View source for 10.27

{{DISPLAYTITLE:Bessel Functions - 10.27 Connection Formulas}}

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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/10.27.E1 10.27.E1] || [[Item:Q3491|<math>\modBesselI{-n}@{z} = \modBesselI{n}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modBesselI{-n}@{z} = \modBesselI{n}@{z}</syntaxhighlight> || <math>\realpart@@{((-n)+k+1)} > 0, \realpart@@{(n+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>BesselI(- n, z) = BesselI(n, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BesselI[- n, z] == BesselI[n, z]</syntaxhighlight> || Failure || Failure || Successful [Tested: 21] || Successful [Tested: 21]
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| [https://dlmf.nist.gov/10.27.E2 10.27.E2] || [[Item:Q3492|<math>\modBesselI{-\nu}@{z} = \modBesselI{\nu}@{z}+(2/\pi)\sin@{\nu\pi}\modBesselK{\nu}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modBesselI{-\nu}@{z} = \modBesselI{\nu}@{z}+(2/\pi)\sin@{\nu\pi}\modBesselK{\nu}@{z}</syntaxhighlight> || <math>\realpart@@{((-\nu)+k+1)} > 0, \realpart@@{(\nu+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>BesselI(- nu, z) = BesselI(nu, z)+(2/Pi)*sin(nu*Pi)*BesselK(nu, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BesselI[- \[Nu], z] == BesselI[\[Nu], z]+(2/Pi)*Sin[\[Nu]*Pi]*BesselK[\[Nu], z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 70]
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| [https://dlmf.nist.gov/10.27.E3 10.27.E3] || [[Item:Q3493|<math>\modBesselK{-\nu}@{z} = \modBesselK{\nu}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modBesselK{-\nu}@{z} = \modBesselK{\nu}@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>BesselK(- nu, z) = BesselK(nu, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BesselK[- \[Nu], z] == BesselK[\[Nu], z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 70]
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| [https://dlmf.nist.gov/10.27.E4 10.27.E4] || [[Item:Q3494|<math>\modBesselK{\nu}@{z} = \tfrac{1}{2}\pi\frac{\modBesselI{-\nu}@{z}-\modBesselI{\nu}@{z}}{\sin@{\nu\pi}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modBesselK{\nu}@{z} = \tfrac{1}{2}\pi\frac{\modBesselI{-\nu}@{z}-\modBesselI{\nu}@{z}}{\sin@{\nu\pi}}</syntaxhighlight> || <math>\realpart@@{((-\nu)+k+1)} > 0, \realpart@@{(\nu+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>BesselK(nu, z) = (1)/(2)*Pi*(BesselI(- nu, z)- BesselI(nu, z))/(sin(nu*Pi))</syntaxhighlight> || <syntaxhighlight lang=mathematica>BesselK[\[Nu], z] == Divide[1,2]*Pi*Divide[BesselI[- \[Nu], z]- BesselI[\[Nu], z],Sin[\[Nu]*Pi]]</syntaxhighlight> || Successful || Successful || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [14 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, -2]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 2]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/10.27.E6 10.27.E6] || [[Item:Q3496|<math>\modBesselI{\nu}@{z} = e^{-\nu\pi i/2}\BesselJ{\nu}@{ze^{+\pi i/2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modBesselI{\nu}@{z} = e^{-\nu\pi i/2}\BesselJ{\nu}@{ze^{+\pi i/2}}</syntaxhighlight> || <math>-\pi \leq +\phase@@{z}, -\pi \leq -\phase@@{z}, +\phase@@{z} \leq \tfrac{1}{2}\pi, -\phase@@{z} \leq \tfrac{1}{2}\pi, \realpart@@{(\nu+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>BesselI(nu, z) = exp(- nu*Pi*I/2)*BesselJ(nu, z*exp(+ Pi*I/2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>BesselI[\[Nu], z] == Exp[- \[Nu]*Pi*I/2]*BesselJ[\[Nu], z*Exp[+ Pi*I/2]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 50] || Successful [Tested: 50]
