4.36: 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/4.36.E1 4.36.E1] || [[Item:Q1915|<math>\sinh@@{z} = z\prod_{n=1}^{\infty}\left(1+\frac{z^{2}}{n^{2}\pi^{2}}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sinh@@{z} = z\prod_{n=1}^{\infty}\left(1+\frac{z^{2}}{n^{2}\pi^{2}}\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sinh(z) = z*product(1 +((z)^(2))/((n)^(2)* (Pi)^(2)), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sinh[z] == z*Product[1 +Divide[(z)^(2),(n)^(2)* (Pi)^(2)], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
| [https://dlmf.nist.gov/4.36.E1 4.36.E1] || <math qid="Q1915">\sinh@@{z} = z\prod_{n=1}^{\infty}\left(1+\frac{z^{2}}{n^{2}\pi^{2}}\right)</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sinh@@{z} = z\prod_{n=1}^{\infty}\left(1+\frac{z^{2}}{n^{2}\pi^{2}}\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sinh(z) = z*product(1 +((z)^(2))/((n)^(2)* (Pi)^(2)), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sinh[z] == z*Product[1 +Divide[(z)^(2),(n)^(2)* (Pi)^(2)], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
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| [https://dlmf.nist.gov/4.36.E2 4.36.E2] || [[Item:Q1916|<math>\cosh@@{z} = \prod_{n=1}^{\infty}\left(1+\frac{4z^{2}}{(2n-1)^{2}\pi^{2}}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\cosh@@{z} = \prod_{n=1}^{\infty}\left(1+\frac{4z^{2}}{(2n-1)^{2}\pi^{2}}\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>cosh(z) = product(1 +(4*(z)^(2))/((2*n - 1)^(2)* (Pi)^(2)), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Cosh[z] == Product[1 +Divide[4*(z)^(2),(2*n - 1)^(2)* (Pi)^(2)], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
| [https://dlmf.nist.gov/4.36.E2 4.36.E2] || <math qid="Q1916">\cosh@@{z} = \prod_{n=1}^{\infty}\left(1+\frac{4z^{2}}{(2n-1)^{2}\pi^{2}}\right)</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\cosh@@{z} = \prod_{n=1}^{\infty}\left(1+\frac{4z^{2}}{(2n-1)^{2}\pi^{2}}\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>cosh(z) = product(1 +(4*(z)^(2))/((2*n - 1)^(2)* (Pi)^(2)), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Cosh[z] == Product[1 +Divide[4*(z)^(2),(2*n - 1)^(2)* (Pi)^(2)], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
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| [https://dlmf.nist.gov/4.36.E3 4.36.E3] || [[Item:Q1917|<math>\coth@@{z} = \frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{1}{z^{2}+n^{2}\pi^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\coth@@{z} = \frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{1}{z^{2}+n^{2}\pi^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>coth(z) = (1)/(z)+ 2*z*sum((1)/((z)^(2)+ (n)^(2)* (Pi)^(2)), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Coth[z] == Divide[1,z]+ 2*z*Sum[Divide[1,(z)^(2)+ (n)^(2)* (Pi)^(2)], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || Successful [Tested: 7] || Successful [Tested: 7]
| [https://dlmf.nist.gov/4.36.E3 4.36.E3] || <math qid="Q1917">\coth@@{z} = \frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{1}{z^{2}+n^{2}\pi^{2}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\coth@@{z} = \frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{1}{z^{2}+n^{2}\pi^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>coth(z) = (1)/(z)+ 2*z*sum((1)/((z)^(2)+ (n)^(2)* (Pi)^(2)), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Coth[z] == Divide[1,z]+ 2*z*Sum[Divide[1,(z)^(2)+ (n)^(2)* (Pi)^(2)], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || Successful [Tested: 7] || Successful [Tested: 7]
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| [https://dlmf.nist.gov/4.36.E4 4.36.E4] || [[Item:Q1918|<math>\csch^{2}@@{z} = \sum_{n=-\infty}^{\infty}\frac{1}{(z-n\pi i)^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\csch^{2}@@{z} = \sum_{n=-\infty}^{\infty}\frac{1}{(z-n\pi i)^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(csch(z))^(2) = sum((1)/((z - n*Pi*I)^(2)), n = - infinity..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(Csch[z])^(2) == Sum[Divide[1,(z - n*Pi*I)^(2)], {n, - Infinity, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || Successful [Tested: 7] || Successful [Tested: 7]
| [https://dlmf.nist.gov/4.36.E4 4.36.E4] || <math qid="Q1918">\csch^{2}@@{z} = \sum_{n=-\infty}^{\infty}\frac{1}{(z-n\pi i)^{2}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\csch^{2}@@{z} = \sum_{n=-\infty}^{\infty}\frac{1}{(z-n\pi i)^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(csch(z))^(2) = sum((1)/((z - n*Pi*I)^(2)), n = - infinity..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(Csch[z])^(2) == Sum[Divide[1,(z - n*Pi*I)^(2)], {n, - Infinity, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || Successful [Tested: 7] || Successful [Tested: 7]
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| [https://dlmf.nist.gov/4.36.E5 4.36.E5] || [[Item:Q1919|<math>\csch@@{z} = \frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{(-1)^{n}}{z^{2}+n^{2}\pi^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\csch@@{z} = \frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{(-1)^{n}}{z^{2}+n^{2}\pi^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>csch(z) = (1)/(z)+ 2*z*sum(((- 1)^(n))/((z)^(2)+ (n)^(2)* (Pi)^(2)), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Csch[z] == Divide[1,z]+ 2*z*Sum[Divide[(- 1)^(n),(z)^(2)+ (n)^(2)* (Pi)^(2)], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || Successful [Tested: 7] || Successful [Tested: 7]
| [https://dlmf.nist.gov/4.36.E5 4.36.E5] || <math qid="Q1919">\csch@@{z} = \frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{(-1)^{n}}{z^{2}+n^{2}\pi^{2}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\csch@@{z} = \frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{(-1)^{n}}{z^{2}+n^{2}\pi^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>csch(z) = (1)/(z)+ 2*z*sum(((- 1)^(n))/((z)^(2)+ (n)^(2)* (Pi)^(2)), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Csch[z] == Divide[1,z]+ 2*z*Sum[Divide[(- 1)^(n),(z)^(2)+ (n)^(2)* (Pi)^(2)], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || Successful [Tested: 7] || Successful [Tested: 7]
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Latest revision as of 11:10, 28 June 2021


