Trigamma function

method. In mathematics, the trigamma function, denoted or , is the second of the polygamma functions, and is defined by : \psi_1(z) = \frac{d^2}{dz^2} \ln\Gamma(z) . It follows from this definition that : \psi_1(z) = \frac{d}{dz} \psi(z) where is the digamma function. It may also be defined as the sum of the series : \psi_1(z) = \sum_{n = 0}^{\infty}\frac{1}{(z + n)^2}, making it a special case of the Hurwitz zeta function : \psi_1(z) = \zeta(2,z). Note that the last two formulas are valid when is not a natural number.

method. In mathematics, the trigamma function, denoted or , is the second of the polygamma functions, and is defined by \psi_1(z) = \frac{d^2}{dz^2} \ln\Gamma(z) . It follows from this definition that \psi_1(z) = \frac{d}{dz} \psi(z) where is the digamma function. It may also be defined as the sum of the series \psi_1(z) = \sum_{n = 0}^{\infty}\frac{1}{(z + n)^2}, making it a special case of the Hurwitz zeta function \psi_1(z) = \zeta(2,z). Note that the last two formulas are valid when 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: \psi_1(z) = \int_0^1\!\!\int_0^x\frac{x^{z-1}}{y(1 - x)}\,dy\,dx using the formula for the sum of a geometric series. Integration over yields: \psi_1(z) = -\int_0^1\frac{x^{z-1}\ln{x}}{1-x}\,dx An asymptotic expansion as a Laurent series can be obtained via the derivative of the asymptotic expansion of the digamma function: \begin{align} \psi_1(z) &\sim {\operatorname{d}\over\operatorname{d}\!z} \left(\ln z - \sum_{n=1}^\infty \frac{B_n}{nz^n}\right) \\ &= \frac{1}{z} + \sum_{n=1}^\infty \frac{B_n}{z^{n+1}} = \sum_{n=0}^{\infty}\frac{B_n}{z^{n+1}} \\ &= \frac{1}{z} + \frac{1}{2z^2} + \frac{1}{6z^3} - \frac{1}{30z^5} + \frac{1}{42z^7} - \frac{1}{30z^9} + \frac{5}{66z^{11}} - \frac{691}{2730z^{13}} + \frac{7}{6z^{15}} \cdots \end{align} where is the th Bernoulli number and we choose . ===Recurrence and reflection formulae=== The trigamma function satisfies the recurrence relation \psi_1(z + 1) = \psi_1(z) - \frac{1}{z^2} and the reflection formula \psi_1(1 - z) + \psi_1(z) = \frac{\pi^2}{\sin^2 \pi z} \, which immediately gives the value for z : \psi_1(\tfrac{1}{2})=\tfrac{\pi^2}{2} . ===Special values=== At positive integer values we have that \psi_1(n) = \frac{\pi^2}{6} - \sum_{k=1}^{n-1} \frac{1}{k^2}, \qquad \psi_1(1) = \frac{\pi^2}{6}, \qquad \psi_1(2) = \frac{\pi^2}{6} - 1, \qquad \psi_1(3) = \frac{\pi^2}{6} - \frac{5}{4}. At positive half integer values we have that \psi_1\left(n+\frac12\right)=\frac{\pi^2}{2}-4\sum_{k=1}^n\frac{1}{(2k-1)^2}, \qquad \psi_1\left(\tfrac12\right) = \frac{\pi^2}{2}, \qquad \psi_1\left(\tfrac32\right) = \frac{\pi^2}{2} - 4 . The trigamma function has other special values such as: \psi_1\left(\tfrac14\right) = \pi^2 + 8G where represents Catalan's constant. There are no roots on the real axis of , but there exist infinitely many pairs of roots for . Each such pair of roots approaches quickly and their imaginary part increases slowly logarithmic with . For example, and are the first two roots with . ===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, \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). ==Appearance== The trigamma function appears in this sum formula: \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). ==See also== Gamma function Digamma function Polygamma function Catalan's constant ==Notes== ==References== Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. . See section §6.4 Eric W. Weisstein. Trigamma Function -- from MathWorld--A Wolfram Web Resource

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