10.17: 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/10.17.E7 10.17.E7] || [[Item:Q3176|<math>z^{\frac{1}{2}} = \exp@{\tfrac{1}{2}\ln@@{|z|}+\tfrac{1}{2}i\phase@@{z}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>z^{\frac{1}{2}} = \exp@{\tfrac{1}{2}\ln@@{|z|}+\tfrac{1}{2}i\phase@@{z}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(z)^((1)/(2)) = exp((1)/(2)*ln(abs(z))+(1)/(2)*I*argument(z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>(z)^(Divide[1,2]) == Exp[Divide[1,2]*Log[Abs[z]]+Divide[1,2]*I*Arg[z]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 7] || Successful [Tested: 7]
| [https://dlmf.nist.gov/10.17.E7 10.17.E7] || <math qid="Q3176">z^{\frac{1}{2}} = \exp@{\tfrac{1}{2}\ln@@{|z|}+\tfrac{1}{2}i\phase@@{z}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>z^{\frac{1}{2}} = \exp@{\tfrac{1}{2}\ln@@{|z|}+\tfrac{1}{2}i\phase@@{z}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(z)^((1)/(2)) = exp((1)/(2)*ln(abs(z))+(1)/(2)*I*argument(z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>(z)^(Divide[1,2]) == Exp[Divide[1,2]*Log[Abs[z]]+Divide[1,2]*I*Arg[z]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 7] || Successful [Tested: 7]
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| [https://dlmf.nist.gov/10.17.E16 10.17.E16] || [[Item:Q3185|<math>\scterminant{p}@{z} = \frac{e^{z}}{2\pi}\EulerGamma@{p}\incGamma@{1-p}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\scterminant{p}@{z} = \frac{e^{z}}{2\pi}\EulerGamma@{p}\incGamma@{1-p}{z}</syntaxhighlight> || <math>\realpart@@{p} > 0</math> || <syntaxhighlight lang=mathematica>(exp(z)/(2*Pi))*GAMMA(p)*GAMMA(1-p,z) = (exp(z))/(2*Pi)*GAMMA(p)*GAMMA(1 - p, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || Successful || Missing Macro Error || - || -
| [https://dlmf.nist.gov/10.17.E16 10.17.E16] || <math qid="Q3185">\scterminant{p}@{z} = \frac{e^{z}}{2\pi}\EulerGamma@{p}\incGamma@{1-p}{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\scterminant{p}@{z} = \frac{e^{z}}{2\pi}\EulerGamma@{p}\incGamma@{1-p}{z}</syntaxhighlight> || <math>\realpart@@{p} > 0</math> || <syntaxhighlight lang=mathematica>(exp(z)/(2*Pi))*GAMMA(p)*GAMMA(1-p,z) = (exp(z))/(2*Pi)*GAMMA(p)*GAMMA(1 - p, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || Successful || Missing Macro Error || - || -