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
4.36.E1 sinh ⁡ z = z ⁢ ∏ n = 1 ∞ ( 1 + z 2 n 2 ⁢ π 2 ) 𝑧 𝑧 superscript subscript product 𝑛 1 1 superscript 𝑧 2 superscript 𝑛 2 superscript 𝜋 2 {\displaystyle{\displaystyle\sinh z=z\prod_{n=1}^{\infty}\left(1+\frac{z^{2}}{% n^{2}\pi^{2}}\right)}}
\sinh@@{z} = z\prod_{n=1}^{\infty}\left(1+\frac{z^{2}}{n^{2}\pi^{2}}\right)		 

sinh(z) = z*product(1 +((z)^(2))/((n)^(2)* (Pi)^(2)), n = 1..infinity)

Sinh[z] == z*Product[1 +Divide[(z)^(2),(n)^(2)* (Pi)^(2)], {n, 1, Infinity}, GenerateConditions->None]
Successful Successful - Successful [Tested: 7]
4.36.E2 cosh ⁡ z = ∏ n = 1 ∞ ( 1 + 4 ⁢ z 2 ( 2 ⁢ n - 1 ) 2 ⁢ π 2 ) 𝑧 superscript subscript product 𝑛 1 1 4 superscript 𝑧 2 superscript 2 𝑛 1 2 superscript 𝜋 2 {\displaystyle{\displaystyle\cosh z=\prod_{n=1}^{\infty}\left(1+\frac{4z^{2}}{% (2n-1)^{2}\pi^{2}}\right)}}
\cosh@@{z} = \prod_{n=1}^{\infty}\left(1+\frac{4z^{2}}{(2n-1)^{2}\pi^{2}}\right)		 

cosh(z) = product(1 +(4*(z)^(2))/((2*n - 1)^(2)* (Pi)^(2)), n = 1..infinity)

