E5 𝗁 n ( 1 ) ⁡ ( z ) = 1 2 ⁢ π / z ⁢ H n + 1 2 ( 1 ) ⁡ ( z ) spherical-Hankel-H-1-Bessel-third-kind 𝑛 𝑧 1 2 𝜋 𝑧 Hankel-H-1-Bessel-third-kind 𝑛 1 2 𝑧 {\displaystyle{\displaystyle{\mathsf{h}^{(1)}_{n}}\left(z\right)=\sqrt{\tfrac{% 1}{2}\pi/z}{H^{(1)}_{n+\frac{1}{2}}}\left(z\right)}}
\sphHankelh{1}{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\HankelH{1}{n+\frac{1}{2}}@{z}		 

Error

SphericalHankelH1[n, z] == Sqrt[Divide[1,2]*Pi/z]*HankelH1[n +Divide[1,2], z]
Missing Macro Error Failure - Successful [Tested: 21]
10.47.E5 1 2 ⁢ π / z ⁢ H n + 1 2 ( 1 ) ⁡ ( z ) = ( - 1 ) n + 1 ⁢ i ⁢ 1 2 ⁢ π / z ⁢ H - n - 1 2 ( 1 ) ⁡ ( z ) 1 2 𝜋 𝑧 Hankel-H-1-Bessel-third-kind 𝑛 1 2 𝑧 superscript 1 𝑛 1 imaginary-unit 1 2 𝜋 𝑧 Hankel-H-1-Bessel-third-kind 𝑛 1 2 𝑧 {\displaystyle{\displaystyle\sqrt{\tfrac{1}{2}\pi/z}{H^{(1)}_{n+\frac{1}{2}}}% \left(z\right)=(-1)^{n+1}\mathrm{i}\sqrt{\tfrac{1}{2}\pi/z}{H^{(1)}_{-n-\frac{% 1}{2}}}\left(z\right)}}
\sqrt{\tfrac{1}{2}\pi/z}\HankelH{1}{n+\frac{1}{2}}@{z} = (-1)^{n+1}\iunit\sqrt{\tfrac{1}{2}\pi/z}\HankelH{1}{-n-\frac{1}{2}}@{z}		 

sqrt((1)/(2)*Pi/z)*HankelH1(n +(1)/(2), z) = (- 1)^(n + 1)* I*sqrt((1)/(2)*Pi/z)*HankelH1(- n -(1)/(2), z)

Sqrt[Divide[1,2]*Pi/z]*HankelH1[n +Divide[1,2], z] == (- 1)^(n + 1)* I*Sqrt[Divide[1,2]*Pi/z]*HankelH1[- n -Divide[1,2], z]
Successful Failure - Successful [Tested: 21]
10.47.E6 𝗁 n ( 2 ) ⁡ ( z ) = 1 2 ⁢ π / z ⁢ H n + 1 2 ( 2 ) ⁡ ( z ) spherical-Hankel-H-2-Bessel-third-kind 𝑛 𝑧 1 2 𝜋 𝑧 Hankel-H-2-Bessel-third-kind 𝑛 1 2 𝑧 {\displaystyle{\displaystyle{\mathsf{h}^{(2)}_{n}}\left(z\right)=\sqrt{\tfrac{% 1}{2}\pi/z}{H^{(2)}_{n+\frac{1}{2}}}\left(z\right)}}
\sphHankelh{2}{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\HankelH{2}{n+\frac{1}{2}}@{z}		 