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| [https://dlmf.nist.gov/10.17.E17 10.17.E17] || [[Item:Q3186|<math>R_{\ell}^{+}(\nu,z) = (-1)^{\ell}2\cos@{\nu\pi}\*\left(\sum_{k=0}^{m-1}(+ i)^{k}\frac{a_{k}(\nu)}{z^{k}}\scterminant{\ell-k}@{- 2iz}+R_{m,\ell}^{+}(\nu,z)\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>R_{\ell}^{+}(\nu,z) = (-1)^{\ell}2\cos@{\nu\pi}\*\left(\sum_{k=0}^{m-1}(+ i)^{k}\frac{a_{k}(\nu)}{z^{k}}\scterminant{\ell-k}@{- 2iz}+R_{m,\ell}^{+}(\nu,z)\right)</syntaxhighlight> || <math>\realpart@@{(\ell-k)} > 0, k \geq 1</math> || <syntaxhighlight lang=mathematica>(R[ell])^(+)(nu , z) = (- 1)^(ell)* 2*cos(nu*Pi)*(sum((+ I)^(k)*(((4*(nu)^(2)- (1)^(2))*(4*(nu)^(2)- (3)^(2)) .. (4*(nu)^(2)-(2*k - 1)^(2)))/(factorial(k)*(8)^(k)))/((z)^(k))*(exp(- 2*I*z)/(2*Pi))*GAMMA(ell - k)*GAMMA(1-ell - k,- 2*I*z), k = 0..m - 1)+ (R[m , ell])^(+)(nu , z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || Error || Missing Macro Error || - || -
| [https://dlmf.nist.gov/10.17.E17 10.17.E17] || <math qid="Q3186">R_{\ell}^{+}(\nu,z) = (-1)^{\ell}2\cos@{\nu\pi}\*\left(\sum_{k=0}^{m-1}(+ i)^{k}\frac{a_{k}(\nu)}{z^{k}}\scterminant{\ell-k}@{- 2iz}+R_{m,\ell}^{+}(\nu,z)\right)</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>R_{\ell}^{+}(\nu,z) = (-1)^{\ell}2\cos@{\nu\pi}\*\left(\sum_{k=0}^{m-1}(+ i)^{k}\frac{a_{k}(\nu)}{z^{k}}\scterminant{\ell-k}@{- 2iz}+R_{m,\ell}^{+}(\nu,z)\right)</syntaxhighlight> || <math>\realpart@@{(\ell-k)} > 0, k \geq 1</math> || <syntaxhighlight lang=mathematica>(R[ell])^(+)(nu , z) = (- 1)^(ell)* 2*cos(nu*Pi)*(sum((+ I)^(k)*(((4*(nu)^(2)- (1)^(2))*(4*(nu)^(2)- (3)^(2)) .. (4*(nu)^(2)-(2*k - 1)^(2)))/(factorial(k)*(8)^(k)))/((z)^(k))*(exp(- 2*I*z)/(2*Pi))*GAMMA(ell - k)*GAMMA(1-ell - k,- 2*I*z), k = 0..m - 1)+ (R[m , ell])^(+)(nu , z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || Error || Missing Macro Error || - || -
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| [https://dlmf.nist.gov/10.17.E17 10.17.E17] || [[Item:Q3186|<math>R_{\ell}^{-}(\nu,z) = (-1)^{\ell}2\cos@{\nu\pi}\*\left(\sum_{k=0}^{m-1}(- i)^{k}\frac{a_{k}(\nu)}{z^{k}}\scterminant{\ell-k}@{+ 2iz}+R_{m,\ell}^{-}(\nu,z)\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>R_{\ell}^{-}(\nu,z) = (-1)^{\ell}2\cos@{\nu\pi}\*\left(\sum_{k=0}^{m-1}(- i)^{k}\frac{a_{k}(\nu)}{z^{k}}\scterminant{\ell-k}@{+ 2iz}+R_{m,\ell}^{-}(\nu,z)\right)</syntaxhighlight> || <math>\realpart@@{(\ell-k)} > 0, k \geq 1</math> || <syntaxhighlight lang=mathematica>(R[ell])^(-)(nu , z) = (- 1)^(ell)* 2*cos(nu*Pi)*(sum((- I)^(k)*(((4*(nu)^(2)- (1)^(2))*(4*(nu)^(2)- (3)^(2)) .. (4*(nu)^(2)-(2*k - 1)^(2)))/(factorial(k)*(8)^(k)))/((z)^(k))*(exp(+ 2*I*z)/(2*Pi))*GAMMA(ell - k)*GAMMA(1-ell - k,+ 2*I*z), k = 0..m - 1)+ (R[m , ell])^(-)(nu , z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || Error || Missing Macro Error || - || -
| [https://dlmf.nist.gov/10.17.E17 10.17.E17] || <math qid="Q3186">R_{\ell}^{-}(\nu,z) = (-1)^{\ell}2\cos@{\nu\pi}\*\left(\sum_{k=0}^{m-1}(- i)^{k}\frac{a_{k}(\nu)}{z^{k}}\scterminant{\ell-k}@{+ 2iz}+R_{m,\ell}^{-}(\nu,z)\right)</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>R_{\ell}^{-}(\nu,z) = (-1)^{\ell}2\cos@{\nu\pi}\*\left(\sum_{k=0}^{m-1}(- i)^{k}\frac{a_{k}(\nu)}{z^{k}}\scterminant{\ell-k}@{+ 2iz}+R_{m,\ell}^{-}(\nu,z)\right)</syntaxhighlight> || <math>\realpart@@{(\ell-k)} > 0, k \geq 1</math> || <syntaxhighlight lang=mathematica>(R[ell])^(-)(nu , z) = (- 1)^(ell)* 2*cos(nu*Pi)*(sum((- I)^(k)*(((4*(nu)^(2)- (1)^(2))*(4*(nu)^(2)- (3)^(2)) .. (4*(nu)^(2)-(2*k - 1)^(2)))/(factorial(k)*(8)^(k)))/((z)^(k))*(exp(+ 2*I*z)/(2*Pi))*GAMMA(ell - k)*GAMMA(1-ell - k,+ 2*I*z), k = 0..m - 1)+ (R[m , ell])^(-)(nu , z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || Error || Missing Macro Error || - || -
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Latest revision as of 11:23, 28 June 2021


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
10.17.E7 z 1 2 = exp ⁡ ( 1 2 ⁢ ln ⁡ | z | + 1 2 ⁢ i ⁢ ph ⁡ z ) superscript 𝑧 1 2 1 2 𝑧 1 2 𝑖 phase 𝑧 {\displaystyle{\displaystyle z^{\frac{1}{2}}=\exp\left(\tfrac{1}{2}\ln|z|+% \tfrac{1}{2}i\operatorname{ph}z\right)}}
z^{\frac{1}{2}} = \exp@{\tfrac{1}{2}\ln@@{|z|}+\tfrac{1}{2}i\phase@@{z}}		 

(z)^((1)/(2)) = exp((1)/(2)*ln(abs(z))+(1)/(2)*I*argument(z))

(z)^(Divide[1,2]) == Exp[Divide[1,2]*Log[Abs[z]]+Divide[1,2]*I*Arg[z]]
Failure Failure Successful [Tested: 7] Successful [Tested: 7]
10.17.E16 G p ⁡ ( z ) = e z 2 ⁢ π ⁢ Γ ⁡ ( p ) ⁢ Γ ⁡ ( 1 - p , z ) rescaled-terminant-function 𝑝 𝑧 superscript 𝑒 𝑧 2 𝜋 Euler-Gamma 𝑝 incomplete-Gamma 1 𝑝 𝑧 {\displaystyle{\displaystyle G_{p}\left(z\right)=\frac{e^{z}}{2\pi}\Gamma\left% (p\right)\Gamma\left(1-p,z\right)}}
\scterminant{p}@{z} = \frac{e^{z}}{2\pi}\EulerGamma@{p}\incGamma@{1-p}{z}		 ℜ ⁡ p > 0 𝑝 0 {\displaystyle{\displaystyle\Re p>0}} 
(exp(z)/(2*Pi))*GAMMA(p)*GAMMA(1-p,z) = (exp(z))/(2*Pi)*GAMMA(p)*GAMMA(1 - p, z)

