The Digamma Function is a mathematical function defined as the logarithmic derivative of the Gamma Function. It is represented by the greek letter digamma, which is similar to a psi: Ψ.

Viewed on a graph, the Digamma Function resembles the tangent function for z < 0 and a less steep logarithmic function for z > 0. It is undefined at negative integers and zero.

The nth derivative of Ψ(z) is called the polygamma function, denoted ψn(z). The digamma function itself is sometimes written ψ0, or just ψ.

The digamma function is sometimes used as the logarithmic derivative of the factorial function, written as F(z) = d/dz ln z!. Since the Gamma Function is essentially the factorial function for natural numbers expanded to real and complex numbers, the two Digamma Functions are related: F(z) = ψ0 (z+1).

Harmonic numbers can be described as the sum of the Digamma Function and the Euler-Mascheroni constant.


Notation note: I couldn't find a character for digamma, so I used a psi. The digamma has bars on the top and bottom of the vertical line. If anyone knows how to put a digamma in, let me know, and I'll fix it. Also, the upper and lower case psis look very similar.

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