4.19: 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/4.19.E7 4.19.E7] || [[Item:Q1686|<math>\ln@{\frac{\sin@@{z}}{z}} = \sum_{n=1}^{\infty}\frac{(-1)^{n}2^{2n-1}\BernoullinumberB{2n}}{n(2n)!}z^{2n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ln@{\frac{\sin@@{z}}{z}} = \sum_{n=1}^{\infty}\frac{(-1)^{n}2^{2n-1}\BernoullinumberB{2n}}{n(2n)!}z^{2n}</syntaxhighlight> || <math>|z| < \pi</math> || <syntaxhighlight lang=mathematica>ln((sin(z))/(z)) = sum(((- 1)^(n)* (2)^(2*n - 1)* bernoulli(2*n))/(n*factorial(2*n))*(z)^(2*n), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Log[Divide[Sin[z],z]] == Sum[Divide[(- 1)^(n)* (2)^(2*n - 1)* BernoulliB[2*n],n*(2*n)!]*(z)^(2*n), {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 7] || Successful [Tested: 7]
| [https://dlmf.nist.gov/4.19.E7 4.19.E7] || <math qid="Q1686">\ln@{\frac{\sin@@{z}}{z}} = \sum_{n=1}^{\infty}\frac{(-1)^{n}2^{2n-1}\BernoullinumberB{2n}}{n(2n)!}z^{2n}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ln@{\frac{\sin@@{z}}{z}} = \sum_{n=1}^{\infty}\frac{(-1)^{n}2^{2n-1}\BernoullinumberB{2n}}{n(2n)!}z^{2n}</syntaxhighlight> || <math>|z| < \pi</math> || <syntaxhighlight lang=mathematica>ln((sin(z))/(z)) = sum(((- 1)^(n)* (2)^(2*n - 1)* bernoulli(2*n))/(n*factorial(2*n))*(z)^(2*n), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Log[Divide[Sin[z],z]] == Sum[Divide[(- 1)^(n)* (2)^(2*n - 1)* BernoulliB[2*n],n*(2*n)!]*(z)^(2*n), {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 7] || Successful [Tested: 7]
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| [https://dlmf.nist.gov/4.19.E8 4.19.E8] || [[Item:Q1687|<math>\ln@{\cos@@{z}} = \sum_{n=1}^{\infty}\frac{(-1)^{n}2^{2n-1}(2^{2n}-1)\BernoullinumberB{2n}}{n(2n)!}z^{2n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ln@{\cos@@{z}} = \sum_{n=1}^{\infty}\frac{(-1)^{n}2^{2n-1}(2^{2n}-1)\BernoullinumberB{2n}}{n(2n)!}z^{2n}</syntaxhighlight> || <math>|z| < \frac{1}{2}\pi</math> || <syntaxhighlight lang=mathematica>ln(cos(z)) = sum(((- 1)^(n)* (2)^(2*n - 1)*((2)^(2*n)- 1)*bernoulli(2*n))/(n*factorial(2*n))*(z)^(2*n), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Log[Cos[z]] == Sum[Divide[(- 1)^(n)* (2)^(2*n - 1)*((2)^(2*n)- 1)*BernoulliB[2*n],n*(2*n)!]*(z)^(2*n), {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Manual Skip! || Successful [Tested: 6]
