In a previous post, we found that . What, then, is ?

Previously, we found that for , . Similarly, as , we see that with , . Setting , we see , for .

Thus

.

As we did previously, we solve the integral in each term with the substitution , which gives us:

.

Plugging into our series,

.

We should recognize that last sum as the Dirichlet eta function. Thus, just as , we see .

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Tags: Dirichlet Eta Function, Gamma Function, Integration, Math, Monday Math

This entry was posted on September 1, 2008 at 2:18 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
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January 19, 2009 at 2:17 am |

[…] . Reversing the order of the double integration, and performing the inner integral, We found previously that , so then (for ) and, in analogy to I(s), we can also find that. Possibly related posts: […]