Error

SphericalHankelH2[n, z] == Sqrt[Divide[1,2]*Pi/z]*HankelH2[n +Divide[1,2], z]
Missing Macro Error Failure - Successful [Tested: 21]
10.47.E6 1 2 ⁢ π / z ⁢ H n + 1 2 ( 2 ) ⁡ ( z ) = ( - 1 ) n ⁢ i ⁢ 1 2 ⁢ π / z ⁢ H - n - 1 2 ( 2 ) ⁡ ( z ) 1 2 𝜋 𝑧 Hankel-H-2-Bessel-third-kind 𝑛 1 2 𝑧 superscript 1 𝑛 imaginary-unit 1 2 𝜋 𝑧 Hankel-H-2-Bessel-third-kind 𝑛 1 2 𝑧 {\displaystyle{\displaystyle\sqrt{\tfrac{1}{2}\pi/z}{H^{(2)}_{n+\frac{1}{2}}}% \left(z\right)=(-1)^{n}\mathrm{i}\sqrt{\tfrac{1}{2}\pi/z}{H^{(2)}_{-n-\frac{1}% {2}}}\left(z\right)}}
\sqrt{\tfrac{1}{2}\pi/z}\HankelH{2}{n+\frac{1}{2}}@{z} = (-1)^{n}\iunit\sqrt{\tfrac{1}{2}\pi/z}\HankelH{2}{-n-\frac{1}{2}}@{z}		 

sqrt((1)/(2)*Pi/z)*HankelH2(n +(1)/(2), z) = (- 1)^(n)* I*sqrt((1)/(2)*Pi/z)*HankelH2(- n -(1)/(2), z)

Sqrt[Divide[1,2]*Pi/z]*HankelH2[n +Divide[1,2], z] == (- 1)^(n)* I*Sqrt[Divide[1,2]*Pi/z]*HankelH2[- n -Divide[1,2], z]
Successful Failure - Successful [Tested: 21]
10.47.E7 𝗂 n ( 1 ) ⁡ ( z ) = 1 2 ⁢ π / z ⁢ I n + 1 2 ⁡ ( z ) spherical-Bessel-I-1 𝑛 𝑧 1 2 𝜋 𝑧 modified-Bessel-first-kind 𝑛 1 2 𝑧 {\displaystyle{\displaystyle{\mathsf{i}^{(1)}_{n}}\left(z\right)=\sqrt{\tfrac{% 1}{2}\pi/z}I_{n+\frac{1}{2}}\left(z\right)}}
\modsphBesseli{1}{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\modBesselI{n+\frac{1}{2}}@{z}		 ℜ ⁡ ( ( n + 1 2 ) + k + 1 ) > 0 𝑛 1 2 𝑘 1 0 {\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0}} 
Error

Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)*(n + 1/2), n] == Sqrt[Divide[1,2]*Pi/z]*BesselI[n +Divide[1,2], z]
Missing Macro Error Failure -
Failed [20 / 21]
Result: Complex[0.06771919180965624, -0.29579816936516184]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.4498252419402129, -0.19064547195046921]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
10.47.E8 𝗂 n ( 2 ) ⁡ ( z ) = 1 2 ⁢ π / z ⁢ I - n - 1 2 ⁡ ( z ) spherical-Bessel-I-2 𝑛 𝑧 1 2 𝜋 𝑧 modified-Bessel-first-kind 𝑛 1 2 𝑧 {\displaystyle{\displaystyle{\mathsf{i}^{(2)}_{n}}\left(z\right)=\sqrt{\tfrac{% 1}{2}\pi/z}I_{-n-\frac{1}{2}}\left(z\right)}}
\modsphBesseli{2}{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\modBesselI{-n-\frac{1}{2}}@{z}		 ℜ ⁡ ( ( - n - 1 2 ) + k + 1 ) > 0 𝑛 1 2 𝑘 1 0 {\displaystyle{\displaystyle\Re((-n-\frac{1}{2})+k+1)>0}} 
Error

Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)*(n + 1/2), n] == Sqrt[Divide[1,2]*Pi/z]*BesselI[- n -Divide[1,2], z]
Missing Macro Error Failure -
Failed [20 / 21]
Result: Complex[-0.41419719140728084, -0.8850762711170854]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.1065867555175597, 2.4569570135519543]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
10.47.E9 𝗄 n ⁡ ( z ) = 1 2 ⁢ π / z ⁢ K n + 1 2 ⁡ ( z ) spherical-Bessel-K 𝑛 𝑧 1 2 𝜋 𝑧 modified-Bessel-second-kind 𝑛 1 2 𝑧 {\displaystyle{\displaystyle\mathsf{k}_{n}\left(z\right)=\sqrt{\tfrac{1}{2}\pi% /z}K_{n+\frac{1}{2}}\left(z\right)}}
\modsphBesselK{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\modBesselK{n+\frac{1}{2}}@{z}		 

