Consider the function , defined by the integral: . Can we find a formula for in terms of known functions?

First, we note that for , we have the series . Thus . Making the change of variable , we find that for , . Now, setting , we see , for .

Thus

.

Now, for the integral in each term, , we make the substitution . Then and , so our integral becomes:

,

and you should recognize that last integral as the gamma function of *x*.

Therefore, we find:

,

and that last sum is also familiar: it is the Riemann zeta function.

Thus, we see that

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Tags: Gamma Function, Math, Monday Math, Riemann Zeta Function

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