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| [https://dlmf.nist.gov/10.27.E6 10.27.E6] || [[Item:Q3496|<math>\modBesselI{\nu}@{z} = e^{+\nu\pi i/2}\BesselJ{\nu}@{ze^{-\pi i/2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modBesselI{\nu}@{z} = e^{+\nu\pi i/2}\BesselJ{\nu}@{ze^{-\pi i/2}}</syntaxhighlight> || <math>-\pi \leq +\phase@@{z}, -\pi \leq -\phase@@{z}, +\phase@@{z} \leq \tfrac{1}{2}\pi, -\phase@@{z} \leq \tfrac{1}{2}\pi, \realpart@@{(\nu+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>BesselI(nu, z) = exp(+ nu*Pi*I/2)*BesselJ(nu, z*exp(- Pi*I/2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>BesselI[\[Nu], z] == Exp[+ \[Nu]*Pi*I/2]*BesselJ[\[Nu], z*Exp[- Pi*I/2]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 50] || Successful [Tested: 50]
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| [https://dlmf.nist.gov/10.27.E7 10.27.E7] || [[Item:Q3497|<math>\modBesselI{\nu}@{z} = \tfrac{1}{2}e^{-\nu\pi i/2}\left(\HankelH{1}{\nu}@{ze^{+\pi i/2}}+\HankelH{2}{\nu}@{ze^{+\pi i/2}}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modBesselI{\nu}@{z} = \tfrac{1}{2}e^{-\nu\pi i/2}\left(\HankelH{1}{\nu}@{ze^{+\pi i/2}}+\HankelH{2}{\nu}@{ze^{+\pi i/2}}\right)</syntaxhighlight> || <math>-\pi \leq +\phase@@{z}, -\pi \leq -\phase@@{z}, +\phase@@{z} \leq \tfrac{1}{2}\pi, -\phase@@{z} \leq \tfrac{1}{2}\pi, \realpart@@{(\nu+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>BesselI(nu, z) = (1)/(2)*exp(- nu*Pi*I/2)*(HankelH1(nu, z*exp(+ Pi*I/2))+ HankelH2(nu, z*exp(+ Pi*I/2)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>BesselI[\[Nu], z] == Divide[1,2]*Exp[- \[Nu]*Pi*I/2]*(HankelH1[\[Nu], z*Exp[+ Pi*I/2]]+ HankelH2[\[Nu], z*Exp[+ Pi*I/2]])</syntaxhighlight> || Failure || Failure || Successful [Tested: 50] || Successful [Tested: 50]
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| [https://dlmf.nist.gov/10.27.E7 10.27.E7] || [[Item:Q3497|<math>\modBesselI{\nu}@{z} = \tfrac{1}{2}e^{+\nu\pi i/2}\left(\HankelH{1}{\nu}@{ze^{-\pi i/2}}+\HankelH{2}{\nu}@{ze^{-\pi i/2}}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modBesselI{\nu}@{z} = \tfrac{1}{2}e^{+\nu\pi i/2}\left(\HankelH{1}{\nu}@{ze^{-\pi i/2}}+\HankelH{2}{\nu}@{ze^{-\pi i/2}}\right)</syntaxhighlight> || <math>-\pi \leq +\phase@@{z}, -\pi \leq -\phase@@{z}, +\phase@@{z} \leq \tfrac{1}{2}\pi, -\phase@@{z} \leq \tfrac{1}{2}\pi, \realpart@@{(\nu+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>BesselI(nu, z) = (1)/(2)*exp(+ nu*Pi*I/2)*(HankelH1(nu, z*exp(- Pi*I/2))+ HankelH2(nu, z*exp(- Pi*I/2)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>BesselI[\[Nu], z] == Divide[1,2]*Exp[+ \[Nu]*Pi*I/2]*(HankelH1[\[Nu], z*Exp[- Pi*I/2]]+ HankelH2[\[Nu], z*Exp[- Pi*I/2]])</syntaxhighlight> || Failure || Failure || Successful [Tested: 50] || Successful [Tested: 50]
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| [https://dlmf.nist.gov/10.27.E9 10.27.E9] || [[Item:Q3499|<math>\pi i\BesselJ{\nu}@{z} = e^{-\nu\pi i/2}\modBesselK{\nu}@{ze^{-\pi i/2}}-e^{\nu\pi i/2}\modBesselK{\nu}@{ze^{\pi i/2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\pi i\BesselJ{\nu}@{z} = e^{-\nu\pi i/2}\modBesselK{\nu}@{ze^{-\pi i/2}}-e^{\nu\pi i/2}\modBesselK{\nu}@{ze^{\pi i/2}}</syntaxhighlight> || <math>|\phase@@{z}| \leq \tfrac{1}{2}\pi, \realpart@@{(\nu+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Pi*I*BesselJ(nu, z) = exp(- nu*Pi*I/2)*BesselK(nu, z*exp(- Pi*I/2))- exp(nu*Pi*I/2)*BesselK(nu, z*exp(Pi*I/2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Pi*I*BesselJ[\[Nu], z] == Exp[- \[Nu]*Pi*I/2]*BesselK[\[Nu], z*Exp[- Pi*I/2]]- Exp[\[Nu]*Pi*I/2]*BesselK[\[Nu], z*Exp[Pi*I/2]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 50] || Successful [Tested: 50]
