20.3. Digamma Function

The digamma function is defined as the logarithmic derivative of the gamma function. It is denoted by the greek letter "ψ" (psi).
If n is a natural number, the digamma function is defined as: ψ(n) = [ ln (Γ (n) ) ]' = Γ'(z) / Γ(z).

The simplest form of the digamma function is ψ(n) = -(γ + 1/n) + Σ[x=1, ∞] [n/(x² + nx)], where γ is the Euler-Mascheroni constant with a value of 0.57721566490153286060651209. When n is a positive integer, ψ(n) can be evaluated as:

$\Psi \left ( n \right )=-\gamma +\sum_{x=1}^{n-1} \left (\frac{1}{x} \right )$

Illustrative Example

Find the digamma function of 5.

$\Psi \left ( 5 \right )$

Solution

$\Psi \left ( 5 \right )=-\gamma +\sum_{x=1}^{4}\left ( \frac{1}{x} \right )$

$\Psi \left ( 5 \right )=-0.5772156649015... +\sum_{x=1}^{4}\left ( \frac{1}{x} \right )$

$\Psi \left ( 5 \right )=-0.5772156649015... +( 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4} )$

$\Psi \left ( 5 \right )=1.5061176684318\overline{33}$

Since it's easy to make a mistake calculating the digamma function by hand, you can use the app to check your work.

To find the digamma function of a natural number, do the following:

1) Switch to the Greek keyboard by tapping the leftmost keyboard button.