Trigamma
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 . It comes up often in application. I just did a quick search and found I've written six posts containing the word digamma."
The derivative of the digamma function ' is the trigamma function.
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].
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 + for > 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[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 + for > 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.
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