Error

Sqrt[1/2 Pi /z] BesselK[n + 1/2, z] == Sqrt[Divide[1,2]*Pi/z]*BesselK[n +Divide[1,2], z]
Missing Macro Error Successful - Successful [Tested: 21]
10.47.E9 1 2 ⁢ π / z ⁢ K n + 1 2 ⁡ ( z ) = 1 2 ⁢ π / z ⁢ K - n - 1 2 ⁡ ( z ) 1 2 𝜋 𝑧 modified-Bessel-second-kind 𝑛 1 2 𝑧 1 2 𝜋 𝑧 modified-Bessel-second-kind 𝑛 1 2 𝑧 {\displaystyle{\displaystyle\sqrt{\tfrac{1}{2}\pi/z}K_{n+\frac{1}{2}}\left(z% \right)=\sqrt{\tfrac{1}{2}\pi/z}K_{-n-\frac{1}{2}}\left(z\right)}}
\sqrt{\tfrac{1}{2}\pi/z}\modBesselK{n+\frac{1}{2}}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\modBesselK{-n-\frac{1}{2}}@{z}		 

sqrt((1)/(2)*Pi/z)*BesselK(n +(1)/(2), z) = sqrt((1)/(2)*Pi/z)*BesselK(- n -(1)/(2), z)

Sqrt[Divide[1,2]*Pi/z]*BesselK[n +Divide[1,2], z] == Sqrt[Divide[1,2]*Pi/z]*BesselK[- n -Divide[1,2], z]
Successful Successful - Successful [Tested: 21]
10.47#Ex1 𝗁 n ( 1 ) ⁡ ( z ) = 𝗃 n ⁡ ( z ) + i ⁢ 𝗒 n ⁡ ( z ) spherical-Hankel-H-1-Bessel-third-kind 𝑛 𝑧 spherical-Bessel-J 𝑛 𝑧 𝑖 spherical-Bessel-Y 𝑛 𝑧 {\displaystyle{\displaystyle{\mathsf{h}^{(1)}_{n}}\left(z\right)=\mathsf{j}_{n% }\left(z\right)+i\mathsf{y}_{n}\left(z\right)}}
\sphHankelh{1}{n}@{z} = \sphBesselJ{n}@{z}+i\sphBesselY{n}@{z}		 ℜ ⁡ ( ( n + 1 2 ) + k + 1 ) > 0 , ℜ ⁡ ( ( - n - 1 2 ) + k + 1 ) > 0 , ℜ ⁡ ( ( - ( - n - 1 2 ) ) + k + 1 ) > 0 , ℜ ⁡ ( ( - ( n + 1 2 ) ) + k + 1 ) > 0 formulae-sequence 𝑛 1 2 𝑘 1 0 formulae-sequence 𝑛 1 2 𝑘 1 0 formulae-sequence 𝑛 1 2 𝑘 1 0 𝑛 1 2 𝑘 1 0 {\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+% 1)>0,\Re((-(-n-\frac{1}{2}))+k+1)>0,\Re((-(n+\frac{1}{2}))+k+1)>0}} 
Error

SphericalHankelH1[n, z] == SphericalBesselJ[n, z]+ I*SphericalBesselY[n, z]
Missing Macro Error Successful - Successful [Tested: 21]
10.47#Ex2 𝗁 n ( 2 ) ⁡ ( z ) = 𝗃 n ⁡ ( z ) - i ⁢ 𝗒 n ⁡ ( z ) spherical-Hankel-H-2-Bessel-third-kind 𝑛 𝑧 spherical-Bessel-J 𝑛 𝑧 𝑖 spherical-Bessel-Y 𝑛 𝑧 {\displaystyle{\displaystyle{\mathsf{h}^{(2)}_{n}}\left(z\right)=\mathsf{j}_{n% }\left(z\right)-i\mathsf{y}_{n}\left(z\right)}}
\sphHankelh{2}{n}@{z} = \sphBesselJ{n}@{z}-i\sphBesselY{n}@{z}		 ℜ ⁡ ( ( n + 1 2 ) + k + 1 ) > 0 , ℜ ⁡ ( ( - n - 1 2 ) + k + 1 ) > 0 , ℜ ⁡ ( ( - ( - n - 1 2 ) ) + k + 1 ) > 0 , ℜ ⁡ ( ( - ( n + 1 2 ) ) + k + 1 ) > 0 formulae-sequence 𝑛 1 2 𝑘 1 0 formulae-sequence 𝑛 1 2 𝑘 1 0 formulae-sequence 𝑛 1 2 𝑘 1 0 𝑛 1 2 𝑘 1 0 {\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+% 1)>0,\Re((-(-n-\frac{1}{2}))+k+1)>0,\Re((-(n+\frac{1}{2}))+k+1)>0}} 
Error

