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  • September 1, 2025
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Guy wearing a ΓΓΓ shirt

The most important mathematical function after the basics is the gamma function. If I could add one function to a calculator that has trig functions, log, and exponential, it would be the gamma function. Or maybe the log of the gamma function; it’s often more useful than the gamma function itself because it doesn’t overflow as easily.

The derivative of the log gamma function is the digamma function, denoted ψ. It comes up often in application. I just did a quick search and found I’ve written six posts containing the word “digamma.”

psi(z) = frac{d}{dz} log Gamma(z)

The derivative of the digamma function ψ′ is the trigamma function.

psi^prime(z) = frac{d}{dz} psi(z) = frac{d^2}{dz^2} log Gamma(z)

The trigamma function, and higher derivatives of the digamma function, appear in applications. I remember, for example, a researcher asking me to add the trigamma function to the mathematical library I wrote for the biostatistics department at MD Anderson.

I was thinking about the trigamma function because I ran across a the following series for the function [1].

psi^prime (z+1) = frac{1}{(z + 1)} + frac{1!}{2} frac{1}{(z+1)^{bar{2}}} + frac{2!}{3}frac{1}{(z+1)^{bar{3}}} + cdots

Note the bars on top of the exponents: the denominators are rising powers of z + 1, not ordinary powers.

The series converges uniformly for Re(z) > −1 + δ for δ > 0 [2]. It series converges quickly for large z.

When I saw the title of the paper I thought it sounded like a Greek fraternity. There is a Tri-Delta fraternity, but as far as I know there is no Tri-Gamma fraternity.

Related posts

  • Computing log gamma differences
  • Two useful asymptotic series
  • Gamma gamma gamma!

[1] Harold Ruben. A Note on the Trigamma Function. The American Mathematical Monthly. Vol 83, No. 8. p. 622.

[2] It may seem unnecessary to say Re(z) > −1 + δ for δ > 0. Couldn’t you just say for Re(z) > −1? Pointwise, yes, but uniform convergence requires the real part of z to be bounded away from −1 by a fixed amount, regardless of the imaginary part of z.

The post Trigamma first appeared on John D. Cook.

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