Gamma Function

The Gamma function, denoted by Γ(z), is one of the most important special functions in mathematics. It was developed by Swiss mathematician Leonhard Euler in the 18th century. The gamma function extends the concept of factorials to non-integer and complex numbers.

The gamma function is defined by the integral:

\Gamma(z) = \int_{0}^{\infty} t^{z-1}e^{-t} \, dt

where z>0 and the integral converges for all complex numbers with positive real part.

For positive integers, it satisfies the relationship:

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

where n is a positive integer.

The gamma function is also known as Euler's integral of the second kind.

Standard Results and Forms

1. Fundamental Properties

Basic Definition:

\Gamma(z) = \int_{0}^{\infty} t^{z-1}e^{-t}dt, \quad \text{Re}(z) > 0

For Positive Integers:

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

Special Values:

\Gamma(1) = 1, \quad \Gamma\left(\frac{1}{2}\right) = \sqrt{\pi}

2. Recurrence Relations

Primary Recurrence Formula:

\Gamma(z) = \Gamma(z - 1) \cdot (z - 1)

or equivalently,

\Gamma(z + 1) = z\Gamma(z)

Proof :

The recurrence relation can be derived using integration by parts. Starting with the definition of the gamma function :

\Gamma(z) = \int_{0}^{\infty} x^{z-1} \exp(-x) \, dx

Using integration by parts with :

• u = x^{z-1}, \quad \text{so } du = (z-1)x^{z-2} \, dx
• dv = \exp(-x)dx, \quad \text{so } v = -\exp(-x)

We get

\Gamma(z) = \left[ -x^{z-1} \exp(-x) \right]_{0}^{\infty} + \int_{0}^{\infty} (z-1)x^{z-2} \exp(-x) \, dx

The boundary term evaluates to zero :

= (0 - 0) + (z - 1) \int_{0}^{\infty} x^{(z-1)-1} \exp(-x) \, dx

= (z-1)\Gamma(z-1)

Therefore:

\Gamma(z) = (z-1)\Gamma(z-1)

3. Reflection Formula (Euler's)

\Gamma(z)\Gamma(1 - z) = \frac{\pi}{\sin(\pi z)}, \quad 0 < z < 1

Relationship to Other Special Functions

Gamma function is related other functions also:

1. Beta Function

B(p, q) = \frac{\Gamma(p)\Gamma(q)}{\Gamma(p + q)}

2. Incomplete Gamma Functions

Lower Incomplete:

\gamma(s, x) = \int_{0}^{x} t^{s-1}e^{-t} \, dt

Upper Incomplete:

\Gamma(s, x) = \int_{x}^{\infty} t^{s-1}e^{-t} \, dt

Relation:

\gamma(s, x) + \Gamma(s, x) = \Gamma(s)

Real-World Applications

The gamma function appears in numerous real-world applications:

• Statistics and Probability: Forms the foundation of the gamma distribution, chi-squared distribution, and beta distribution used in reliability analysis and risk modeling.
• Physics: Essential in quantum mechanics, statistical mechanics, and scattering theory. Used in calculating energy distributions and particle interactions.
• Engineering: Applied in signal processing, control systems, and reliability engineering for modeling failure times and system performance.
• Hydrology: Used to model rainfall patterns and flood frequency analysis due to its ability to handle only positive values.
• Queuing Theory: Models waiting times and service times in various systems like customer service centers and network traffic.

Examples of Gamma Function

Example 1 : Evaluate: Γ(5)

Solution:

Using the property Γ(n) = (n−1)!

= Γ(5)

= (5-1)!

= 4!

= 24

Example 2 : Evaluate: \Gamma\left(\frac{1}{6}\right) \Gamma\left(\frac{5}{6}\right)

Solution:

Using Euler's reflection formula with z = 1 / 6 :

=\Gamma\left(\frac{1}{6}\right)\Gamma\left(1-\frac{1}{6}\right)\\ = \Gamma\left(\frac{1}{6}\right)\Gamma\left(\frac{5}{6}\right)\\=\frac{\pi}{\sin\left(\frac{\pi}{6}\right)}\\ = \frac{\pi}{\frac{1}{2}}\\ = 2\pi