Now that we know how to find the derivatives of the gamma function, we can also find the partial derivatives of the beta function. We know that

.

Thus

.

We see from the symmetry between *x* and *y* that

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Finding the second partial derivatives, we see first that

.

Similarly,

.

So we just need the mixed partial derivative

These have a number of applications to definite integrals involving logarithms.

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Tags: Beta Function, Digamma Function, Gamma Function, Math, Monday Math, Partial Derivatives, Polygamma Function

This entry was posted on November 3, 2008 at 3:53 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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November 10, 2008 at 12:07 am |

[…] Math 45 As we noted previously, the derivatives of the beta function can be used in solving various integration problems. First, […]