Cosh[z] == Product[1 +Divide[4*(z)^(2),(2*n - 1)^(2)* (Pi)^(2)], {n, 1, Infinity}, GenerateConditions->None]
Successful Successful - Successful [Tested: 7]
4.36.E3 coth ⁡ z = 1 z + 2 ⁢ z ⁢ ∑ n = 1 ∞ 1 z 2 + n 2 ⁢ π 2 hyperbolic-cotangent 𝑧 1 𝑧 2 𝑧 superscript subscript 𝑛 1 1 superscript 𝑧 2 superscript 𝑛 2 superscript 𝜋 2 {\displaystyle{\displaystyle\coth z=\frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{1}{% z^{2}+n^{2}\pi^{2}}}}
\coth@@{z} = \frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{1}{z^{2}+n^{2}\pi^{2}}		 

coth(z) = (1)/(z)+ 2*z*sum((1)/((z)^(2)+ (n)^(2)* (Pi)^(2)), n = 1..infinity)

Coth[z] == Divide[1,z]+ 2*z*Sum[Divide[1,(z)^(2)+ (n)^(2)* (Pi)^(2)], {n, 1, Infinity}, GenerateConditions->None]
Failure Successful Successful [Tested: 7] Successful [Tested: 7]
4.36.E4 csch 2 ⁡ z = ∑ n = - ∞ ∞ 1 ( z - n ⁢ π ⁢ i ) 2 2 𝑧 superscript subscript 𝑛 1 superscript 𝑧 𝑛 𝜋 𝑖 2 {\displaystyle{\displaystyle{\operatorname{csch}^{2}}z=\sum_{n=-\infty}^{% \infty}\frac{1}{(z-n\pi i)^{2}}}}
\csch^{2}@@{z} = \sum_{n=-\infty}^{\infty}\frac{1}{(z-n\pi i)^{2}}		 

(csch(z))^(2) = sum((1)/((z - n*Pi*I)^(2)), n = - infinity..infinity)

(Csch[z])^(2) == Sum[Divide[1,(z - n*Pi*I)^(2)], {n, - Infinity, Infinity}, GenerateConditions->None]
Failure Successful Successful [Tested: 7] Successful [Tested: 7]
4.36.E5 csch ⁡ z = 1 z + 2 ⁢ z ⁢ ∑ n = 1 ∞ ( - 1 ) n z 2 + n 2 ⁢ π 2 𝑧 1 𝑧 2 𝑧 superscript subscript 𝑛 1 superscript 1 𝑛 superscript 𝑧 2 superscript 𝑛 2 superscript 𝜋 2 {\displaystyle{\displaystyle\operatorname{csch}z=\frac{1}{z}+2z\sum_{n=1}^{% \infty}\frac{(-1)^{n}}{z^{2}+n^{2}\pi^{2}}}}
\csch@@{z} = \frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{(-1)^{n}}{z^{2}+n^{2}\pi^{2}}		 

csch(z) = (1)/(z)+ 2*z*sum(((- 1)^(n))/((z)^(2)+ (n)^(2)* (Pi)^(2)), n = 1..infinity)

Csch[z] == Divide[1,z]+ 2*z*Sum[Divide[(- 1)^(n),(z)^(2)+ (n)^(2)* (Pi)^(2)], {n, 1, Infinity}, GenerateConditions->None]
Failure Successful Successful [Tested: 7] Successful [Tested: 7]
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