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psi

Psi (polygamma) function

Y = psi(X)
Y = psi(k,X)

Description

Y = psi(X) evaluates the ψ function for each element of array X. X must be real and nonnegative. The ψ function, also known as the digamma function, is the logarithmic derivative of the gamma function

$\begin{array}{c}\psi \left(x\right)=\text{digamma}\left(x\right)\\ =\frac{d\left(\mathrm{log}\left(\Gamma \left(x\right)\right)\right)}{dx}\\ =\frac{d\left(\Gamma \left(x\right)\right)/dx}{\Gamma \left(x\right)}\end{array}$

Y = psi(k,X) evaluates the kth derivative of ψ at the elements of X. psi(0,X) is the digamma function, psi(1,X) is the trigamma function, psi(2,X) is the tetragamma function, etc.

Examples

Example 1

Use the psi function to calculate Euler's constant, γ.

```format long
-psi(1)
ans =
0.57721566490153

-psi(0,1)
ans =
0.57721566490153```

Example 2

The trigamma function of 2, psi(1,2), is the same as (π2/6) – 1.

```format long
psi(1,2)
ans =
0.64493406684823

pi^2/6 - 1
ans =
0.64493406684823```

References

[1] Abramowitz, M. and I. A. Stegun, Handbook of Mathematical Functions, Dover Publications, 1965, Sections 6.3 and 6.4.