SphericalHankelH2[n, z] == SphericalBesselJ[n, z]- I*SphericalBesselY[n, z]
Missing Macro Error Successful - Successful [Tested: 21]
10.47.E11 𝗄 n ⁡ ( z ) = ( - 1 ) n + 1 ⁢ 1 2 ⁢ π ⁢ ( 𝗂 n ( 1 ) ⁡ ( z ) - 𝗂 n ( 2 ) ⁡ ( z ) ) spherical-Bessel-K 𝑛 𝑧 superscript 1 𝑛 1 1 2 𝜋 spherical-Bessel-I-1 𝑛 𝑧 spherical-Bessel-I-2 𝑛 𝑧 {\displaystyle{\displaystyle\mathsf{k}_{n}\left(z\right)=(-1)^{n+1}\tfrac{1}{2% }\pi\left({\mathsf{i}^{(1)}_{n}}\left(z\right)-{\mathsf{i}^{(2)}_{n}}\left(z% \right)\right)}}
\modsphBesselK{n}@{z} = (-1)^{n+1}\tfrac{1}{2}\pi\left(\modsphBesseli{1}{n}@{z}-\modsphBesseli{2}{n}@{z}\right)		 ℜ ⁡ ( ( n + 1 2 ) + k + 1 ) > 0 , ℜ ⁡ ( ( - n - 1 2 ) + k + 1 ) > 0 formulae-sequence 𝑛 1 2 𝑘 1 0 𝑛 1 2 𝑘 1 0 {\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+% 1)>0}} 
Error

Sqrt[1/2 Pi /z] BesselK[n + 1/2, z] == (- 1)^(n + 1)*Divide[1,2]*Pi*(Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)*(n + 1/2), n]- Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)*(n + 1/2), n])
Missing Macro Error Failure -
Failed [20 / 21]
Result: Complex[-0.7569924845794465, -0.925635877692591]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-1.0316385731075524, -4.1588442590402455]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
10.47#Ex3 𝗂 n ( 1 ) ⁡ ( z ) = i - n ⁢ 𝗃 n ⁡ ( i ⁢ z ) spherical-Bessel-I-1 𝑛 𝑧 superscript 𝑖 𝑛 spherical-Bessel-J 𝑛 𝑖 𝑧 {\displaystyle{\displaystyle{\mathsf{i}^{(1)}_{n}}\left(z\right)=i^{-n}\mathsf% {j}_{n}\left(iz\right)}}
\modsphBesseli{1}{n}@{z} = i^{-n}\sphBesselJ{n}@{iz}		 ℜ ⁡ ( ( n + 1 2 ) + k + 1 ) > 0 , ℜ ⁡ ( ( - n - 1 2 ) + k + 1 ) > 0 , ℜ ⁡ ( ( - ( - n - 1 2 ) ) + k + 1 ) > 0 formulae-sequence 𝑛 1 2 𝑘 1 0 formulae-sequence 𝑛 1 2 𝑘 1 0 𝑛 1 2 𝑘 1 0 {\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+% 1)>0,\Re((-(-n-\frac{1}{2}))+k+1)>0}} 
Error

Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)*(n + 1/2), n] == (I)^(- n)* SphericalBesselJ[n, I*z]
Missing Macro Error Failure -
Failed [20 / 21]
Result: Complex[0.06771919180965624, -0.2957981693651618]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.44982524194021284, -0.19064547195046921]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
10.47#Ex4 𝗂 n ( 2 ) ⁡ ( z ) = i - n - 1 ⁢ 𝗒 n ⁡ ( i ⁢ z ) spherical-Bessel-I-2 𝑛 𝑧 superscript 𝑖 𝑛 1 spherical-Bessel-Y 𝑛 𝑖 𝑧 {\displaystyle{\displaystyle{\mathsf{i}^{(2)}_{n}}\left(z\right)=i^{-n-1}% \mathsf{y}_{n}\left(iz\right)}}
\modsphBesseli{2}{n}@{z} = i^{-n-1}\sphBesselY{n}@{iz}		 ℜ ⁡ ( ( - n - 1 2 ) + k + 1 ) > 0 , ℜ ⁡ ( ( n + 1 2 ) + k + 1 ) > 0 , ℜ ⁡ ( ( - ( n + 1 2 ) ) + k + 1 ) > 0 formulae-sequence 𝑛 1 2 𝑘 1 0 formulae-sequence 𝑛 1 2 𝑘 1 0 𝑛 1 2 𝑘 1 0 {\displaystyle{\displaystyle\Re((-n-\frac{1}{2})+k+1)>0,\Re((n+\frac{1}{2})+k+% 1)>0,\Re((-(n+\frac{1}{2}))+k+1)>0}} 
Error

Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)*(n + 1/2), n] == (I)^(- n - 1)* SphericalBesselY[n, I*z]
Missing Macro Error Failure -
Failed [20 / 21]
Result: Complex[-0.41419719140728045, -0.8850762711170859]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.1065867555175588, 2.456957013551956]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
10.47.E13 𝗄 n ⁡ ( z ) = - 1 2 ⁢ π ⁢ i n ⁢ 𝗁 n ( 1 ) ⁡ ( i ⁢ z ) spherical-Bessel-K 𝑛 𝑧 1 2 𝜋 superscript 𝑖 𝑛 spherical-Hankel-H-1-Bessel-third-kind 𝑛 𝑖 𝑧 {\displaystyle{\displaystyle\mathsf{k}_{n}\left(z\right)=-\tfrac{1}{2}\pi i^{n% }{\mathsf{h}^{(1)}_{n}}\left(iz\right)}}
\modsphBesselK{n}@{z} = -\tfrac{1}{2}\pi i^{n}\sphHankelh{1}{n}@{iz}		 

Error

Sqrt[1/2 Pi /z] BesselK[n + 1/2, z] == -Divide[1,2]*Pi*(I)^(n)* SphericalHankelH1[n, I*z]
Missing Macro Error Failure - Successful [Tested: 21]
10.47.E13 - 1 2 ⁢ π ⁢ i n ⁢ 𝗁 n ( 1 ) ⁡ ( i ⁢ z ) = - 1 2 ⁢ π ⁢ i - n ⁢ 𝗁 n ( 2 ) ⁡ ( - i ⁢ z ) 1 2 𝜋 superscript 𝑖 𝑛 spherical-Hankel-H-1-Bessel-third-kind 𝑛 𝑖 𝑧 1 2 𝜋 superscript 𝑖 𝑛 spherical-Hankel-H-2-Bessel-third-kind 𝑛 𝑖 𝑧 {\displaystyle{\displaystyle-\tfrac{1}{2}\pi i^{n}{\mathsf{h}^{(1)}_{n}}\left(% iz\right)=-\tfrac{1}{2}\pi i^{-n}{\mathsf{h}^{(2)}_{n}}\left(-iz\right)}}
-\tfrac{1}{2}\pi i^{n}\sphHankelh{1}{n}@{iz} = -\tfrac{1}{2}\pi i^{-n}\sphHankelh{2}{n}@{-iz}		 

