Polygamma Function ψ⁽ⁿ⁾(x) Calculator

Enter an order n and argument x to evaluate the polygamma function ψ⁽ⁿ⁾(x), which becomes the digamma function when n=0, using recurrence and asymptotic expansion.

Input

The polygamma function ψ⁽ⁿ⁾(x) is the (n+1)-th derivative of ln Γ(x). Enter an order n and an argument x. With n=0 it becomes the digamma function.

Order n

Non-negative integer (0 is digamma)

Argument x

Non-positive integers are poles

Result

Value of ψ⁽0⁾(x)

0.7031566406

argument x = 2.5

Digamma ψ(x)

0.70315664

Order n

Recurrence steps

The argument was raised to 10.5 with the recurrence, where the asymptotic expansion was evaluated.

How it works

The polygamma function ψ⁽ⁿ⁾(x) is the (n+1)-th derivative of the log-gamma function ln Γ(x). When n=0 it equals ψ(x)=Γ′(x)/Γ(x), the digamma function.
The value is obtained by raising the argument x to at least 10 with the recurrence relation, evaluating the asymptotic (Stirling-type) series with Bernoulli numbers, then shifting back.
Negative arguments are moved to the positive side through the recurrence. When x is a non-positive integer the function has a pole and cannot be evaluated.
The order n must be a non-negative integer. ψ⁽¹⁾ is the trigamma function and ψ⁽²⁾ is the tetragamma function.
Reference checks: ψ(1)=−γ (Euler constant), ψ⁽¹⁾(1)=π²/6, ψ⁽²⁾(1)=−2ζ(3).

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