View source for 10.47

{{DISPLAYTITLE:Bessel Functions - 10.47 Definitions and Basic Properties}}

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! scope="col" style="position: sticky; top: 0;" | DLMF 
! scope="col" style="position: sticky; top: 0;" | Formula 
! scope="col" style="position: sticky; top: 0;" | Constraints 
! scope="col" style="position: sticky; top: 0;" | Maple
! scope="col" style="position: sticky; top: 0;" | Mathematica
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Maple
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Maple
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/10.47.E1 10.47.E1] || <math qid="Q3669">z^{2}\deriv[2]{w}{z}+2z\deriv{w}{z}+\left(z^{2}-n(n+1)\right)w = 0</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>z^{2}\deriv[2]{w}{z}+2z\deriv{w}{z}+\left(z^{2}-n(n+1)\right)w = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(z)^(2)* diff(w, [z$(2)])+ 2*z*diff(w, z)+((z)^(2)- n*(n + 1))*w = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>(z)^(2)* D[w, {z, 2}]+ 2*z*D[w, z]+((z)^(2)- n*(n + 1))*w == 0</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [210 / 210]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -1.732050808+.3733632160e-9*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -5.196152424-2.000000000*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [210 / 210]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-1.7320508075688772, 1.1102230246251565*^-16]
Test Values: {Rule[n, 1], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-5.196152422706633, -1.9999999999999996]
Test Values: {Rule[n, 2], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/10.47.E2 10.47.E2] || <math qid="Q3670">z^{2}\deriv[2]{w}{z}+2z\deriv{w}{z}-\left(z^{2}+n(n+1)\right)w = 0</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>z^{2}\deriv[2]{w}{z}+2z\deriv{w}{z}-\left(z^{2}+n(n+1)\right)w = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(z)^(2)* diff(w, [z$(2)])+ 2*z*diff(w, z)-((z)^(2)+ n*(n + 1))*w = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>(z)^(2)* D[w, {z, 2}]+ 2*z*D[w, z]-((z)^(2)+ n*(n + 1))*w == 0</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [210 / 210]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [210 / 210]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/10.47.E3 10.47.E3] || <math qid="Q3671">\sphBesselJ{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\BesselJ{n+\frac{1}{2}}@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphBesselJ{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\BesselJ{n+\frac{1}{2}}@{z}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>SphericalBesselJ[n, z] == Sqrt[Divide[1,2]*Pi/z]*BesselJ[n +Divide[1,2], z]</syntaxhighlight> || Missing Macro Error || Failure || Skip - symbolical successful subtest || Successful [Tested: 21]
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| [https://dlmf.nist.gov/10.47.E3 10.47.E3] || <math qid="Q3671">\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}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-n-\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>sqrt((1)/(2)*Pi/z)*BesselJ(n +(1)/(2), z) = (- 1)^(n)*sqrt((1)/(2)*Pi/z)*BesselY(- n -(1)/(2), z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Failure || Failure || Successful [Tested: 21] || Successful [Tested: 21]
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| [https://dlmf.nist.gov/10.47.E4 10.47.E4] || <math qid="Q3672">\sphBesselY{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\BesselY{n+\frac{1}{2}}@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphBesselY{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\BesselY{n+\frac{1}{2}}@{z}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-(n+\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>SphericalBesselY[n, z] == Sqrt[Divide[1,2]*Pi/z]*BesselY[n +Divide[1,2], z]</syntaxhighlight> || Missing Macro Error || Failure || Skip - symbolical successful subtest || Successful [Tested: 21]
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| [https://dlmf.nist.gov/10.47.E4 10.47.E4] || <math qid="Q3672">\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}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-(n+\frac{1}{2}))+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>sqrt((1)/(2)*Pi/z)*BesselY(n +(1)/(2), z) = (- 1)^(n + 1)*sqrt((1)/(2)*Pi/z)*BesselJ(- n -(1)/(2), z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Failure || Failure || Successful [Tested: 21] || Successful [Tested: 21]