| [https://dlmf.nist.gov/4.19.E8 4.19.E8] || <math qid="Q1687">\ln@{\cos@@{z}} = \sum_{n=1}^{\infty}\frac{(-1)^{n}2^{2n-1}(2^{2n}-1)\BernoullinumberB{2n}}{n(2n)!}z^{2n}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ln@{\cos@@{z}} = \sum_{n=1}^{\infty}\frac{(-1)^{n}2^{2n-1}(2^{2n}-1)\BernoullinumberB{2n}}{n(2n)!}z^{2n}</syntaxhighlight> || <math>|z| < \frac{1}{2}\pi</math> || <syntaxhighlight lang=mathematica>ln(cos(z)) = sum(((- 1)^(n)* (2)^(2*n - 1)*((2)^(2*n)- 1)*bernoulli(2*n))/(n*factorial(2*n))*(z)^(2*n), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Log[Cos[z]] == Sum[Divide[(- 1)^(n)* (2)^(2*n - 1)*((2)^(2*n)- 1)*BernoulliB[2*n],n*(2*n)!]*(z)^(2*n), {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Manual Skip! || Successful [Tested: 6]
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| [https://dlmf.nist.gov/4.19.E9 4.19.E9] || [[Item:Q1688|<math>\ln@{\frac{\tan@@{z}}{z}} = \sum_{n=1}^{\infty}\frac{(-1)^{n-1}2^{2n}(2^{2n-1}-1)\BernoullinumberB{2n}}{n(2n)!}z^{2n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ln@{\frac{\tan@@{z}}{z}} = \sum_{n=1}^{\infty}\frac{(-1)^{n-1}2^{2n}(2^{2n-1}-1)\BernoullinumberB{2n}}{n(2n)!}z^{2n}</syntaxhighlight> || <math>|z| < \frac{1}{2}\pi</math> || <syntaxhighlight lang=mathematica>ln((tan(z))/(z)) = sum(((- 1)^(n - 1)* (2)^(2*n)*((2)^(2*n - 1)- 1)*bernoulli(2*n))/(n*factorial(2*n))*(z)^(2*n), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Log[Divide[Tan[z],z]] == Sum[Divide[(- 1)^(n - 1)* (2)^(2*n)*((2)^(2*n - 1)- 1)*BernoulliB[2*n],n*(2*n)!]*(z)^(2*n), {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Manual Skip! || Successful [Tested: 6]
| [https://dlmf.nist.gov/4.19.E9 4.19.E9] || <math qid="Q1688">\ln@{\frac{\tan@@{z}}{z}} = \sum_{n=1}^{\infty}\frac{(-1)^{n-1}2^{2n}(2^{2n-1}-1)\BernoullinumberB{2n}}{n(2n)!}z^{2n}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ln@{\frac{\tan@@{z}}{z}} = \sum_{n=1}^{\infty}\frac{(-1)^{n-1}2^{2n}(2^{2n-1}-1)\BernoullinumberB{2n}}{n(2n)!}z^{2n}</syntaxhighlight> || <math>|z| < \frac{1}{2}\pi</math> || <syntaxhighlight lang=mathematica>ln((tan(z))/(z)) = sum(((- 1)^(n - 1)* (2)^(2*n)*((2)^(2*n - 1)- 1)*bernoulli(2*n))/(n*factorial(2*n))*(z)^(2*n), n = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Log[Divide[Tan[z],z]] == Sum[Divide[(- 1)^(n - 1)* (2)^(2*n)*((2)^(2*n - 1)- 1)*BernoulliB[2*n],n*(2*n)!]*(z)^(2*n), {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Manual Skip! || Successful [Tested: 6]
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Latest revision as of 11:06, 28 June 2021


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
