10.28: Difference between revisions

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! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/10.28.E1 10.28.E1] || [[Item:Q3502|<math>\Wronskian@{\modBesselI{\nu}@{z},\modBesselI{-\nu}@{z}} = \modBesselI{\nu}@{z}\modBesselI{-\nu-1}@{z}-\modBesselI{\nu+1}@{z}\modBesselI{-\nu}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Wronskian@{\modBesselI{\nu}@{z},\modBesselI{-\nu}@{z}} = \modBesselI{\nu}@{z}\modBesselI{-\nu-1}@{z}-\modBesselI{\nu+1}@{z}\modBesselI{-\nu}@{z}</syntaxhighlight> || <math>\realpart@@{(\nu+k+1)} > 0, \realpart@@{((-\nu)+k+1)} > 0, \realpart@@{((-\nu-1)+k+1)} > 0, \realpart@@{((\nu+1)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>(BesselI(nu, z))*diff(BesselI(- nu, z), z)-diff(BesselI(nu, z), z)*(BesselI(- nu, z)) = BesselI(nu, z)*BesselI(- nu - 1, z)- BesselI(nu + 1, z)*BesselI(- nu, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Wronskian[{BesselI[\[Nu], z], BesselI[- \[Nu], z]}, z] == BesselI[\[Nu], z]*BesselI[- \[Nu]- 1, z]- BesselI[\[Nu]+ 1, z]*BesselI[- \[Nu], z]</syntaxhighlight> || Successful || Successful || Skip - symbolical successful subtest || Successful [Tested: 70]
| [https://dlmf.nist.gov/10.28.E1 10.28.E1] || <math qid="Q3502">\Wronskian@{\modBesselI{\nu}@{z},\modBesselI{-\nu}@{z}} = \modBesselI{\nu}@{z}\modBesselI{-\nu-1}@{z}-\modBesselI{\nu+1}@{z}\modBesselI{-\nu}@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Wronskian@{\modBesselI{\nu}@{z},\modBesselI{-\nu}@{z}} = \modBesselI{\nu}@{z}\modBesselI{-\nu-1}@{z}-\modBesselI{\nu+1}@{z}\modBesselI{-\nu}@{z}</syntaxhighlight> || <math>\realpart@@{(\nu+k+1)} > 0, \realpart@@{((-\nu)+k+1)} > 0, \realpart@@{((-\nu-1)+k+1)} > 0, \realpart@@{((\nu+1)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>(BesselI(nu, z))*diff(BesselI(- nu, z), z)-diff(BesselI(nu, z), z)*(BesselI(- nu, z)) = BesselI(nu, z)*BesselI(- nu - 1, z)- BesselI(nu + 1, z)*BesselI(- nu, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Wronskian[{BesselI[\[Nu], z], BesselI[- \[Nu], z]}, z] == BesselI[\[Nu], z]*BesselI[- \[Nu]- 1, z]- BesselI[\[Nu]+ 1, z]*BesselI[- \[Nu], z]</syntaxhighlight> || Successful || Successful || Skip - symbolical successful subtest || Successful [Tested: 70]
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| [https://dlmf.nist.gov/10.28.E1 10.28.E1] || [[Item:Q3502|<math>\modBesselI{\nu}@{z}\modBesselI{-\nu-1}@{z}-\modBesselI{\nu+1}@{z}\modBesselI{-\nu}@{z} = -2\sin@{\nu\pi}/(\pi z)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modBesselI{\nu}@{z}\modBesselI{-\nu-1}@{z}-\modBesselI{\nu+1}@{z}\modBesselI{-\nu}@{z} = -2\sin@{\nu\pi}/(\pi z)</syntaxhighlight> || <math>\realpart@@{(\nu+k+1)} > 0, \realpart@@{((-\nu)+k+1)} > 0, \realpart@@{((-\nu-1)+k+1)} > 0, \realpart@@{((\nu+1)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>BesselI(nu, z)*BesselI(- nu - 1, z)- BesselI(nu + 1, z)*BesselI(- nu, z) = - 2*sin(nu*Pi)/(Pi*z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BesselI[\[Nu], z]*BesselI[- \[Nu]- 1, z]- BesselI[\[Nu]+ 1, z]*BesselI[- \[Nu], z] == - 2*Sin[\[Nu]*Pi]/(Pi*z)</syntaxhighlight> || Failure || Successful || Successful [Tested: 70] || Successful [Tested: 70]