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| [https://dlmf.nist.gov/10.47.E5 10.47.E5] || <math qid="Q3673">\sphHankelh{1}{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\HankelH{1}{n+\frac{1}{2}}@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphHankelh{1}{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\HankelH{1}{n+\frac{1}{2}}@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>SphericalHankelH1[n, z] == Sqrt[Divide[1,2]*Pi/z]*HankelH1[n +Divide[1,2], z]</syntaxhighlight> || Missing Macro Error || Failure || - || Successful [Tested: 21]
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| [https://dlmf.nist.gov/10.47.E5 10.47.E5] || <math qid="Q3673">\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}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sqrt((1)/(2)*Pi/z)*HankelH1(n +(1)/(2), z) = (- 1)^(n + 1)* I*sqrt((1)/(2)*Pi/z)*HankelH1(- n -(1)/(2), z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 21]
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| [https://dlmf.nist.gov/10.47.E6 10.47.E6] || <math qid="Q3674">\sphHankelh{2}{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\HankelH{2}{n+\frac{1}{2}}@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphHankelh{2}{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\HankelH{2}{n+\frac{1}{2}}@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>SphericalHankelH2[n, z] == Sqrt[Divide[1,2]*Pi/z]*HankelH2[n +Divide[1,2], z]</syntaxhighlight> || Missing Macro Error || Failure || - || Successful [Tested: 21]
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| [https://dlmf.nist.gov/10.47.E6 10.47.E6] || <math qid="Q3674">\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}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sqrt((1)/(2)*Pi/z)*HankelH2(n +(1)/(2), z) = (- 1)^(n)* I*sqrt((1)/(2)*Pi/z)*HankelH2(- n -(1)/(2), z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 21]
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| [https://dlmf.nist.gov/10.47.E7 10.47.E7] || <math qid="Q3675">\modsphBesseli{1}{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\modBesselI{n+\frac{1}{2}}@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modsphBesseli{1}{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\modBesselI{n+\frac{1}{2}}@{z}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [20 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.06771919180965624, -0.29579816936516184]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.4498252419402129, -0.19064547195046921]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/10.47.E8 10.47.E8] || <math qid="Q3676">\modsphBesseli{2}{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\modBesselI{-n-\frac{1}{2}}@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modsphBesseli{2}{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\modBesselI{-n-\frac{1}{2}}@{z}</syntaxhighlight> || <math>\realpart@@{((-n-\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [20 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.41419719140728084, -0.8850762711170854]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[1.1065867555175597, 2.4569570135519543]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/10.47.E9 10.47.E9] || <math qid="Q3677">\modsphBesselK{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\modBesselK{n+\frac{1}{2}}@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modsphBesselK{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\modBesselK{n+\frac{1}{2}}@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[1/2 Pi /z] BesselK[n + 1/2, z] == Sqrt[Divide[1,2]*Pi/z]*BesselK[n +Divide[1,2], z]</syntaxhighlight> || Missing Macro Error || Successful || - || Successful [Tested: 21]
