**Gamma Function Formula (Table of Contents)**

## What is the Gamma Function Formula?

In the mathematic world, from the secondary school itself, people will be aware of the three Greek symbols Alpha, Beta, and Gamma. These are predominantly used in the field of Physics and also widely used in Algebraic mathematical functions and integral functions. The gamma function is also often known as the well-known factorial symbol. It was hosted by the famous mathematician L. Euler (Swiss Mathematician 1707 – 1783) as a natural extension of the factorial operation from positive integers to real and even complex values of an argument. This Gamma function is calculated using the following formulae:

To identify Gamma function

**Γ (s) = (s-1)!**

Where,

**s:** Positive Integer

To identify Gamma function for other integers:

**r(s) = ∫**

^{∞}_{0}t^{s-1}e^{-t}dt,This can be further simplified as follows:

**Γ(s) = (s-1) Γ (s-1)**

Where,

**s:** positive real number and s should be always greater than 0.

**Example of Gamma Function Formula**

Let’s take an example to understand the calculation of the Gamma Function in a better manner.

#### Gamma Function Formula – Example #1

**Assume if the number is a ‘s’ and also it is a positive integer, then the gamma function will be the factorial of the number. This is mentioned as s! = 1*2*3… (s − 1)*s. For example, 4! = 1 × 2 × 3 × 4 = 24. However, this formula is not a valid one if s is not an integer although.**

Solution:

Γ(4) = (4-1)! = **6**

#### Gamma Function Formula – Example #2

**Evaluate Gamma Function Value for: Γ (3/2) / Γ (5/2)**

Solution:

**Step 1:** Identify whether the number is integer. In this case, it is not an integer

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**Step 2:** Apply the simplified version of second formula:

**Γ (5/2) = (s-1) Γ (s-1)**

- Γ (5/2) = ((5/2)-1) Γ ((5/2)-1)
- Γ (5/2) =
**(3/2) Γ (3/2)**

**Step 3:** Now apply the value for (5/2) in the original equation

- Γ (3/2) / Γ (5/2)
- = Γ (3/2) / (3/2) Γ (3/2)

**Step 4:** Cancelling Γ (3/2) as it is part of both numerator and denominator, the final value will be (3/2)

Hence the conclusion for the equation will be **Γ**** (3/2) / ****Γ**** (5/2) = (3/2)**

### Explanation

The formula for Gamma Function Formula can be calculated by using the following steps:

**Step 1:** Identify whether the input value is an integer or real number.

**Step 2:** If it is an integer, then we have to go with 1^{st} formulae i.e. identifying the factorial of the integer value – 1.

In the mathematics world, the word “Factorial” denotes the product (multiplication) of positive integers which is < or = the input integer. It is always denoted by that integer followed by an exclamatory note. Thus, any factorial (s) is written S!, and if s = 4 meaning, it is 4! Which is equal to 1 × 2 × 3 × 4. Always note that zero factorial is always equal to 1!!

**Step 3:** If it is a positive real number, then we have to go with 2^{nd} formulae. For example, we need to identify the Γ value for the number (5/2). Applying to the simplified formulae, it will be **(s-1) Γ (s-1)**. In this case, (5/2)-1 Γ (5/2)-1 which will be further simplified as (3/2) Γ (3/2)

### Relevance and Use of Gamma Function Formula

The integral in Γ(1) is convergent which can be proved in an easy way. There are some unique properties of the gamma function. Below are the properties specific to the gamma function:

It is proved that:

- Γ(s + 1) = sΓ(s), since
- Γ(s + 1) = lim T→∞ (Integral of 0 → T) e
^{−t}t^{p}dt - = p (Integral 0 → ∞) e
^{−t}t^{p-1}dt - = pΓ(p)
- Γ(1) = 1 (inconsequential proof)
- If s = n, a positive integer, then Γ(n + 1) = n!

Gamma function is applicable for all complex numbers (A complex number is defined as a number in common. It has two parts. The first one is known as real and followed by an imaginary number. The common imaginary number that is used as part of mathematics is the small letter “i” and the square of i is always equal to the constant -1) and it is not defined for negative integers including zero. All-natural numbers will be known as a discrete set and all positive real numbers will be known as a continuous set. Please note that the Gamma function will behave like a normal factorial for all natural numbers. However, the leeway to the real positive numbers makes it useful for modeling sites that include continuous change, with vital applications to differential and integral calculus, differential equations, complex analysis, and statistics

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