# Gamma function (infinite product representation)

• math8
In summary, the Gamma function is a special function in mathematics that extends the factorial function to complex numbers. It is defined as an integral and has an infinite product representation that is useful for complex numbers. It has a close relationship with the factorial function and has various applications in different fields. The Gamma function also has important properties, such as the reflection formula and the duplication formula.

#### math8

I have come across this expression in some notes

$$\Gamma$$ (z) = $$\frac{1}{z}$$ $$\prod$$ $$\frac{(1+ \frac{1}{n})^{z}}{1+ \frac{z}{n}}$$

Do you think it's accurate? I have some doubts because I have looked for it on wokipedia, and I couldn't find it.

Do you mean

$$\Gamma(z)=\frac{1}{z}\prod_{n=1}^{\infty} \frac{\left(1+\frac{1}{n}\right)^z}{1+\frac{z}{n}}$$

If so, then yes, it's correct for all complex numbers $z$ except for zero and negative integers.

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Thanks a lot :)

## What is the Gamma function and how is it defined?

The Gamma function, denoted by Γ, is a special function in mathematics that generalizes the factorial function to complex numbers. It is defined as the integral of tx-1e-tdt from 0 to infinity.

## What is the infinite product representation of the Gamma function?

The infinite product representation of the Gamma function is given by Γ(x)=limn→∞ n!x/x(x+1)(x+2)...(x+n). This representation is useful for calculating the value of the Gamma function for complex numbers.

## What is the relationship between the Gamma function and the factorial function?

The Gamma function is an extension of the factorial function to complex numbers. For positive integers n, Γ(n)=(n-1)!. This relationship is the basis for the infinite product representation of the Gamma function.

## What are some applications of the Gamma function?

The Gamma function has many applications in mathematics, physics, and engineering. It is used in the calculation of definite integrals, in the theory of probability and statistics, in the study of special functions, and in the development of mathematical models.

## What are some properties of the Gamma function?

The Gamma function has several important properties, such as the reflection formula, which relates the values of Γ(x) and Γ(1-x), and the duplication formula, which relates the values of Γ(x) and Γ(2x). It also satisfies the recurrence relation Γ(x+1)=xΓ(x) and has poles at the non-positive integers.