| [https://dlmf.nist.gov/10.28.E1 10.28.E1] || <math qid="Q3502">\modBesselI{\nu}@{z}\modBesselI{-\nu-1}@{z}-\modBesselI{\nu+1}@{z}\modBesselI{-\nu}@{z} = -2\sin@{\nu\pi}/(\pi z)</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modBesselI{\nu}@{z}\modBesselI{-\nu-1}@{z}-\modBesselI{\nu+1}@{z}\modBesselI{-\nu}@{z} = -2\sin@{\nu\pi}/(\pi z)</syntaxhighlight> || <math>\realpart@@{(\nu+k+1)} > 0, \realpart@@{((-\nu)+k+1)} > 0, \realpart@@{((-\nu-1)+k+1)} > 0, \realpart@@{((\nu+1)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>BesselI(nu, z)*BesselI(- nu - 1, z)- BesselI(nu + 1, z)*BesselI(- nu, z) = - 2*sin(nu*Pi)/(Pi*z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BesselI[\[Nu], z]*BesselI[- \[Nu]- 1, z]- BesselI[\[Nu]+ 1, z]*BesselI[- \[Nu], z] == - 2*Sin[\[Nu]*Pi]/(Pi*z)</syntaxhighlight> || Failure || Successful || Successful [Tested: 70] || Successful [Tested: 70]
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| [https://dlmf.nist.gov/10.28.E2 10.28.E2] || [[Item:Q3503|<math>\Wronskian@{\modBesselK{\nu}@{z},\modBesselI{\nu}@{z}} = \modBesselI{\nu}@{z}\modBesselK{\nu+1}@{z}+\modBesselI{\nu+1}@{z}\modBesselK{\nu}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Wronskian@{\modBesselK{\nu}@{z},\modBesselI{\nu}@{z}} = \modBesselI{\nu}@{z}\modBesselK{\nu+1}@{z}+\modBesselI{\nu+1}@{z}\modBesselK{\nu}@{z}</syntaxhighlight> || <math>\realpart@@{(\nu+k+1)} > 0, \realpart@@{((\nu+1)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>(BesselK(nu, z))*diff(BesselI(nu, z), z)-diff(BesselK(nu, z), z)*(BesselI(nu, z)) = BesselI(nu, z)*BesselK(nu + 1, z)+ BesselI(nu + 1, z)*BesselK(nu, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Wronskian[{BesselK[\[Nu], z], BesselI[\[Nu], z]}, z] == BesselI[\[Nu], z]*BesselK[\[Nu]+ 1, z]+ BesselI[\[Nu]+ 1, z]*BesselK[\[Nu], z]</syntaxhighlight> || Successful || Successful || Skip - symbolical successful subtest || Successful [Tested: 70]
| [https://dlmf.nist.gov/10.28.E2 10.28.E2] || <math qid="Q3503">\Wronskian@{\modBesselK{\nu}@{z},\modBesselI{\nu}@{z}} = \modBesselI{\nu}@{z}\modBesselK{\nu+1}@{z}+\modBesselI{\nu+1}@{z}\modBesselK{\nu}@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Wronskian@{\modBesselK{\nu}@{z},\modBesselI{\nu}@{z}} = \modBesselI{\nu}@{z}\modBesselK{\nu+1}@{z}+\modBesselI{\nu+1}@{z}\modBesselK{\nu}@{z}</syntaxhighlight> || <math>\realpart@@{(\nu+k+1)} > 0, \realpart@@{((\nu+1)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>(BesselK(nu, z))*diff(BesselI(nu, z), z)-diff(BesselK(nu, z), z)*(BesselI(nu, z)) = BesselI(nu, z)*BesselK(nu + 1, z)+ BesselI(nu + 1, z)*BesselK(nu, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Wronskian[{BesselK[\[Nu], z], BesselI[\[Nu], z]}, z] == BesselI[\[Nu], z]*BesselK[\[Nu]+ 1, z]+ BesselI[\[Nu]+ 1, z]*BesselK[\[Nu], z]</syntaxhighlight> || Successful || Successful || Skip - symbolical successful subtest || Successful [Tested: 70]
