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We can define a set of numbers, the so called Bernoulli Numbers, via the equation,
![$\displaystyle \frac{x}{e^x - 1} = \sum_{n=0}^{\infty}\frac{B_n}{n!}x^2$](img8.png) |
(2.1) |
Complex analysis allows us to extract the
![$\displaystyle B_n = \frac{n!}{2\pi i}\oint_C \frac{1}{e^z - 1} \frac{dz}{z^n}$](img10.png) |
(2.2) |
We can construct a contour which avoids the central pole (wrapping over the positive
real axis, around a clockwise infinitesimal circle centered at the origin, and back along
the positive real axis to rejoin the the main contour of clockwise orientation. We then
have,
![$\displaystyle B_n = \frac{n!}{2\pi i}(-2\pi i)\sum_{p} Res[\pm p2\pi i]$](img11.png) |
(2.3) |
where
![$\displaystyle Res(f;z_0)=\lim_{z\rightarrow z_0}\frac{1}{(m-1)!}\frac{d^{m-1}}{dz^{m-1}}[(z-z_0)^mf(z)]$](img12.png) |
(2.4) |
With an application of l'Hopital's rule,
![$\displaystyle \lim_{z\rightarrow p2\pi i}\frac{z-p2\pi i}{e^z - 1} = \lim_{z\rightarrow p2\pi i}\frac{1}{e^z}=1$](img13.png) |
(2.5) |
Thus we note that the odd residues beyond one will cancel each other out
![$\displaystyle B_n = \frac{n!}{2\pi i } (-2\pi i) 2 \sum_{p=1}^{\infty}\frac{1}{...
... = -\frac{(-1)^{n/2}2n!}{4\pi^2}\left(\sum^{\infty}_{p=1}p^{-n}=\zeta(n)\right)$](img14.png) |
(2.6) |
For example,
, and so on.
Bernoulli functions are derived in the same way, defined by
![$\displaystyle \frac{x e^{xs}}{e^x - 1} = \sum^{\infty}_{n=0}B_n(s)\frac{x^n}{n!}$](img16.png) |
(2.7) |
This is easily solved by expanding the extra
factor and relating the results to the previous
results for the Bernoulli numbers, i.e.,
![$\displaystyle B_n(s)=\frac{n!}{2\pi i}\oint \frac{e^z}{e^z -1}\frac{dz}{z^n} = ...
...eft(\frac{1}{z^n} + \frac{s}{z^{n-1}} + \frac{s^2}{2!z^{n-2}} + \cdots\right)dz$](img18.png) |
(2.8) |
For example,
![$\displaystyle B_3(s) = B_3 + B_2 \frac{3!}{2!} s + B_1 \frac{3!}{1!2!}s^2 + s^3 \frac{3!}{0!3!}B_0 = s^3 -\frac{3}{2}s^2 + \frac{1}{2}s$](img19.png) |
(2.9) |
Next: Euler-Maclaurin Integration Formula
Up: The legal way I:
Previous: The legal way I:
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2006-09-15