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| [https://dlmf.nist.gov/10.27.E10 10.27.E10] || [[Item:Q3500|<math>-\pi\BesselY{\nu}@{z} = e^{-\nu\pi i/2}\modBesselK{\nu}@{ze^{-\pi i/2}}+e^{\nu\pi i/2}\modBesselK{\nu}@{ze^{\pi i/2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>-\pi\BesselY{\nu}@{z} = e^{-\nu\pi i/2}\modBesselK{\nu}@{ze^{-\pi i/2}}+e^{\nu\pi i/2}\modBesselK{\nu}@{ze^{\pi i/2}}</syntaxhighlight> || <math>|\phase@@{z}| \leq \tfrac{1}{2}\pi, \realpart@@{(\nu+k+1)} > 0, \realpart@@{((-\nu)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>- Pi*BesselY(nu, z) = exp(- nu*Pi*I/2)*BesselK(nu, z*exp(- Pi*I/2))+ exp(nu*Pi*I/2)*BesselK(nu, z*exp(Pi*I/2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>- Pi*BesselY[\[Nu], z] == Exp[- \[Nu]*Pi*I/2]*BesselK[\[Nu], z*Exp[- Pi*I/2]]+ Exp[\[Nu]*Pi*I/2]*BesselK[\[Nu], z*Exp[Pi*I/2]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 50] || Successful [Tested: 50]
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| [https://dlmf.nist.gov/10.27.E11 10.27.E11] || [[Item:Q3501|<math>\BesselY{\nu}@{z} = e^{+(\nu+1)\pi i/2}\modBesselI{\nu}@{ze^{-\pi i/2}}-(2/\pi)e^{-\nu\pi i/2}\modBesselK{\nu}@{ze^{-\pi i/2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BesselY{\nu}@{z} = e^{+(\nu+1)\pi i/2}\modBesselI{\nu}@{ze^{-\pi i/2}}-(2/\pi)e^{-\nu\pi i/2}\modBesselK{\nu}@{ze^{-\pi i/2}}</syntaxhighlight> || <math>-\tfrac{1}{2}\pi \leq +\phase@@{z}, -\tfrac{1}{2}\pi \leq -\phase@@{z}, +\phase@@{z} \leq \pi, -\phase@@{z} \leq \pi, \realpart@@{(\nu+k+1)} > 0, \realpart@@{((-\nu)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>BesselY(nu, z) = exp(+(nu + 1)*Pi*I/2)*BesselI(nu, z*exp(- Pi*I/2))-(2/Pi)*exp(- nu*Pi*I/2)*BesselK(nu, z*exp(- Pi*I/2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>BesselY[\[Nu], z] == Exp[+(\[Nu]+ 1)*Pi*I/2]*BesselI[\[Nu], z*Exp[- Pi*I/2]]-(2/Pi)*Exp[- \[Nu]*Pi*I/2]*BesselK[\[Nu], z*Exp[- Pi*I/2]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 50] || Successful [Tested: 50]
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| [https://dlmf.nist.gov/10.27.E11 10.27.E11] || [[Item:Q3501|<math>\BesselY{\nu}@{z} = e^{-(\nu+1)\pi i/2}\modBesselI{\nu}@{ze^{+\pi i/2}}-(2/\pi)e^{+\nu\pi i/2}\modBesselK{\nu}@{ze^{+\pi i/2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BesselY{\nu}@{z} = e^{-(\nu+1)\pi i/2}\modBesselI{\nu}@{ze^{+\pi i/2}}-(2/\pi)e^{+\nu\pi i/2}\modBesselK{\nu}@{ze^{+\pi i/2}}</syntaxhighlight> || <math>-\tfrac{1}{2}\pi \leq +\phase@@{z}, -\tfrac{1}{2}\pi \leq -\phase@@{z}, +\phase@@{z} \leq \pi, -\phase@@{z} \leq \pi, \realpart@@{(\nu+k+1)} > 0, \realpart@@{((-\nu)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>BesselY(nu, z) = exp(-(nu + 1)*Pi*I/2)*BesselI(nu, z*exp(+ Pi*I/2))-(2/Pi)*exp(+ nu*Pi*I/2)*BesselK(nu, z*exp(+ Pi*I/2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>BesselY[\[Nu], z] == Exp[-(\[Nu]+ 1)*Pi*I/2]*BesselI[\[Nu], z*Exp[+ Pi*I/2]]-(2/Pi)*Exp[+ \[Nu]*Pi*I/2]*BesselK[\[Nu], z*Exp[+ Pi*I/2]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 50] || Successful [Tested: 50]
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