naginterfaces.library.specfun.psi_​deriv_​complex

naginterfaces.library.specfun.psi_deriv_complex(z, k)

psi_deriv_complex returns the value of the $k$th derivative of the psi function $\psi \left(z\right)$ for complex $z$ and $k=0,1,\dots ,4$.

For full information please refer to the NAG Library document for s14af

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/s/s14aff.html

Parameters

zcomplex

The argument $z$ of the function.

kint

The function ${\psi }^{\left(k\right)}\left(z\right)$ to be evaluated.

Returns

pkzcomplex

The value of the $k$th derivative of the psi function.

Raises

NagValueError

(errno $1$)

On entry, $Re\left(z\right)$ is ‘too close’ to a non-positive integer when $Im\left(z\right)=0.0$: $Re\left(z\right)=⟨value⟩$, $nint\left(Re\left(z\right)\right)=⟨value⟩$.

(errno $1$)

On entry, $k=⟨value⟩$.

Constraint: $k\le 4$.

(errno $1$)

On entry, $k=⟨value⟩$.

Constraint: $k\ge 0$.

(errno $2$)

Evaluation abandoned due to likelihood of overflow.

Notes

psi_deriv_complex evaluates an approximation to the $k$th derivative of the psi function $\psi \left(z\right)$ given by

$${\psi }^{\left(k\right)}\left(z\right)=\frac{d^k}{d{z}^k}\psi \left(z\right)=\frac{d^k}{d{z}^k}\left(\frac{d}{dz}loge\left(\Gamma \left(z\right)\right)\right)\text{,}$$

where $z=x+iy$ is complex provided $y\ne 0$ and $k=0,1,\dots ,4$. If $y=0$, $z$ is real and thus ${\psi }^{\left(k\right)}\left(z\right)$ is singular when $z=0,-1,-2,\dots$.

Note that ${\psi }^{\left(k\right)}\left(z\right)$ is also known as the polygamma function. Specifically, ${\psi }^{\left(0\right)}\left(z\right)$ is often referred to as the digamma function and ${\psi }^{\left(1\right)}\left(z\right)$ as the trigamma function in the literature. Further details can be found in Abramowitz and Stegun (1972).

psi_deriv_complex is based on a modification of the method proposed by Kölbig (1972).

To obtain the value of ${\psi }^{\left(k\right)}\left(z\right)$ when $z$ is real, psi_deriv_real() can be used.

References

Abramowitz, M and Stegun, I A, 1972, Handbook of Mathematical Functions, (3rd Edition), Dover Publications

Kölbig, K S, 1972, Programs for computing the logarithm of the gamma function, and the digamma function, for complex arguments, Comp. Phys. Comm. (4), 221–226