Trigamma function: Difference between revisions
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===Computation and approximation=== |
===Computation and approximation=== |
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An easy method to approximate the trigamma function is to take the derivative of the [[Digamma function#Asymptotic expansion|asymptotic expansion of the digamma function]]. |
An easy method to approximate the trigamma function is to take the derivative of the [[Digamma function#Asymptotic expansion|asymptotic expansion of the digamma function]].<blockquote><math>\begin{align} |
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\psi_1(x) |
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&\sim {\operatorname{d}\over\operatorname{d}\!x} (\ln x - \sum_{n=1}^\infty \frac{B_n}{nx^n}) \\ |
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&= \frac{1}{x} + \sum_{n=1}^\infty \frac{B_n}{x^{n+1}} \\ |
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\end{align}</math></blockquote>where ''{{mvar|''B''<sub>''k''</sub>}}'' is the ''{{mvar|''k''}}''th [[Bernoulli number]]. |
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==Appearance== |
==Appearance== |
Revision as of 15:14, 7 June 2024
![](http://upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Psi1.png/300px-Psi1.png)
In mathematics, the trigamma function, denoted ψ1(z) or ψ(1)(z), is the second of the polygamma functions, and is defined by
- .
It follows from this definition that
where ψ(z) is the digamma function. It may also be defined as the sum of the series
making it a special case of the Hurwitz zeta function
Note that the last two formulas are valid when 1 − z is not a natural number.
Calculation
A double integral representation, as an alternative to the ones given above, may be derived from the series representation:
using the formula for the sum of a geometric series. Integration over y yields:
An asymptotic expansion as a Laurent series is
if we have chosen B1 = 1/2, i.e. the Bernoulli numbers of the second kind.
Recurrence and reflection formulae
The trigamma function satisfies the recurrence relation
and the reflection formula
which immediately gives the value for z = 1/2: .
Special values
At positive half integer values we have that
Moreover, the trigamma function has the following special values:
where G represents Catalan's constant and n is a positive integer.
There are no roots on the real axis of ψ1, but there exist infinitely many pairs of roots zn, zn for Re z < 0. Each such pair of roots approaches Re zn = −n + 1/2 quickly and their imaginary part increases slowly logarithmic with n. For example, z1 = −0.4121345... + 0.5978119...i and z2 = −1.4455692... + 0.6992608...i are the first two roots with Im(z) > 0.
Relation to the Clausen function
The digamma function at rational arguments can be expressed in terms of trigonometric functions and logarithm by the digamma theorem. A similar result holds for the trigamma function but the circular functions are replaced by Clausen's function. Namely,[1]
Computation and approximation
An easy method to approximate the trigamma function is to take the derivative of the asymptotic expansion of the digamma function.
where Bk is the kth Bernoulli number.
Appearance
The trigamma function appears in this sum formula:[2]
See also
Notes
- ^ Lewin, L., ed. (1991). Structural properties of polylogarithms. American Mathematical Society. ISBN 978-0821816349.
- ^ Mező, István (2013). "Some infinite sums arising from the Weierstrass Product Theorem". Applied Mathematics and Computation. 219 (18): 9838–9846. doi:10.1016/j.amc.2013.03.122.
References
- Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. ISBN 0-486-61272-4. See section §6.4
- Eric W. Weisstein. Trigamma Function -- from MathWorld--A Wolfram Web Resource