Definition

The digamma function, denoted as , is the logarithm derivative of the Gamma Function. It is also the polygamma function of first order:

Digamma function is also Holomorphic on with simple poles where is non-positive integer.

Properties

Using we can derive a fundamental property for digamma function:

Finally, differentiate both sides we get:

By using the recurrence relation for we can get a relation with Harmonic Series:

This equivalents to:

Applying Stirling’s formula to log-gamma function :

From this approximation, the asymptotic behavior of follows:

Then taking the limit as approaches to infinity, we get:

From and , we have the popular Euler–Mascheroni Constant:

The same way we did for from :

Substracting from :

By doing the same with for :

Taking the limit both sides of eq to obtain:

Thanks to Trigamma Function a.k.a can be defined as:

Putting to :