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| [https://dlmf.nist.gov/10.47.E9 10.47.E9] || <math qid="Q3677">\sqrt{\tfrac{1}{2}\pi/z}\modBesselK{n+\frac{1}{2}}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\modBesselK{-n-\frac{1}{2}}@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sqrt{\tfrac{1}{2}\pi/z}\modBesselK{n+\frac{1}{2}}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\modBesselK{-n-\frac{1}{2}}@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sqrt((1)/(2)*Pi/z)*BesselK(n +(1)/(2), z) = sqrt((1)/(2)*Pi/z)*BesselK(- n -(1)/(2), z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[Divide[1,2]*Pi/z]*BesselK[n +Divide[1,2], z] == Sqrt[Divide[1,2]*Pi/z]*BesselK[- n -Divide[1,2], z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 21]
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| [https://dlmf.nist.gov/10.47#Ex1 10.47#Ex1] || <math qid="Q3678">\sphHankelh{1}{n}@{z} = \sphBesselJ{n}@{z}+i\sphBesselY{n}@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphHankelh{1}{n}@{z} = \sphBesselJ{n}@{z}+i\sphBesselY{n}@{z}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-n-\frac{1}{2}))+k+1)} > 0, \realpart@@{((-(n+\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>SphericalHankelH1[n, z] == SphericalBesselJ[n, z]+ I*SphericalBesselY[n, z]</syntaxhighlight> || Missing Macro Error || Successful || - || Successful [Tested: 21]
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| [https://dlmf.nist.gov/10.47#Ex2 10.47#Ex2] || <math qid="Q3679">\sphHankelh{2}{n}@{z} = \sphBesselJ{n}@{z}-i\sphBesselY{n}@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphHankelh{2}{n}@{z} = \sphBesselJ{n}@{z}-i\sphBesselY{n}@{z}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-n-\frac{1}{2}))+k+1)} > 0, \realpart@@{((-(n+\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>SphericalHankelH2[n, z] == SphericalBesselJ[n, z]- I*SphericalBesselY[n, z]</syntaxhighlight> || Missing Macro Error || Successful || - || Successful [Tested: 21]
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| [https://dlmf.nist.gov/10.47.E11 10.47.E11] || <math qid="Q3680">\modsphBesselK{n}@{z} = (-1)^{n+1}\tfrac{1}{2}\pi\left(\modsphBesseli{1}{n}@{z}-\modsphBesseli{2}{n}@{z}\right)</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modsphBesselK{n}@{z} = (-1)^{n+1}\tfrac{1}{2}\pi\left(\modsphBesseli{1}{n}@{z}-\modsphBesseli{2}{n}@{z}\right)</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>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])</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [20 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.7569924845794465, -0.925635877692591]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-1.0316385731075524, -4.1588442590402455]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/10.47#Ex3 10.47#Ex3] || <math qid="Q3681">\modsphBesseli{1}{n}@{z} = i^{-n}\sphBesselJ{n}@{iz}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modsphBesseli{1}{n}@{z} = i^{-n}\sphBesselJ{n}@{iz}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-n-\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)*(n + 1/2), n] == (I)^(- n)* SphericalBesselJ[n, I*z]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [20 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.06771919180965624, -0.2957981693651618]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.44982524194021284, -0.19064547195046921]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/10.47#Ex4 10.47#Ex4] || <math qid="Q3682">\modsphBesseli{2}{n}@{z} = i^{-n-1}\sphBesselY{n}@{iz}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modsphBesseli{2}{n}@{z} = i^{-n-1}\sphBesselY{n}@{iz}</syntaxhighlight> || <math>\realpart@@{((-n-\frac{1}{2})+k+1)} > 0, \realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-(n+\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)*(n + 1/2), n] == (I)^(- n - 1)* SphericalBesselY[n, I*z]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [20 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.41419719140728045, -0.8850762711170859]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[1.1065867555175588, 2.456957013551956]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/10.47.E13 10.47.E13] || <math qid="Q3683">\modsphBesselK{n}@{z} = -\tfrac{1}{2}\pi i^{n}\sphHankelh{1}{n}@{iz}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modsphBesselK{n}@{z} = -\tfrac{1}{2}\pi i^{n}\sphHankelh{1}{n}@{iz}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[1/2 Pi /z] BesselK[n + 1/2, z] == -Divide[1,2]*Pi*(I)^(n)* SphericalHankelH1[n, I*z]</syntaxhighlight> || Missing Macro Error || Failure || - || Successful [Tested: 21]
