Definition of a Bernoulli Number(& Contour Integral)
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So I'm a little confused on where the Bernoulli numbers come from. Most of the definitions involve contour integrals, which I can't define and understand as well as I thought, though I do know regular integration. So, where do the Bernoulli numbers come from? Also, what is a contour integral really?
integration derivatives
asked Dec 3 '18 at 19:58
Xavier StantonXavier Stanton
311211
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add a comment |
$begingroup$
So I'm a little confused on where the Bernoulli numbers come from. Most of the definitions involve contour integrals, which I can't define and understand as well as I thought, though I do know regular integration. So, where do the Bernoulli numbers come from? Also, what is a contour integral really?
integration derivatives
asked Dec 3 '18 at 19:58
Xavier StantonXavier Stanton
311211
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According to Wikipedia Bernoulli number from Jakob Bernoulli. The article does not mention any contour integrals. Do you have a reference?
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– Somos
Dec 3 '18 at 20:01
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In Wolfram Math Alpha, they gave a definition of the Bernoulli numbers in terms of contour integrals
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– Xavier Stanton
Dec 3 '18 at 21:05
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What do you mean by "come from"?
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– Qiaochu Yuan
Dec 3 '18 at 23:16
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As in, how are the Bernoulli numbers calculated?
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– Xavier Stanton
Dec 4 '18 at 0:28
add a comment |
$begingroup$
So I'm a little confused on where the Bernoulli numbers come from. Most of the definitions involve contour integrals, which I can't define and understand as well as I thought, though I do know regular integration. So, where do the Bernoulli numbers come from? Also, what is a contour integral really?
integration derivatives
asked Dec 3 '18 at 19:58
Xavier StantonXavier Stanton
311211
$endgroup$
So I'm a little confused on where the Bernoulli numbers come from. Most of the definitions involve contour integrals, which I can't define and understand as well as I thought, though I do know regular integration. So, where do the Bernoulli numbers come from? Also, what is a contour integral really?
integration derivatives
integration derivatives
asked Dec 3 '18 at 19:58
Xavier StantonXavier Stanton
311211
asked Dec 3 '18 at 19:58
Xavier StantonXavier Stanton
311211
asked Dec 3 '18 at 19:58
Xavier StantonXavier Stanton
311211
asked Dec 3 '18 at 19:58
Xavier StantonXavier Stanton
311211
asked Dec 3 '18 at 19:58
Xavier StantonXavier Stanton
311211
311211
$begingroup$
According to Wikipedia Bernoulli number from Jakob Bernoulli. The article does not mention any contour integrals. Do you have a reference?
$endgroup$
– Somos
Dec 3 '18 at 20:01
$begingroup$
In Wolfram Math Alpha, they gave a definition of the Bernoulli numbers in terms of contour integrals
$endgroup$
– Xavier Stanton
Dec 3 '18 at 21:05
$begingroup$
What do you mean by "come from"?
$endgroup$
– Qiaochu Yuan
Dec 3 '18 at 23:16
$begingroup$
As in, how are the Bernoulli numbers calculated?
$endgroup$
– Xavier Stanton
Dec 4 '18 at 0:28
add a comment |
$begingroup$
According to Wikipedia Bernoulli number from Jakob Bernoulli. The article does not mention any contour integrals. Do you have a reference?
$endgroup$
– Somos
Dec 3 '18 at 20:01
$begingroup$
In Wolfram Math Alpha, they gave a definition of the Bernoulli numbers in terms of contour integrals
$endgroup$
– Xavier Stanton
Dec 3 '18 at 21:05
$begingroup$
What do you mean by "come from"?
$endgroup$
– Qiaochu Yuan
Dec 3 '18 at 23:16
$begingroup$
As in, how are the Bernoulli numbers calculated?
$endgroup$
– Xavier Stanton
Dec 4 '18 at 0:28
$begingroup$
According to Wikipedia Bernoulli number from Jakob Bernoulli. The article does not mention any contour integrals. Do you have a reference?
$endgroup$
– Somos
Dec 3 '18 at 20:01
$begingroup$
According to Wikipedia Bernoulli number from Jakob Bernoulli. The article does not mention any contour integrals. Do you have a reference?