Error
Successful Missing Macro Error - -
10.17.E17 R ℓ + ⁢ ( ν , z ) = ( - 1 ) ℓ ⁢ 2 ⁢ cos ⁡ ( ν ⁢ π ) ⁢ ( ∑ k = 0 m - 1 ( + i ) k ⁢ a k ⁢ ( ν ) z k ⁢ G ℓ - k ⁡ ( - 2 ⁢ i ⁢ z ) + R m , ℓ + ⁢ ( ν , z ) ) superscript subscript 𝑅 ℓ 𝜈 𝑧 superscript 1 ℓ 2 𝜈 𝜋 superscript subscript 𝑘 0 𝑚 1 superscript 𝑖 𝑘 subscript 𝑎 𝑘 𝜈 superscript 𝑧 𝑘 rescaled-terminant-function ℓ 𝑘 2 𝑖 𝑧 superscript subscript 𝑅 𝑚 ℓ 𝜈 𝑧 {\displaystyle{\displaystyle R_{\ell}^{+}(\nu,z)=(-1)^{\ell}2\cos\left(\nu\pi% \right)\*\left(\sum_{k=0}^{m-1}(+i)^{k}\frac{a_{k}(\nu)}{z^{k}}G_{\ell-k}\left% (-2iz\right)+R_{m,\ell}^{+}(\nu,z)\right)}}
R_{\ell}^{+}(\nu,z) = (-1)^{\ell}2\cos@{\nu\pi}\*\left(\sum_{k=0}^{m-1}(+ i)^{k}\frac{a_{k}(\nu)}{z^{k}}\scterminant{\ell-k}@{- 2iz}+R_{m,\ell}^{+}(\nu,z)\right)		 ℜ ⁡ ( ℓ - k ) > 0 , k ≥ 1 formulae-sequence ℓ 𝑘 0 𝑘 1 {\displaystyle{\displaystyle\Re(\ell-k)>0,k\geq 1}} 
(R[ell])^(+)(nu , z) = (- 1)^(ell)* 2*cos(nu*Pi)*(sum((+ I)^(k)*(((4*(nu)^(2)- (1)^(2))*(4*(nu)^(2)- (3)^(2)) .. (4*(nu)^(2)-(2*k - 1)^(2)))/(factorial(k)*(8)^(k)))/((z)^(k))*(exp(- 2*I*z)/(2*Pi))*GAMMA(ell - k)*GAMMA(1-ell - k,- 2*I*z), k = 0..m - 1)+ (R[m , ell])^(+)(nu , z))

Error
Error Missing Macro Error - -
10.17.E17 R ℓ - ⁢ ( ν , z ) = ( - 1 ) ℓ ⁢ 2 ⁢ cos ⁡ ( ν ⁢ π ) ⁢ ( ∑ k = 0 m - 1 ( - i ) k ⁢ a k ⁢ ( ν ) z k ⁢ G ℓ - k ⁡ ( + 2 ⁢ i ⁢ z ) + R m , ℓ - ⁢ ( ν , z ) ) superscript subscript 𝑅 ℓ 𝜈 𝑧 superscript 1 ℓ 2 𝜈 𝜋 superscript subscript 𝑘 0 𝑚 1 superscript 𝑖 𝑘 subscript 𝑎 𝑘 𝜈 superscript 𝑧 𝑘 rescaled-terminant-function ℓ 𝑘 2 𝑖 𝑧 superscript subscript 𝑅 𝑚 ℓ 𝜈 𝑧 {\displaystyle{\displaystyle R_{\ell}^{-}(\nu,z)=(-1)^{\ell}2\cos\left(\nu\pi% \right)\*\left(\sum_{k=0}^{m-1}(-i)^{k}\frac{a_{k}(\nu)}{z^{k}}G_{\ell-k}\left% (+2iz\right)+R_{m,\ell}^{-}(\nu,z)\right)}}
R_{\ell}^{-}(\nu,z) = (-1)^{\ell}2\cos@{\nu\pi}\*\left(\sum_{k=0}^{m-1}(- i)^{k}\frac{a_{k}(\nu)}{z^{k}}\scterminant{\ell-k}@{+ 2iz}+R_{m,\ell}^{-}(\nu,z)\right)		 ℜ ⁡ ( ℓ - k ) > 0 , k ≥ 1 formulae-sequence ℓ 𝑘 0 𝑘 1 {\displaystyle{\displaystyle\Re(\ell-k)>0,k\geq 1}} 
(R[ell])^(-)(nu , z) = (- 1)^(ell)* 2*cos(nu*Pi)*(sum((- I)^(k)*(((4*(nu)^(2)- (1)^(2))*(4*(nu)^(2)- (3)^(2)) .. (4*(nu)^(2)-(2*k - 1)^(2)))/(factorial(k)*(8)^(k)))/((z)^(k))*(exp(+ 2*I*z)/(2*Pi))*GAMMA(ell - k)*GAMMA(1-ell - k,+ 2*I*z), k = 0..m - 1)+ (R[m , ell])^(-)(nu , z))

Error
Error Missing Macro Error - -
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