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| [https://dlmf.nist.gov/10.28.E2 10.28.E2] || [[Item:Q3503|<math>\modBesselI{\nu}@{z}\modBesselK{\nu+1}@{z}+\modBesselI{\nu+1}@{z}\modBesselK{\nu}@{z} = 1/z</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modBesselI{\nu}@{z}\modBesselK{\nu+1}@{z}+\modBesselI{\nu+1}@{z}\modBesselK{\nu}@{z} = 1/z</syntaxhighlight> || <math>\realpart@@{(\nu+k+1)} > 0, \realpart@@{((\nu+1)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>BesselI(nu, z)*BesselK(nu + 1, z)+ BesselI(nu + 1, z)*BesselK(nu, z) = 1/z</syntaxhighlight> || <syntaxhighlight lang=mathematica>BesselI[\[Nu], z]*BesselK[\[Nu]+ 1, z]+ BesselI[\[Nu]+ 1, z]*BesselK[\[Nu], z] == 1/z</syntaxhighlight> || Failure || Successful || Successful [Tested: 70] || Successful [Tested: 70]
| [https://dlmf.nist.gov/10.28.E2 10.28.E2] || <math qid="Q3503">\modBesselI{\nu}@{z}\modBesselK{\nu+1}@{z}+\modBesselI{\nu+1}@{z}\modBesselK{\nu}@{z} = 1/z</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modBesselI{\nu}@{z}\modBesselK{\nu+1}@{z}+\modBesselI{\nu+1}@{z}\modBesselK{\nu}@{z} = 1/z</syntaxhighlight> || <math>\realpart@@{(\nu+k+1)} > 0, \realpart@@{((\nu+1)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>BesselI(nu, z)*BesselK(nu + 1, z)+ BesselI(nu + 1, z)*BesselK(nu, z) = 1/z</syntaxhighlight> || <syntaxhighlight lang=mathematica>BesselI[\[Nu], z]*BesselK[\[Nu]+ 1, z]+ BesselI[\[Nu]+ 1, z]*BesselK[\[Nu], z] == 1/z</syntaxhighlight> || Failure || Successful || Successful [Tested: 70] || Successful [Tested: 70]
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Latest revision as of 11:25, 28 June 2021


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
10.28.E1 𝒲 ⁡ { I ν ⁡ ( z ) , I - ν ⁡ ( z ) } = I ν ⁡ ( z ) ⁢ I - ν - 1 ⁡ ( z ) - I ν + 1 ⁡ ( z ) ⁢ I - ν ⁡ ( z ) Wronskian modified-Bessel-first-kind 𝜈 𝑧 modified-Bessel-first-kind 𝜈 𝑧 modified-Bessel-first-kind 𝜈 𝑧 modified-Bessel-first-kind 𝜈 1 𝑧 modified-Bessel-first-kind 𝜈 1 𝑧 modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle\mathscr{W}\left\{I_{\nu}\left(z\right),I_{-\nu}% \left(z\right)\right\}=I_{\nu}\left(z\right)I_{-\nu-1}\left(z\right)-I_{\nu+1}% \left(z\right)I_{-\nu}\left(z\right)}}
\Wronskian@{\modBesselI{\nu}@{z},\modBesselI{-\nu}@{z}} = \modBesselI{\nu}@{z}\modBesselI{-\nu-1}@{z}-\modBesselI{\nu+1}@{z}\modBesselI{-\nu}@{z}		 ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( - ν ) + k + 1 ) > 0 , ℜ ⁡ ( ( - ν - 1 ) + k + 1 ) > 0 , ℜ ⁡ ( ( ν + 1 ) + k + 1 ) > 0 formulae-sequence 𝜈 𝑘 1 0 formulae-sequence 𝜈 𝑘 1 0 formulae-sequence 𝜈 1 𝑘 1 0 𝜈 1 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0,\Re((-\nu-1)+k+1)% >0,\Re((\nu+1)+k+1)>0}} 
(BesselI(nu, z))*diff(BesselI(- nu, z), z)-diff(BesselI(nu, z), z)*(BesselI(- nu, z)) = BesselI(nu, z)*BesselI(- nu - 1, z)- BesselI(nu + 1, z)*BesselI(- nu, z)

Wronskian[{BesselI[\[Nu], z], BesselI[- \[Nu], z]}, z] == BesselI[\[Nu], z]*BesselI[- \[Nu]- 1, z]- BesselI[\[Nu]+ 1, z]*BesselI[- \[Nu], z]
Successful Successful Skip - symbolical successful subtest Successful [Tested: 70]