$endgroup$
– Somos
Dec 3 '18 at 20:01
$begingroup$
In Wolfram Math Alpha, they gave a definition of the Bernoulli numbers in terms of contour integrals
$endgroup$
– Xavier Stanton
Dec 3 '18 at 21:05
$begingroup$
In Wolfram Math Alpha, they gave a definition of the Bernoulli numbers in terms of contour integrals
$endgroup$
– Xavier Stanton
Dec 3 '18 at 21:05
$begingroup$
What do you mean by "come from"?
$endgroup$
– Qiaochu Yuan
Dec 3 '18 at 23:16
$begingroup$
What do you mean by "come from"?
$endgroup$
– Qiaochu Yuan
Dec 3 '18 at 23:16
$begingroup$
As in, how are the Bernoulli numbers calculated?
$endgroup$
– Xavier Stanton
Dec 4 '18 at 0:28
$begingroup$
As in, how are the Bernoulli numbers calculated?
$endgroup$
– Xavier Stanton
Dec 4 '18 at 0:28
add a comment |
1 Answer
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You probably mean the same as equation (2) of the MathWorld Bernoulli Number article, essentially equivalent to the exponential generating function in equation (1). Read the Wikipedia article Laurent series for the equivalence. Of course, there are easier ways of calculating Bernoulli numbers going back before Jakob Bernoulli. See the Wikipedia article Bernoulli number for more details and references.
answered Dec 3 '18 at 22:01
SomosSomos
13.5k11135
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
You probably mean the same as equation (2) of the MathWorld Bernoulli Number article, essentially equivalent to the exponential generating function in equation (1). Read the Wikipedia article Laurent series for the equivalence. Of course, there are easier ways of calculating Bernoulli numbers going back before Jakob Bernoulli. See the Wikipedia article Bernoulli number for more details and references.
answered Dec 3 '18 at 22:01
SomosSomos
13.5k11135
$endgroup$
add a comment |
$begingroup$
You probably mean the same as equation (2) of the MathWorld Bernoulli Number article, essentially equivalent to the exponential generating function in equation (1). Read the Wikipedia article Laurent series for the equivalence. Of course, there are easier ways of calculating Bernoulli numbers going back before Jakob Bernoulli. See the Wikipedia article Bernoulli number for more details and references.
answered Dec 3 '18 at 22:01
SomosSomos
13.5k11135
$endgroup$
add a comment |
$begingroup$
You probably mean the same as equation (2) of the MathWorld Bernoulli Number article, essentially equivalent to the exponential generating function in equation (1). Read the Wikipedia article Laurent series for the equivalence. Of course, there are easier ways of calculating Bernoulli numbers going back before Jakob Bernoulli. See the Wikipedia article Bernoulli number for more details and references.
answered Dec 3 '18 at 22:01
SomosSomos
13.5k11135
$endgroup$
You probably mean the same as equation (2) of the MathWorld Bernoulli Number article, essentially equivalent to the exponential generating function in equation (1). Read the Wikipedia article Laurent series for the equivalence. Of course, there are easier ways of calculating Bernoulli numbers going back before Jakob Bernoulli. See the Wikipedia article Bernoulli number for more details and references.
answered Dec 3 '18 at 22:01
SomosSomos
13.5k11135
edited Dec 4 '18 at 0:54
answered Dec 3 '18 at 22:01
SomosSomos
13.5k11135
answered Dec 3 '18 at 22:01
SomosSomos
13.5k11135
answered Dec 3 '18 at 22:01
SomosSomos
13.5k11135
13.5k11135
add a comment |
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$begingroup$
According to Wikipedia Bernoulli number from Jakob Bernoulli. The article does not mention any contour integrals. Do you have a reference?
$endgroup$
– Somos
Dec 3 '18 at 20:01
$begingroup$
In Wolfram Math Alpha, they gave a definition of the Bernoulli numbers in terms of contour integrals
$endgroup$
– Xavier Stanton
Dec 3 '18 at 21:05
$begingroup$
What do you mean by "come from"?
$endgroup$
– Qiaochu Yuan
Dec 3 '18 at 23:16
$begingroup$
As in, how are the Bernoulli numbers calculated?
$endgroup$
– Xavier Stanton
Dec 4 '18 at 0:28