Calculate Inverse Gamma Function

Calculator and formula for computing the inverse (reciprocal) Gamma function

Inverse Gamma Function Calculator

Reciprocal Gamma Function

The 1/Γ(x) or reciprocal Gamma function is a holomorphic function without poles and important in complex analysis.

Real number for inverse Gamma function
Result
1/Γ(x):

Inverse Gamma Function Curve

Mouse pointer on the graph shows the values.
The inverse Gamma function is holomorphic everywhere (no poles).

Formulas for the Inverse Gamma Function

Definition
\[\frac{1}{\Gamma(x)} = \text{reciprocal Gamma function}\]

Reciprocal of the Eulerian Gamma function

Holomorphy
\[\frac{1}{\Gamma(z)} \text{ is holomorphic on } \mathbb{C}\]

No singularities in the complex plane

Product Representation
\[\frac{1}{\Gamma(z)} = z e^{\gamma z} \prod_{n=1}^{\infty} \left(1 + \frac{z}{n}\right) e^{-z/n}\]

Weierstrass product formula

Zeros
\[\frac{1}{\Gamma(x)} = 0 \text{ at } x = 0, -1, -2, -3, ...\]

Zeros at negative integers

Properties

Special Values
1/Γ(1) = 1 1/Γ(2) = 1 1/Γ(0) = 0 1/Γ(-1) = 0
Holomorphy
Analytic everywhere

No poles in the complex plane

Asymptotics
\[\frac{1}{\Gamma(x)} \sim \frac{x^{x-1/2} e^{-x}}{\sqrt{2\pi}}\]

for large |x|

Applications

Complex analysis, number theory, mathematical physics and approximation theory.

Detailed Description of the Inverse Gamma Function

Mathematical Definition

The inverse or reciprocal Gamma function is defined as the reciprocal of the Eulerian Gamma function. Unlike the Gamma function itself, it is an entire function without poles in the complex plane.

Definition: f(x) = 1/Γ(x)
Using the Calculator

Enter the argument x and click 'Calculate'. The function is defined everywhere but has zeros at negative integers.

Historical Background

The reciprocal Gamma function was systematically studied by Karl Weierstrass, who developed its product representation. It plays an important role in complex analysis and the theory of entire functions.

Properties and Applications

Mathematical Applications
  • Complex analysis (entire functions)
  • Analytic number theory
  • Approximation theory and interpolation
  • Asymptotic expansions
Physical Applications
  • Quantum field theory (Feynman diagrams)
  • Statistical mechanics (partition functions)
  • Mathematical physics (integral equations)
  • Scattering theory (resonances)
Special Properties
  • Holomorphy: Entire function without poles
  • Zeros: Simple zeros at x = 0, -1, -2, ...
  • Growth: Exponential decay for large |x|
  • Symmetry: Special reflection properties
Interesting Facts
  • 1/Γ(x) is the only entire function with zeros at negative integers
  • Weierstrass product formula characterizes it uniquely
  • Important for the theory of L-functions in number theory
  • Generalizations to multiple Gamma functions are possible

Calculation Examples

Example 1

1/Γ(1) = 1

Since Γ(1) = 1

Example 2

1/Γ(0) = 0

Zero at x = 0

Example 3

1/Γ(1/2) = 1/√π ≈ 0.5642

Half-integer value

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