Trigamma 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.

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.

The trigamma function satisfies the recurrence relation and the reflection formula which immediately gives the value for z = ⁠1/2⁠:

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.

Namely,[1] The trigamma function appears in this sum formula:[2]