Re: [math-fun] Real index Bernoulli numbers?
Are there extensions of Bernoulli numbers to real (or complex) index?
Is there one that is generally accepted as the natural way to do it?
It would be nice if there were a natural real-index Bernoulli function that took real values.
I read somewhere that Ramanujan used to refer to real order Bernoulli numbers in his notebooks. Does anyone know what definition he used?
--The generating function for Bernoulli(n)/n! is z/[exp(z)-1]. So we can use Cauchy residue theorem to write Bernoulli(n) = Gamma(n+1)/(2*pi*i) (* int z^(1-n) / [exp(z)-1] dz where the contour of integration encircles the origin going counterclockwise and is contained within the strip |Im(z)|<2*pi. Actually, it need not be contained in this strip, if it is shaped right. All we really need is that |Im()|<2*pi when Re(z)=0 and that if the contour goes to infinity it must do so in good directions. I would suggest a parabola-like path which starts at someplace around +i+RealInfinity, stays above the positive real axis, crosses the negative real axis, then staying below the positive real axis goes to someplace near -i+RealInfinity. That way we do not care if the integrand is only defined on a complex z-plane SLIT along the positive real axis. Define the ln function in this slit way and the power function via A^B = exp(B*ln(A)). And then observe that this definition will produce a unique Bernoulli(n) value when n is any complex number, not required to be an integer. Furthermore by reflection symmetry of everything about the real axis we see it always produces real values if n is real because the imaginary part cancels out. --Warren D. Smith.
--Same post as before, but now I corrected some typos, sorry.
D Asimov: Are there extensions of Bernoulli numbers to real (or complex) index?
Is there one that is generally accepted as the natural way to do it?
It would be nice if there were a natural real-index Bernoulli function that took real values.
I read somewhere that Ramanujan used to refer to real order Bernoulli numbers in his notebooks. Does anyone know what definition he used?
--The generating function for Bernoulli(n)/n! is z/[exp(z)-1]. So we can use Cauchy residue theorem to write Bernoulli(n) = Gamma(n+1)/(2*pi*i) * int z^(1-n) / [exp(z)-1] dz where the contour of integration encircles the origin going counterclockwise and is contained within the strip |Im(z)|<2*pi. Actually, it need not be contained in this strip, if it is shaped right. All we really need is that |Im(z)|<2*pi when Re(z)=0 and that if the contour goes to infinity it must do so in good directions. I would suggest a parabola-like contour which starts at someplace around +i+RealInfinity, stays above the positive real axis, crosses the negative real axis, then staying below the positive real axis goes to someplace near -i+RealInfinity. That way we do not care if the integrand is only defined on a complex z-plane SLIT along the positive real axis. Define the ln function in this slit way and the power function via A^B = exp(B*ln(A)). One concrete choice for the coutour could be the parabola x = y^2 - 1. And then observe that this definition will produce a unique Bernoulli(n) value when n is any complex number, not required to be an integer. Furthermore by reflection symmetry of everything about the real axis we see it always produces real values if n is real because the imaginary part cancels out. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
Turns out I still had typos. Trying again. Goal is to devise a function Bernoulli(n) defined for complex n (not nec'ly integers) and such that reals map to reals, and agrees with Bernulli numbers when n=0,1,2,3,... http://en.wikipedia.org/wiki/Bernoulli_number My original suggestion with typos hopefully now finally corrected, using a ln(z) definition slit along the positive real axis (which is not the standard definition), and A^B = exp(B*ln(A)), was Bernoulli(n) = Gamma(n+1)/(2*pi*i) * integral z^(-n) / [exp(z)-1] dz using an anticlockwise contour of integration like x = y^2 - 1. However, saying that about the nonstandard ln was somewhat cavalier of me. If we use the standard ln(z) definition with slit along the negative real axis, then Bernoulli(n) = Gamma(n+1)/(4*pi*i) * integral [ exp(n*[pi*i-ln(-z)]) + exp(n*[-pi*i-ln(-z)]) ] / [exp(z)-1] dz would work where the average of the two exp's (and the reflection symmetry of the contour) serves to cancel out imaginary parts so that Bernoulli(n) is real if n is real. However, Asimov may wish to reconsider his desire that it map real-->real. To explain, consider the functions sin(x) and cos(x), which both map real-->real. Do you not think exp(i*x) is somehow more fundamental? If so you might prefer something like Bernoulli(n) = Gamma(n+1)/(2*pi*i) * integral exp(n*[pi*i-ln(-z)]) / [exp(z)-1] dz which does not map reals to reals, but its real part is my previous definition...
