Calculate Digamma Function

Online calculator and formulas for calculating the Digamma function (Psi function)

Digamma Function Calculator

Digamma (Psi) Function

The ψ(x) or Digamma function is the logarithmic derivative of the Gamma function and an important special function.

Real number > 0 for the Digamma function
Result
ψ(x):

Digamma Function Curve

Mouse pointer on the graph shows the values.
The Digamma function has poles at negative integers.

Digamma Function Formulas

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

Logarithmic derivative of the Gamma function

Recurrence Formula
\[\psi(x+1) = \psi(x) + \frac{1}{x}\]

Basic recurrence relation

Integral Representation
\[\psi(x) = -\gamma + \int_0^1 \frac{1-t^{x-1}}{1-t} dt\]

γ is the Euler-Mascheroni constant

Reflection Formula
\[\psi(1-x) - \psi(x) = \pi \cot(\pi x)\]

Symmetry relation

Special Values

Important Values
ψ(1) = -γ ψ(2) = 1-γ ψ(1/2) = -γ-2ln(2)
Euler-Mascheroni Constant
γ ≈ 0.5772156649

The fundamental mathematical constant

Asymptotic
\[\psi(x) \sim \ln(x) - \frac{1}{2x}\]

for large x

Comprehensive Description of the Digamma Function

Mathematical Definition

The Digamma function, also called the Psi function, is the logarithmic derivative of the Gamma function. It plays a central role in analytic number theory and special functions.

Definition: ψ(x) = d/dx [ln Γ(x)] = Γ'(x)/Γ(x)
Using the Calculator

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

Historical Background

The Digamma function was systematically studied by Euler and later by Legendre. The name "Digamma" comes from the ancient Greek letter Ϝ (Digamma), which looks like the modern form ψ.

Properties and Applications

Mathematical Applications
  • Analytic number theory (Riemann zeta function)
  • Asymptotic expansions and Stirling's formula
  • Hypergeometric functions
  • Harmonic numbers and Euler sums
Physical Applications
  • Quantum field theory (renormalization)
  • Statistical mechanics (distribution functions)
  • Condensed matter (critical phenomena)
  • Mathematical physics (integral equations)
Special Properties
  • Poles: At x = 0, -1, -2, -3, ... with residue -1
  • Monotonicity: Strictly monotonically increasing for x > 0
  • Convexity: Convex for x > 0
  • Duplication Formula: Special identities
Interesting Facts
  • ψ(x) is the unique meromorphic function with residues -1 at negative integers
  • Connection to harmonic numbers: ψ(n+1) = -γ + H_n
  • The Trigamma function is the derivative of the Digamma function
  • Important for computing Beta and Zeta functions

Calculation Examples

Example 1

ψ(1) = -γ ≈ -0.5772

The negative Euler-Mascheroni constant

Example 2

ψ(2) = 1 - γ ≈ 0.4228

Important special case for natural numbers

Example 3

ψ(1/2) = -γ - 2ln(2) ≈ -1.9635

Half-integer value with logarithmic term

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