Trigamma function

Short description: Mathematical function

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

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

[math]\displaystyle{ \psi_1(z) = \frac{d^2}{dz^2} \ln\Gamma(z) }[/math].

It follows from this definition that

[math]\displaystyle{ \psi_1(z) = \frac{d}{dz} \psi(z) }[/math]

where ψ(z) is the digamma function. It may also be defined as the sum of the series

[math]\displaystyle{ \psi_1(z) = \sum_{n = 0}^{\infty}\frac{1}{(z + n)^2}, }[/math]

making it a special case of the Hurwitz zeta function

[math]\displaystyle{ \psi_1(z) = \zeta(2,z). }[/math]

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:

[math]\displaystyle{ \psi_1(z) = \int_0^1\!\!\int_0^x\frac{x^{z-1}}{y(1 - x)}\,dy\,dx }[/math]

using the formula for the sum of a geometric series. Integration over y yields:

[math]\displaystyle{ \psi_1(z) = -\int_0^1\frac{x^{z-1}\ln{x}}{1-x}\,dx }[/math]

An asymptotic expansion as a Laurent series is

[math]\displaystyle{ \psi_1(z) = \frac{1}{z} + \frac{1}{2z^2} + \sum_{k=1}^{\infty}\frac{B_{2k}}{z^{2k+1}} = \sum_{k=0}^{\infty}\frac{B_k}{z^{k+1}} }[/math]

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

[math]\displaystyle{ \psi_1(z + 1) = \psi_1(z) - \frac{1}{z^2} }[/math]

and the reflection formula

[math]\displaystyle{ \psi_1(1 - z) + \psi_1(z) = \frac{\pi^2}{\sin^2 \pi z} \, }[/math]

which immediately gives the value for z = 1/2: [math]\displaystyle{ \psi_1(\tfrac{1}{2})=\tfrac{\pi^2}{2} }[/math].

Special values

At positive half integer values we have that

[math]\displaystyle{ \psi_1\left(n+\frac12\right)=\frac{\pi^2}{2}-4\sum_{k=1}^n\frac{1}{(2k-1)^2}. }[/math]

Moreover, the trigamma function has the following special values:

[math]\displaystyle{ \begin{align} \psi_1\left(\tfrac14\right) &= \pi^2 + 8G \quad & \psi_1\left(\tfrac12\right) &= \frac{\pi^2}{2} & \psi_1(1) &= \frac{\pi^2}{6} \\[6px] \psi_1\left(\tfrac32\right) &= \frac{\pi^2}{2} - 4 & \psi_1(2) &= \frac{\pi^2}{6} - 1 \quad \end{align} }[/math]

where G represents Catalan's constant.

There are no roots on the real axis of ψ₁, but there exist infinitely many pairs of roots z_n, z_n for Re z < 0. Each such pair of roots approaches Re z_n = −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,

[math]\displaystyle{ \psi_1\left(\frac{p}{q}\right)=\frac{\pi^2}{2\sin^2(\pi p/q)}+2q\sum_{m=1}^{(q-1)/2}\sin\left(\frac{2\pi mp}{q}\right)\textrm{Cl}_2\left(\frac{2\pi m}{q}\right). }[/math]

Computation and approximation

An easy method to approximate the trigamma function is to take the derivative of the asymptotic expansion of the digamma function.

[math]\displaystyle{ \psi_1(x) \approx \frac{1}{x} + \frac{1}{2x^2} + \frac{1}{6x^3} - \frac{1}{30x^5} + \frac{1}{42x^7} - \frac{1}{30x^9} + \frac{5}{66x^{11}} - \frac{691}{2730x^{13}} + \frac{7}{6x^{15}} }[/math]

Appearance

The trigamma function appears in this sum formula:

[math]\displaystyle{ \sum_{n=1}^\infty\frac{n^2-\frac12}{\left(n^2+\frac12\right)^2}\left(\psi_1\bigg(n-\frac{i}{\sqrt{2}}\bigg)+\psi_1\bigg(n+\frac{i}{\sqrt{2}}\bigg)\right)= -1+\frac{\sqrt{2}}{4}\pi\coth\frac{\pi}{\sqrt{2}}-\frac{3\pi^2}{4\sinh^2\frac{\pi}{\sqrt{2}}}+\frac{\pi^4}{12\sinh^4\frac{\pi}{\sqrt{2}}}\left(5+\cosh\pi\sqrt{2}\right). }[/math]

See also

• Gamma function
• Digamma function
• Polygamma function
• Catalan's constant

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