Trigamma function

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. 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. The trigamma function satisfies the recurrence relation and the reflection formula which immediately gives the value for z = 1/2: . At positive half integer values we have that Moreover, the trigamma function has the following special values: where G represents Catalan's constant. There are no roots on the real axis of ψ1, but there exist infinitely many pairs of roots 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. 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, An easy method to approximate the trigamma function is to take the derivative of the asymptotic expansion of the digamma function.