Gamma Function Calculator
The Gamma Function
The Gamma function, denoted as \( \Gamma(z) \), is an extension of the factorial function to complex and real numbers.
Definition
For a complex number \(z\) with a positive real part, the Gamma function is defined by the integral:
\( \Gamma(z) = \int_0^\infty t^{z-1}e^{-t}dt \)
Factorial Relationship
For any positive integer \(n\):
\( \Gamma(n) = (n-1)! \)
For example, \( \Gamma(5) = 4! = 24 \).
Key Properties
- \( \Gamma(z+1) = z\Gamma(z) \)
- \( \Gamma(1/2) = \sqrt{\pi} \)
- The function has simple poles (goes to infinity) at all non-positive integers (0, -1, -2, ...).
Calculate the Gamma function \( \Gamma(z) \) for real and complex numbers.
Gamma(z)
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