# Gamma function

The Gamma function is extremely tightly linked to the notion of factorial.  While you move on to the Gamma function properties, first let us tell you what the game function operation is.

#### Input

$$a=3.2$$

#### Solution

$$\normalsize Gamma\ function\ \Gamma(a)=2.4239654799354$$

#### Formula

$$\normalsize Gamma\ function\ \Gamma(a)$$ $$(1)\ \Gamma(a)= {\large\int_{\tiny 0}^ {\tiny \infty}}t^{a-1}e^{-t}dt, Re(a)\gt 0$$ $$(2)\ \Gamma(a)={\large\frac{\Gamma(a+1)}{a}}, \Gamma(a)\Gamma(1-a)={\large\frac{\pi}{\sin(\pi a)}}$$ $$(3)\ \Gamma(n+1)=n!, \Gamma({\large\frac{1}{2}})={\large\sqrt{\pi}}\\$$

# What is the Gamma function? Gamma function properties

The Gamma function is extremely tightly linked to the notion of factorial.  While you move on to the Gamma function properties, first let us tell you what the game function operation is.

Factorial operator

Let n be a positive integer. The factorial of n (denoted by n!) is that the product of all integers between (and including) 1 and n:

n! = 1 ⋅ 2 ⋅.......⋅ (n-1) ⋅ n.

For instance

5! = 1 ⋅ 2 ⋅ 3 ⋅ 4 ⋅ 5 = 120.

Factorials are essential in statistics and probability. they need many uses, too many to say here, so allow us to just mention permutations & combinations, binomial coefficients alongside Pascal's triangle, and therefore the binomial probability distribution. If you would like to find out more about factorials, especially their applications, visit our dedicated factorial calculator.

What is essential for us is that the incontrovertible fact that the factorial is merely defined for positive integers. Are there any thanks to extending the factorial to all or any positive numbers? How about all real (or even complex) numbers? We will consider this because of the following interpolation problem:

Can we discover a smooth curve that connects the points (x, y) given by y = x! at the positive integer values for x?

There are several approaches to unravel this problem, and therefore the Gamma function is the hottest one.

## What are the Gamma function and factorials

Gamma may be a function (denoted by the Greek letter 𝚪) that permits us to increase the notion of factorial well beyond positive integer numbers. Formally, the Gamma function formula is given by an integral (see a subsequent section for more details). Most significantly, the Gamma function and factorials are linked via the relationship:

(n) = (n - 1)!

So it's indeed an answer to the interpolation problem described above. Below you'll see the graph of the Gamma function (n + 1) = n! for all positive numbers (the red dots are the values of the factorial at each integer of x):

Is anyone know that... this function was defined by Swiss mathematician Euler within the half of the 18th century, while the name Gamma and notation 𝚪 was introduced by the French mathematician Adrian-Marie Legendre at the start of the 19th century.

### How to use this Gamma function calculator?

Are you continue to wondering the way to calculate the Gamma function? Well, the simplest answer is: use our Gamma function calculator! Here we briefly describe the way to use it:

Choose the sort of argument you would like our Gamma function calculator to use. By default, we set it to real arguments, but complex numbers are available also.

Enter the argument within the appropriate field(s).

The calculator uses very precise Gamma function approximations to work out the result. It appears just beneath the argument. Enjoy!

If you would like to extend the precision of calculations, attend the advanced mode. By default, this Gamma function calculator uses five sig figs to display the result.

How to use this calculator

This tool is really easy to use and even our tool has a really very simple layout so that it will be easy to understand for people.

Now as you can see this tool doesn’t require any registration and it's a totally free online tool that anyone can use from anywhere. They can use it on their phone and they can use it on a desktop however they feel comfortable.

You can see on your screen you have only one text box in this tool right but you have other different numbers.

You must be confused about how you are going to type other numbers but you don’t need to worry too much because we have already mentioned how you will type numbers in it.

Just typing space or typing a comma at the end of each number will be fine.

And then you will be able to calculate your Function map single arrow.

Simply click on the calculate button to get the answer.

Tips: you should bookmark this tool so that you can use it in the future.

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