|- 
| [https://dlmf.nist.gov/10.47.E13 10.47.E13] || <math qid="Q3683">-\tfrac{1}{2}\pi i^{n}\sphHankelh{1}{n}@{iz} = -\tfrac{1}{2}\pi i^{-n}\sphHankelh{2}{n}@{-iz}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>-\tfrac{1}{2}\pi i^{n}\sphHankelh{1}{n}@{iz} = -\tfrac{1}{2}\pi i^{-n}\sphHankelh{2}{n}@{-iz}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>-Divide[1,2]*Pi*(I)^(n)* SphericalHankelH1[n, I*z] == -Divide[1,2]*Pi*(I)^(- n)* SphericalHankelH2[n, - I*z]</syntaxhighlight> || Missing Macro Error || Failure || - || Successful [Tested: 21]
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/10.47.E14 10.47.E14] || <math qid="Q3685">\displaystyle\sphBesselJ{n}@{-z} = (-1)^{n}\sphBesselJ{n}@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\displaystyle\sphBesselJ{n}@{-z} = (-1)^{n}\sphBesselJ{n}@{z}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-n-\frac{1}{2}))+k+1)} > 0</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Error</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">SphericalBesselJ[n, - z] == (- 1)^(n)* SphericalBesselJ[n, z]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/10.47.E14 10.47.E14] || <math qid="Q3685">\displaystyle\sphBesselY{n}@{-z} = (-1)^{n+1}\sphBesselY{n}@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\displaystyle\sphBesselY{n}@{-z} = (-1)^{n+1}\sphBesselY{n}@{z}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-(n+\frac{1}{2}))+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Error</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">SphericalBesselY[n, - z] == (- 1)^(n + 1)* SphericalBesselY[n, z]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/10.47.E15 10.47.E15] || <math qid="Q3687">\displaystyle\sphHankelh{1}{n}@{-z} = (-1)^{n}\sphHankelh{2}{n}@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\displaystyle\sphHankelh{1}{n}@{-z} = (-1)^{n}\sphHankelh{2}{n}@{z}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Error</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">SphericalHankelH1[n, - z] == (- 1)^(n)* SphericalHankelH2[n, z]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/10.47.E15 10.47.E15] || <math qid="Q3687">\displaystyle\sphHankelh{2}{n}@{-z} = (-1)^{n}\sphHankelh{1}{n}@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\displaystyle\sphHankelh{2}{n}@{-z} = (-1)^{n}\sphHankelh{1}{n}@{z}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Error</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">SphericalHankelH2[n, - z] == (- 1)^(n)* SphericalHankelH1[n, z]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/10.47.E16 10.47.E16] || <math qid="Q3689">\displaystyle\modsphBesseli{1}{n}@{-z} = (-1)^{n}\modsphBesseli{1}{n}@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\displaystyle\modsphBesseli{1}{n}@{-z} = (-1)^{n}\modsphBesseli{1}{n}@{z}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Error</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">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]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/10.47.E16 10.47.E16] || <math qid="Q3689">\displaystyle\modsphBesseli{2}{n}@{-z} = (-1)^{n+1}\modsphBesseli{2}{n}@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\displaystyle\modsphBesseli{2}{n}@{-z} = (-1)^{n+1}\modsphBesseli{2}{n}@{z}</syntaxhighlight> || <math>\realpart@@{((-n-\frac{1}{2})+k+1)} > 0</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Error</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">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]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- 
| [https://dlmf.nist.gov/10.47.E17 10.47.E17] || <math qid="Q3690">\modsphBesselK{n}@{-z} = -\tfrac{1}{2}\pi\left(\modsphBesseli{1}{n}@{z}+\modsphBesseli{2}{n}@{z}\right)</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modsphBesselK{n}@{-z} = -\tfrac{1}{2}\pi\left(\modsphBesseli{1}{n}@{z}+\modsphBesseli{2}{n}@{z}\right)</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>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])</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.5442463690831921, -1.8549132335154932]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[2.444806248586177, 3.5599138449204935]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|}
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