`psi`

Digamma function and its higher derivatives (polygamma).

Syntax

Y = psi(X)

Returns the digamma function, the derivative of log(gamma(X)).

Y = psi(k, X)

Returns the k-th derivative of the digamma function (polygamma of order k).

Description

Because gamma extends the factorial, psi extends the harmonic numbers: psi(n+1) equals the sum 1 + 1/2 + ... + 1/n minus the Euler-Mascheroni constant, and psi(1) is that constant negated, about -0.5772. It has poles at zero and at the negative integers.

The higher orders arise in series expansions: psi(1, 1) equals pi^2/6, the sum of the reciprocal squares.

Input arguments

k: Nonnegative integer scalar, the order of the derivative. psi(0, X) is the same as psi(X).
X: Real array.

Output arguments

Y: Array of the same size as X.

Example

Digamma and trigamma

psi([1 2 0.5])
psi(1, 1)
pi^2/6

Output

ans = 1×3

   -0.5772    0.4228   -1.9635
ans = 1.6449
ans = 1.6449

psi