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Bernoulli Numbers

We can define a set of numbers, the so called Bernoulli Numbers, via the equation,

 (2.1)

Complex analysis allows us to extract the

 (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,

 (2.3)

where

 (2.4)

With an application of l'Hopital's rule,

 (2.5)

Thus we note that the odd residues beyond one will cancel each other out

 (2.6)

For example, , and so on. Bernoulli functions are derived in the same way, defined by

 (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.,

 (2.8)

For example,

 (2.9)

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root 2006-09-15