Gamma Function Calculator

What is the Gamma Function?

The gamma function, denoted by Γ(z), is an extension of the factorial function to complex and real number arguments. It is a fundamental function in mathematics with numerous applications in probability, statistics, and physics.

Formula and Its Meaning

The gamma function is defined for all complex numbers except the non-positive integers. For complex numbers with a positive real part, it is defined via a convergent improper integral:

\[\Gamma(z) = \int_0^\infty t^{z-1} e^{-t} dt\]

Where:

• z is a complex number with a positive real part
• e is Euler's number (approximately 2.71828)
• t is the variable of integration

For positive integers, the gamma function is related to factorials:

\[\Gamma(n) = (n-1)!\]

Calculation Steps

1. For positive real numbers, we use numerical integration techniques to approximate the integral.
2. For complex numbers or improved efficiency, we often use the Lanczos approximation.
3. Special cases (like integers and half-integers) can be calculated using specific formulas.

Example Calculation

Let's calculate Γ(3.5):

1. We can use the property Γ(z+1) = z · Γ(z)
2. Γ(3.5) = 2.5 · Γ(2.5)
3. Γ(2.5) = √π · 1.5 / 2^1.5 ≈ 1.329340388
4. Therefore, Γ(3.5) ≈ 2.5 · 1.329340388 ≈ 3.323350970

Visual Representation

This graph represents the gamma function for positive real numbers. The function has a minimum at x ≈ 1.46163, and it grows rapidly for both small positive values and large values of x.