4.19.E7 ln ⁡ ( sin ⁡ z z ) = ∑ n = 1 ∞ ( - 1 ) n ⁢ 2 2 ⁢ n - 1 ⁢ B 2 ⁢ n n ⁢ ( 2 ⁢ n ) ! ⁢ z 2 ⁢ n 𝑧 𝑧 superscript subscript 𝑛 1 superscript 1 𝑛 superscript 2 2 𝑛 1 Bernoulli-number-B 2 𝑛 𝑛 2 𝑛 superscript 𝑧 2 𝑛 {\displaystyle{\displaystyle\ln\left(\frac{\sin z}{z}\right)=\sum_{n=1}^{% \infty}\frac{(-1)^{n}2^{2n-1}B_{2n}}{n(2n)!}z^{2n}}}
\ln@{\frac{\sin@@{z}}{z}} = \sum_{n=1}^{\infty}\frac{(-1)^{n}2^{2n-1}\BernoullinumberB{2n}}{n(2n)!}z^{2n}		 | z | < π 𝑧 𝜋 {\displaystyle{\displaystyle|z|<\pi}} 
ln((sin(z))/(z)) = sum(((- 1)^(n)* (2)^(2*n - 1)* bernoulli(2*n))/(n*factorial(2*n))*(z)^(2*n), n = 1..infinity)

Log[Divide[Sin[z],z]] == Sum[Divide[(- 1)^(n)* (2)^(2*n - 1)* BernoulliB[2*n],n*(2*n)!]*(z)^(2*n), {n, 1, Infinity}, GenerateConditions->None]
Failure Failure Successful [Tested: 7] Successful [Tested: 7]
4.19.E8 ln ⁡ ( cos ⁡ z ) = ∑ n = 1 ∞ ( - 1 ) n ⁢ 2 2 ⁢ n - 1 ⁢ ( 2 2 ⁢ n - 1 ) ⁢ B 2 ⁢ n n ⁢ ( 2 ⁢ n ) ! ⁢ z 2 ⁢ n 𝑧 superscript subscript 𝑛 1 superscript 1 𝑛 superscript 2 2 𝑛 1 superscript 2 2 𝑛 1 Bernoulli-number-B 2 𝑛 𝑛 2 𝑛 superscript 𝑧 2 𝑛 {\displaystyle{\displaystyle\ln\left(\cos z\right)=\sum_{n=1}^{\infty}\frac{(-% 1)^{n}2^{2n-1}(2^{2n}-1)B_{2n}}{n(2n)!}z^{2n}}}
\ln@{\cos@@{z}} = \sum_{n=1}^{\infty}\frac{(-1)^{n}2^{2n-1}(2^{2n}-1)\BernoullinumberB{2n}}{n(2n)!}z^{2n}		 | z | < 1 2 ⁢ π 𝑧 1 2 𝜋 {\displaystyle{\displaystyle|z|<\frac{1}{2}\pi}} 
ln(cos(z)) = sum(((- 1)^(n)* (2)^(2*n - 1)*((2)^(2*n)- 1)*bernoulli(2*n))/(n*factorial(2*n))*(z)^(2*n), n = 1..infinity)

Log[Cos[z]] == Sum[Divide[(- 1)^(n)* (2)^(2*n - 1)*((2)^(2*n)- 1)*BernoulliB[2*n],n*(2*n)!]*(z)^(2*n), {n, 1, Infinity}, GenerateConditions->None]
Failure Failure Manual Skip! Successful [Tested: 6]
4.19.E9 ln ⁡ ( tan ⁡ z z ) = ∑ n = 1 ∞ ( - 1 ) n - 1 ⁢ 2 2 ⁢ n ⁢ ( 2 2 ⁢ n - 1 - 1 ) ⁢ B 2 ⁢ n n ⁢ ( 2 ⁢ n ) ! ⁢ z 2 ⁢ n 𝑧 𝑧 superscript subscript 𝑛 1 superscript 1 𝑛 1 superscript 2 2 𝑛 superscript 2 2 𝑛 1 1 Bernoulli-number-B 2 𝑛 𝑛 2 𝑛 superscript 𝑧 2 𝑛 {\displaystyle{\displaystyle\ln\left(\frac{\tan z}{z}\right)=\sum_{n=1}^{% \infty}\frac{(-1)^{n-1}2^{2n}(2^{2n-1}-1)B_{2n}}{n(2n)!}z^{2n}}}
\ln@{\frac{\tan@@{z}}{z}} = \sum_{n=1}^{\infty}\frac{(-1)^{n-1}2^{2n}(2^{2n-1}-1)\BernoullinumberB{2n}}{n(2n)!}z^{2n}		 | z | < 1 2 ⁢ π 𝑧 1 2 𝜋 {\displaystyle{\displaystyle|z|<\frac{1}{2}\pi}} 
ln((tan(z))/(z)) = sum(((- 1)^(n - 1)* (2)^(2*n)*((2)^(2*n - 1)- 1)*bernoulli(2*n))/(n*factorial(2*n))*(z)^(2*n), n = 1..infinity)

Log[Divide[Tan[z],z]] == Sum[Divide[(- 1)^(n - 1)* (2)^(2*n)*((2)^(2*n - 1)- 1)*BernoulliB[2*n],n*(2*n)!]*(z)^(2*n), {n, 1, Infinity}, GenerateConditions->None]
Failure Failure Manual Skip! Successful [Tested: 6]
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