Error

-Divide[1,2]*Pi*(I)^(n)* SphericalHankelH1[n, I*z] == -Divide[1,2]*Pi*(I)^(- n)* SphericalHankelH2[n, - I*z]
Missing Macro Error Failure - Successful [Tested: 21]
10.47.E14 𝗃 n ⁡ ( - z ) = ( - 1 ) n ⁢ 𝗃 n ⁡ ( z ) spherical-Bessel-J 𝑛 𝑧 superscript 1 𝑛 spherical-Bessel-J 𝑛 𝑧 {\displaystyle{\displaystyle\displaystyle\mathsf{j}_{n}\left(-z\right)=(-1)^{n% }\mathsf{j}_{n}\left(z\right)}}
\displaystyle\sphBesselJ{n}@{-z} = (-1)^{n}\sphBesselJ{n}@{z}		 ℜ ⁡ ( ( n + 1 2 ) + k + 1 ) > 0 , ℜ ⁡ ( ( - n - 1 2 ) + k + 1 ) > 0 , ℜ ⁡ ( ( - ( - n - 1 2 ) ) + k + 1 ) > 0 formulae-sequence 𝑛 1 2 𝑘 1 0 formulae-sequence 𝑛 1 2 𝑘 1 0 𝑛 1 2 𝑘 1 0 {\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+% 1)>0,\Re((-(-n-\frac{1}{2}))+k+1)>0}} 
Error
 
SphericalBesselJ[n, - z] == (- 1)^(n)* SphericalBesselJ[n, z]
 Skipped - no semantic math Skipped - no semantic math - -
10.47.E14 𝗒 n ⁡ ( - z ) = ( - 1 ) n + 1 ⁢ 𝗒 n ⁡ ( z ) spherical-Bessel-Y 𝑛 𝑧 superscript 1 𝑛 1 spherical-Bessel-Y 𝑛 𝑧 {\displaystyle{\displaystyle\displaystyle\mathsf{y}_{n}\left(-z\right)=(-1)^{n% +1}\mathsf{y}_{n}\left(z\right)}}
\displaystyle\sphBesselY{n}@{-z} = (-1)^{n+1}\sphBesselY{n}@{z}		 ℜ ⁡ ( ( n + 1 2 ) + k + 1 ) > 0 , ℜ ⁡ ( ( - ( n + 1 2 ) ) + k + 1 ) > 0 , ℜ ⁡ ( ( - n - 1 2 ) + k + 1 ) > 0 formulae-sequence 𝑛 1 2 𝑘 1 0 formulae-sequence 𝑛 1 2 𝑘 1 0 𝑛 1 2 𝑘 1 0 {\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-(n+\frac{1}{2}))+% k+1)>0,\Re((-n-\frac{1}{2})+k+1)>0}} 
Error
 
SphericalBesselY[n, - z] == (- 1)^(n + 1)* SphericalBesselY[n, z]
 Skipped - no semantic math Skipped - no semantic math - -
10.47.E15 𝗁 n ( 1 ) ⁡ ( - z ) = ( - 1 ) n ⁢ 𝗁 n ( 2 ) ⁡ ( z ) spherical-Hankel-H-1-Bessel-third-kind 𝑛 𝑧 superscript 1 𝑛 spherical-Hankel-H-2-Bessel-third-kind 𝑛 𝑧 {\displaystyle{\displaystyle\displaystyle{\mathsf{h}^{(1)}_{n}}\left(-z\right)% =(-1)^{n}{\mathsf{h}^{(2)}_{n}}\left(z\right)}}
\displaystyle\sphHankelh{1}{n}@{-z} = (-1)^{n}\sphHankelh{2}{n}@{z}		 

Error
 
SphericalHankelH1[n, - z] == (- 1)^(n)* SphericalHankelH2[n, z]
 Skipped - no semantic math Skipped - no semantic math - -
10.47.E15 𝗁 n ( 2 ) ⁡ ( - z ) = ( - 1 ) n ⁢ 𝗁 n ( 1 ) ⁡ ( z ) spherical-Hankel-H-2-Bessel-third-kind 𝑛 𝑧 superscript 1 𝑛 spherical-Hankel-H-1-Bessel-third-kind 𝑛 𝑧 {\displaystyle{\displaystyle\displaystyle{\mathsf{h}^{(2)}_{n}}\left(-z\right)% =(-1)^{n}{\mathsf{h}^{(1)}_{n}}\left(z\right)}}
\displaystyle\sphHankelh{2}{n}@{-z} = (-1)^{n}\sphHankelh{1}{n}@{z}		 

