Problem 1. Differentiate the gamma function
We begin with the integral definition of the Gamma function
To differentiate under the integral, we use Leibniz Rule .
First we find
Then we evaluate the limit
Problem 2. Show that where is the Euler-Mascheroni constant.
To evaluate , we set
We replace with
Let and we get
is the harmonic number. By definition is the Euler-Mascheroni constant; therefore,
Check out my other notes at Proof, Disproof, and Derivation