Finding the derivative of the gamma function is not an easy task. However, there are reasons to do so. In particular, mathematicians define the digamma function (which I’ve mentioned here), the logarithmic derivative of the gamma function:

This is continuous except for the poles at zero and the negative integers.

We can use the Weierstrass form to find some useful expressions for the digamma function. We have:

,

where *γ* is the Euler-Mascheroni constant, and thus

.

Taking the derivative with respect to *z*, we get

Now, note immediately what happens if *z* is a positive integer *n*; the terms , *k*=1,2,3,… are canceled by the terms for *k*=*n*,*n*+1,*n*+2,…, and thus the sum collapses for positive integer n to the finite sum

, where *H*_{n} are the harmonic numbers. Thus for positive integer *n*,

and so we see immediately that *ψ*_{0}(1)=-*γ* and *ψ*_{0}(2)=1-*γ* (and thus the only positive zero of *ψ*_{0} lies between these two).

Using the series form

,

we see

,

and so we have

as a recurrence relation for *ψ*_{0}.

Now, recall the Euler reflection formula we derived previously:

Taking the logarithm of both sides, and then differentiating,

.

Applying a similar process to the Legendre duplication formula:

Plugging in *z*=1/2, we get

In fact, Gauss proved a formula (Gauss’s DIgamma Theorem) for the exact value of the digamma function for any rational value between 0 and 1: given 0<*p*<*q*,

.

Lastly, recall that . From this, and that , we expect that should approach , and thus approach , as z goes to infinity. Now, using the Stirling’s approximation for the gamma function,

we see

,

which actually lies sandwiched between and .

### Like this:

Like Loading...

*Related*

Tags: Digamma Function, Euler Reflection Formula, Euler-Mascheroni, Gamma Function, Gauss's DIgamma Theorem, Legendre Duplication, Logarithmic Derivative, Math, Monday Math

This entry was posted on October 13, 2008 at 12:11 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
You can leave a response, or trackback from your own site.

October 20, 2008 at 12:15 am |

[…] ? How about the general , for non-negative integer n? First, recall that . Thus, Recall from the previous Monday Math post the digamma function: , and thus . In particular, , with γ the Euler-Mascheroni […]

October 27, 2008 at 12:19 am |

[…] Math 43 Now, we’ve introduced the digamma function, which allows us to find the derivative of the gamma function. But what about higher order […]

November 15, 2008 at 7:44 am |

I recomend my article “Construction of the digamma function by derivative definition” in http://arxiv.org/abs/0804.1081

My regards,