Derivative of polygamma function

Question

I am working on my Matlab homework and I have to make a derivative of function $f(x)=\psi (x)\cdot \sin (x)$ , where $\psi(x)$ is polygamma function. What the derivative of $\psi(x)$ will be?

Answer 1

Using $\psi(x) = \frac{\Gamma'(x)}{\Gamma(x)}$, the derivative $\psi'(x) = \frac{\Gamma(x)\Gamma''(x)-\Gamma'(x)^2}{\Gamma(x)^2} = \frac{\Gamma''(x)}{\Gamma(x)}-\psi(x)^2$,

With $\Gamma(x)$ defined as per usual,

$\Gamma'(x) = $ "take differential operator inside integral" $ = \int_0^\infty \ln(t)t^{x-1} e^{-t} dt$

$\Gamma''(x) = \int_0^\infty \ln(t)^2t^{x-1} e^{-t} dt$