10.28.E1 I ν ⁡ ( z ) ⁢ I - ν - 1 ⁡ ( z ) - I ν + 1 ⁡ ( z ) ⁢ I - ν ⁡ ( z ) = - 2 ⁢ sin ⁡ ( ν ⁢ π ) / ( π ⁢ z ) modified-Bessel-first-kind 𝜈 𝑧 modified-Bessel-first-kind 𝜈 1 𝑧 modified-Bessel-first-kind 𝜈 1 𝑧 modified-Bessel-first-kind 𝜈 𝑧 2 𝜈 𝜋 𝜋 𝑧 {\displaystyle{\displaystyle I_{\nu}\left(z\right)I_{-\nu-1}\left(z\right)-I_{% \nu+1}\left(z\right)I_{-\nu}\left(z\right)=-2\sin\left(\nu\pi\right)/(\pi z)}}
\modBesselI{\nu}@{z}\modBesselI{-\nu-1}@{z}-\modBesselI{\nu+1}@{z}\modBesselI{-\nu}@{z} = -2\sin@{\nu\pi}/(\pi z)		 ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( - ν ) + k + 1 ) > 0 , ℜ ⁡ ( ( - ν - 1 ) + k + 1 ) > 0 , ℜ ⁡ ( ( ν + 1 ) + k + 1 ) > 0 formulae-sequence 𝜈 𝑘 1 0 formulae-sequence 𝜈 𝑘 1 0 formulae-sequence 𝜈 1 𝑘 1 0 𝜈 1 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0,\Re((-\nu-1)+k+1)% >0,\Re((\nu+1)+k+1)>0}} 
BesselI(nu, z)*BesselI(- nu - 1, z)- BesselI(nu + 1, z)*BesselI(- nu, z) = - 2*sin(nu*Pi)/(Pi*z)

BesselI[\[Nu], z]*BesselI[- \[Nu]- 1, z]- BesselI[\[Nu]+ 1, z]*BesselI[- \[Nu], z] == - 2*Sin[\[Nu]*Pi]/(Pi*z)
Failure Successful Successful [Tested: 70] Successful [Tested: 70]
10.28.E2 𝒲 ⁡ { K ν ⁡ ( z ) , I ν ⁡ ( z ) } = I ν ⁡ ( z ) ⁢ K ν + 1 ⁡ ( z ) + I ν + 1 ⁡ ( z ) ⁢ K ν ⁡ ( z ) Wronskian modified-Bessel-second-kind 𝜈 𝑧 modified-Bessel-first-kind 𝜈 𝑧 modified-Bessel-first-kind 𝜈 𝑧 modified-Bessel-second-kind 𝜈 1 𝑧 modified-Bessel-first-kind 𝜈 1 𝑧 modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle\mathscr{W}\left\{K_{\nu}\left(z\right),I_{\nu}% \left(z\right)\right\}=I_{\nu}\left(z\right)K_{\nu+1}\left(z\right)+I_{\nu+1}% \left(z\right)K_{\nu}\left(z\right)}}
\Wronskian@{\modBesselK{\nu}@{z},\modBesselI{\nu}@{z}} = \modBesselI{\nu}@{z}\modBesselK{\nu+1}@{z}+\modBesselI{\nu+1}@{z}\modBesselK{\nu}@{z}		 ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( ν + 1 ) + k + 1 ) > 0 formulae-sequence 𝜈 𝑘 1 0 𝜈 1 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((\nu+1)+k+1)>0}} 
(BesselK(nu, z))*diff(BesselI(nu, z), z)-diff(BesselK(nu, z), z)*(BesselI(nu, z)) = BesselI(nu, z)*BesselK(nu + 1, z)+ BesselI(nu + 1, z)*BesselK(nu, z)

Wronskian[{BesselK[\[Nu], z], BesselI[\[Nu], z]}, z] == BesselI[\[Nu], z]*BesselK[\[Nu]+ 1, z]+ BesselI[\[Nu]+ 1, z]*BesselK[\[Nu], z]
Successful Successful Skip - symbolical successful subtest Successful [Tested: 70]
10.28.E2 I ν ⁡ ( z ) ⁢ K ν + 1 ⁡ ( z ) + I ν + 1 ⁡ ( z ) ⁢ K ν ⁡ ( z ) = 1 / z modified-Bessel-first-kind 𝜈 𝑧 modified-Bessel-second-kind 𝜈 1 𝑧 modified-Bessel-first-kind 𝜈 1 𝑧 modified-Bessel-second-kind 𝜈 𝑧 1 𝑧 {\displaystyle{\displaystyle I_{\nu}\left(z\right)K_{\nu+1}\left(z\right)+I_{% \nu+1}\left(z\right)K_{\nu}\left(z\right)=1/z}}
\modBesselI{\nu}@{z}\modBesselK{\nu+1}@{z}+\modBesselI{\nu+1}@{z}\modBesselK{\nu}@{z} = 1/z		 ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( ν + 1 ) + k + 1 ) > 0 formulae-sequence 𝜈 𝑘 1 0 𝜈 1 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((\nu+1)+k+1)>0}} 
BesselI(nu, z)*BesselK(nu + 1, z)+ BesselI(nu + 1, z)*BesselK(nu, z) = 1/z

BesselI[\[Nu], z]*BesselK[\[Nu]+ 1, z]+ BesselI[\[Nu]+ 1, z]*BesselK[\[Nu], z] == 1/z
Failure Successful Successful [Tested: 70] Successful [Tested: 70]
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