Gamma Function Calculator
Calculate the gamma function Γ(z) for real numbers with step-by-step solutions
Γ(z) = ∫₀^∞ t^(z-1)e^(-t) dt
For integers: Γ(n) = (n-1)! | Special: Γ(1/2) = √π ≈ 1.7724538509
Note: Γ(z) is undefined at z = 0, -1, -2, ... Uses Γ(z+1) = z! for any real number z
Quick Examples
Undefined Value (Pole)
The Gamma function has a pole at z = . It is undefined at 0 and negative integers.
Γ() !
= ()! =
Reciprocal 1/Γ(z)
Log Gamma ln(Γ(z))
Input Type
Step-by-Step Solution
Identify the input value
z =
Apply the Gamma function Γ(z) Apply generalized factorial z! = Γ(z+1)
For integer z: Γ() = ()! Using Lanczos approximation for numerical computation
Result
Γ() = ! =
Special Values Reference
| z | Γ(z) | Equivalent |
|---|---|---|
| 1 | 1 | 0! = 1 |
| 2 | 1 | 1! = 1 |
| 3 | 2 | 2! = 2 |
| 1/2 | √π ≈ 1.7724538 | Special value |
| 3/2 | √π/2 ≈ 0.8862269 | (1/2)! = (1/2)√π |
| -1/2 | -2√π ≈ -3.5449077 | Reflection formula |
| 0, -1, -2, ... | Undefined | Poles of Γ(z) |
If you like this calculator
Please help us simply by sharing it. It will help us a lot!
Related Calculators
Other calculators you might find useful.
Molarity Calculator
Calculate solution concentration, moles, volume, and perform dilution calculations
Surface Area Calculator
Calculate the surface area of cubes, spheres, cylinders, cones, and other 3D shapes instantly
Error Function Calculator
Calculate the error function erf(x) with step-by-step solutions, inverse erf, and probability applications
Matrix Inverse Calculator
Calculate the inverse of 2x2 and 3x3 matrices with step-by-step solutions and determinant calculations
Percentage Change Calculator
Calculate the percentage change between two values with step-by-step solutions
P-Value Calculator
Calculate p-values from z-scores and t-scores for hypothesis testing
About Gamma Function Calculator
What is the Gamma Function?
The Gamma function Γ(z) is a fundamental mathematical function that extends the factorial function to complex and real numbers. It is defined for all complex numbers except non-positive integers.
Mathematical Definition
For complex numbers with positive real part (ℜ(z) > 0), the Gamma function is defined by the integral:
Γ(z) = ∫₀^∞ t^(z-1)e^(-t) dt
Relation to Factorial
For any positive integer n:
Γ(n) = (n-1)!
Or equivalently: Γ(n+1) = n!
Examples:
- Γ(1) = 0! = 1
- Γ(2) = 1! = 1
- Γ(3) = 2! = 2
- Γ(4) = 3! = 6
- Γ(5) = 4! = 24
Key Properties
Recursive Property
Γ(z+1) = z × Γ(z)
This allows computing Gamma values for any real or complex number.
Special Values
- Γ(1/2) = √π ≈ 1.7724538509
- Γ(3/2) = (1/2)√π ≈ 0.8862269255
- Γ(-1/2) = -2√π ≈ -3.5449077018
Euler's Reflection Formula
For non-integer values of z:
Γ(z) × Γ(1-z) = π / sin(πz)
Poles of the Gamma Function
The Gamma function has simple poles (undefined points) at:
- z = 0, -1, -2, -3, ... (all non-positive integers)
Applications
- Probability and Statistics - Appears in gamma, beta, chi-squared distributions
- Complex Analysis - Fundamental in analytic number theory
- Physics - Quantum mechanics, statistical mechanics
- Combinatorics - Generalized binomial coefficients
- Engineering - Signal processing, control theory
Stirling's Approximation
For large values of z:
Γ(z) ≈ √(2π/z) × (z/e)^z
Common Mistakes
- Confusing Γ(n) with n! - Remember: Γ(n) = (n-1)!, not n!
- Using non-positive integers - Γ(z) is undefined at z = 0, -1, -2, ...
- Forgetting the shift - The gamma function is shifted by 1 from factorial