In mathematics, the trigamma function, denoted ψ1(z) or ψ(1)(z), is the second of the polygamma functions, and is defined by

Color representation of the trigamma function, ψ1(z), in a rectangular region of the complex plane. It is generated using the domain coloring method.
.

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

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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 can be obtained via the derivative of the asymptotic expansion of the digamma function:

 

where Bn is the nth Bernoulli number and we choose B1 = 1/2.

Recurrence and reflection formulae

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The trigamma function satisfies the recurrence relation

 

and the reflection formula

 

which immediately gives the value for z = 1/2:  .

Special values

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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

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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]

 

Appearance

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The trigamma function appears in this sum formula:[2]

 

See also

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Notes

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  1. ^ Lewin, L., ed. (1991). Structural properties of polylogarithms. American Mathematical Society. ISBN 978-0821816349.
  2. ^ 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

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