Calculate Trigamma Function

Online calculator and formulas for computing the Trigamma function (second derivative of the Digamma function)

Trigamma Function Calculator

Using the Calculator

Enter the argument x and click 'Calculate'. The function has poles at x = 0, -1, -2, -3, ... and is strictly decreasing for x > 0.

Real number ≠ 0, -1, -2, -3, ... for the Trigamma function
Result
ψ'(x):

Trigamma Function Curve

Trigamma (Polygamma) Function

The ψ'(x) or Trigamma function is the second derivative of the logarithm of the Gamma function and an important Polygamma function.

Mouse pointer on the graph shows the values.
The Trigamma function has poles at x = 0, -1, -2, -3, ... and is strictly decreasing for x > 0.

Formulas for the Trigamma Function

Definition
\[\psi'(x) = \frac{d^2}{dx^2} \ln \Gamma(x) = \frac{d}{dx} \psi(x)\]

Second derivative of the logarithm of the Gamma function

Recurrence Formula
\[\psi'(x+1) = \psi'(x) - \frac{1}{x^2}\]

Basic recurrence relation

Series Expansion
\[\psi'(x) = \sum_{n=0}^{\infty} \frac{1}{(x+n)^2}\]

For Re(x) > 0

Integral Representation
\[\psi'(x) = \int_0^{\infty} \frac{t e^{-xt}}{1-e^{-t}} dt\]

For Re(x) > 0

Asymptotic Expansion
\[\psi'(x) \sim \frac{1}{x} + \frac{1}{2x^2} + \frac{1}{6x^3} + ...\]

For large |x|

Polygamma Generalization
\[\psi^{(n)}(x) = (-1)^{n+1} n! \sum_{k=0}^{\infty} \frac{1}{(x+k)^{n+1}}\]

Trigamma is the case n = 1

Properties

Special Values
ψ'(1) = π²/6 ψ'(1/2) = π²/2 ψ'(2) = π²/6 - 1
Monotonicity
Strictly decreasing

for x > 0

Convexity
\[\psi'(x) > 0 \text{ for all } x > 0\]

Always positive for positive arguments

Applications

Statistical physics, number theory, asymptotic analysis and quantum mechanics.

Detailed Description of the Trigamma Function

Mathematical Definition

The Trigamma function is the second derivative of the logarithm of the Gamma function and belongs to the family of Polygamma functions. It is closely related to the Digamma function and plays an important role in analysis and mathematical physics.

Definition: ψ'(x) = d²/dx² [ln Γ(x)] = d/dx [ψ(x)]
Historical Background

The Trigamma function was systematically studied by Euler and later by Gauss. The name derives from the Greek "tri" (three) because it is the third in the hierarchy of Gamma-related functions (after Gamma and Digamma).

Special Properties
  • Monotonicity: Strictly decreasing for x > 0
  • Positivity: ψ'(x) > 0 for all x > 0
  • Poles: Second-order poles at x = 0, -1, -2, ...
  • Asymptotics: ψ'(x) ~ 1/x for large x

Properties and Applications

Mathematical Applications
  • Asymptotic expansions and Stirling formulas
  • Analytic number theory (L-functions)
  • Harmonic numbers and Riemann zeta function
  • Infinite series and integrals
Physical Applications
  • Statistical physics (fluctuation theory)
  • Quantum mechanics (energy levels)
  • Condensed matter (critical phenomena)
  • Mathematical physics (path integrals)
Interesting Facts
  • ψ'(1) = π²/6 (related to the Riemann zeta function ζ(2))
  • Connection to harmonic numbers: ψ'(n+1) = ζ(2) - H_n^{(2)}
  • Important for the variance of Gamma-distributed random variables
  • Appears in quantum field theory loop calculations

Calculation Examples

Example 1

ψ'(1) = π²/6 ≈ 1.6449

Famous value, related to ζ(2)

Example 2

ψ'(1/2) = π²/2 ≈ 4.9348

Half-integer value

Example 3

ψ'(2) = π²/6 - 1 ≈ 0.6449

Recursive relation

Connections to Other Functions

Riemann Zeta Function

The Trigamma function is closely connected to the Riemann zeta function:

\[\psi'(1) = \zeta(2) = \frac{\pi^2}{6}\]

This connection makes it important for analytic number theory.

Harmonic Numbers

Relationship with generalized harmonic numbers:

\[\psi'(n+1) = \zeta(2) - H_n^{(2)}\]

Where H_n^{(2)} are the harmonic numbers of second order.

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