Error
 
SphericalHankelH2[n, - z] == (- 1)^(n)* SphericalHankelH1[n, z]
 Skipped - no semantic math Skipped - no semantic math - -
10.47.E16 𝗂 n ( 1 ) ⁡ ( - z ) = ( - 1 ) n ⁢ 𝗂 n ( 1 ) ⁡ ( z ) spherical-Bessel-I-1 𝑛 𝑧 superscript 1 𝑛 spherical-Bessel-I-1 𝑛 𝑧 {\displaystyle{\displaystyle\displaystyle{\mathsf{i}^{(1)}_{n}}\left(-z\right)% =(-1)^{n}{\mathsf{i}^{(1)}_{n}}\left(z\right)}}
\displaystyle\modsphBesseli{1}{n}@{-z} = (-1)^{n}\modsphBesseli{1}{n}@{z}		 ℜ ⁡ ( ( n + 1 2 ) + k + 1 ) > 0 𝑛 1 2 𝑘 1 0 {\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0}} 
Error
 
Sqrt[Divide[Pi, - z]/2] BesselI[(-1)^(1-1)*(n + 1/2), n] == (- 1)^(n)* Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)*(n + 1/2), n]
 Skipped - no semantic math Skipped - no semantic math - -
10.47.E16 𝗂 n ( 2 ) ⁡ ( - z ) = ( - 1 ) n + 1 ⁢ 𝗂 n ( 2 ) ⁡ ( z ) spherical-Bessel-I-2 𝑛 𝑧 superscript 1 𝑛 1 spherical-Bessel-I-2 𝑛 𝑧 {\displaystyle{\displaystyle\displaystyle{\mathsf{i}^{(2)}_{n}}\left(-z\right)% =(-1)^{n+1}{\mathsf{i}^{(2)}_{n}}\left(z\right)}}
\displaystyle\modsphBesseli{2}{n}@{-z} = (-1)^{n+1}\modsphBesseli{2}{n}@{z}		 ℜ ⁡ ( ( - n - 1 2 ) + k + 1 ) > 0 𝑛 1 2 𝑘 1 0 {\displaystyle{\displaystyle\Re((-n-\frac{1}{2})+k+1)>0}} 
Error
 
Sqrt[Divide[Pi, - z]/2] BesselI[(-1)^(2-1)*(n + 1/2), n] == (- 1)^(n + 1)* Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)*(n + 1/2), n]
 Skipped - no semantic math Skipped - no semantic math - -
10.47.E17 𝗄 n ⁡ ( - z ) = - 1 2 ⁢ π ⁢ ( 𝗂 n ( 1 ) ⁡ ( z ) + 𝗂 n ( 2 ) ⁡ ( z ) ) spherical-Bessel-K 𝑛 𝑧 1 2 𝜋 spherical-Bessel-I-1 𝑛 𝑧 spherical-Bessel-I-2 𝑛 𝑧 {\displaystyle{\displaystyle\mathsf{k}_{n}\left(-z\right)=-\tfrac{1}{2}\pi% \left({\mathsf{i}^{(1)}_{n}}\left(z\right)+{\mathsf{i}^{(2)}_{n}}\left(z\right% )\right)}}
\modsphBesselK{n}@{-z} = -\tfrac{1}{2}\pi\left(\modsphBesseli{1}{n}@{z}+\modsphBesseli{2}{n}@{z}\right)		 ℜ ⁡ ( ( n + 1 2 ) + k + 1 ) > 0 , ℜ ⁡ ( ( - n - 1 2 ) + k + 1 ) > 0 formulae-sequence 𝑛 1 2 𝑘 1 0 𝑛 1 2 𝑘 1 0 {\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+% 1)>0}} 
Error

Sqrt[1/2 Pi /- z] BesselK[n + 1/2, - z] == -Divide[1,2]*Pi*(Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)*(n + 1/2), n]+ Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)*(n + 1/2), n])
Missing Macro Error Failure -
Failed [21 / 21]
Result: Complex[-0.5442463690831921, -1.8549132335154932]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[2.444806248586177, 3.5599138449204935]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
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