I wonder if this is related to the "standard" extension of Bernoulli number to have p-adic index. First, if zeta denotes the Riemann zeta function and k is an integer >=2 we have zeta(1-k) = -B_k/k In the p-adic integers, the ordinary positive integers are dense. The Kummer congruence shows that -B_k/k is p-adically continuous,so there is a unique p-adic continuous function which interpolates these values. It turns out to be p-adic analytic. Here's a nice survey about it: http://www.rac.es/ficheros/doc/00261.pdf Victor On Wed, Jan 7, 2015 at 3:42 PM, Warren D Smith <warren.wds@gmail.com> wrote:
Turns out I still had typos. Trying again. Goal is to devise a function Bernoulli(n) defined for complex n (not nec'ly integers) and such that reals map to reals, and agrees with Bernulli numbers when n=0,1,2,3,... http://en.wikipedia.org/wiki/Bernoulli_number
My original suggestion with typos hopefully now finally corrected, using a ln(z) definition slit along the positive real axis (which is not the standard definition), and A^B = exp(B*ln(A)), was
Bernoulli(n) = Gamma(n+1)/(2*pi*i) * integral z^(-n) / [exp(z)-1] dz
using an anticlockwise contour of integration like x = y^2 - 1. However, saying that about the nonstandard ln was somewhat cavalier of me. If we use the standard ln(z) definition with slit along the negative real axis, then
Bernoulli(n) = Gamma(n+1)/(4*pi*i) * integral [ exp(n*[pi*i-ln(-z)]) + exp(n*[-pi*i-ln(-z)]) ] / [exp(z)-1] dz
would work where the average of the two exp's (and the reflection symmetry of the contour) serves to cancel out imaginary parts so that Bernoulli(n) is real if n is real.
However, Asimov may wish to reconsider his desire that it map real-->real. To explain, consider the functions sin(x) and cos(x), which both map real-->real. Do you not think exp(i*x) is somehow more fundamental? If so you might prefer something like
Bernoulli(n) = Gamma(n+1)/(2*pi*i) * integral exp(n*[pi*i-ln(-z)]) / [exp(z)-1] dz
which does not map reals to reals, but its real part is my previous definition...
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Since we're on the topic of Bernoulli numbers, Barry Mazur's lectures on their universality is worth reading: http://wstein.org/wiki/attachments/2008%282f%29480a/Bernoulli.pdf On Wed, Jan 7, 2015 at 4:09 PM, Victor Miller <victorsmiller@gmail.com> wrote:
I wonder if this is related to the "standard" extension of Bernoulli number to have p-adic index. First, if zeta denotes the Riemann zeta function and k is an integer >=2 we have
zeta(1-k) = -B_k/k
In the p-adic integers, the ordinary positive integers are dense. The Kummer congruence shows that -B_k/k is p-adically continuous,so there is a unique p-adic continuous function which interpolates these values. It turns out to be p-adic analytic. Here's a nice survey about it: http://www.rac.es/ficheros/doc/00261.pdf
Victor
On Wed, Jan 7, 2015 at 3:42 PM, Warren D Smith <warren.wds@gmail.com> wrote:
Turns out I still had typos. Trying again. Goal is to devise a function Bernoulli(n) defined for complex n (not nec'ly integers) and such that reals map to reals, and agrees with Bernulli numbers when n=0,1,2,3,... http://en.wikipedia.org/wiki/Bernoulli_number
My original suggestion with typos hopefully now finally corrected, using a ln(z) definition slit along the positive real axis (which is not the standard definition), and A^B = exp(B*ln(A)), was
Bernoulli(n) = Gamma(n+1)/(2*pi*i) * integral z^(-n) / [exp(z)-1] dz
using an anticlockwise contour of integration like x = y^2 - 1. However, saying that about the nonstandard ln was somewhat cavalier of me. If we use the standard ln(z) definition with slit along the negative real axis, then
Bernoulli(n) = Gamma(n+1)/(4*pi*i) * integral [ exp(n*[pi*i-ln(-z)]) + exp(n*[-pi*i-ln(-z)]) ] / [exp(z)-1] dz
would work where the average of the two exp's (and the reflection symmetry of the contour) serves to cancel out imaginary parts so that Bernoulli(n) is real if n is real.
However, Asimov may wish to reconsider his desire that it map real-->real. To explain, consider the functions sin(x) and cos(x), which both map real-->real. Do you not think exp(i*x) is somehow more fundamental? If so you might prefer something like
Bernoulli(n) = Gamma(n+1)/(2*pi*i) * integral exp(n*[pi*i-ln(-z)]) / [exp(z)-1] dz
which does not map reals to reals, but its real part is my previous definition...
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Perhaps somebody will produce a plot of the bernoulli(n) function now? Or some simple transform of it, such as (2*pi)^n * bernoulli(n) / Gamma(n+1) which would produce nicer plots...
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