The expression you've given is the definition of the Gamma function, often denoted as $\Gamma(x)$. It is defined as:

$\Gamma(x) = \int_0^\infty e^{-t} t^{x-1} \, dt$

This is valid for $x > 0$. The Gamma function generalizes the factorial function to non-integer values. Specifically, for a positive integer $n$, we have:

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

For example:

• $\Gamma(1) = 1$
• $\Gamma(2) = 1$
• $\Gamma(3) = 2$
• $\Gamma(4) = 6$, and so on.

Key Properties:

1. Recurrence Relation: $\Gamma(x+1) = x \Gamma(x)$

2. Reflection Formula (for $x > 0$): $\Gamma(x) \Gamma(1-x) = \frac{\pi}{\sin(\pi x)}$

3. Relation to Factorials: $\Gamma(n) = (n-1)!$

The Gamma function also has applications in various fields like probability theory, combinatorics, and complex analysis.

Would you like to explore more about the properties of the Gamma function or its applications?

Follow-up questions:

1. How does the Gamma function relate to the factorial of non-integer values?
2. Can the Gamma function be extended to complex numbers?
3. How is the Gamma function used in probability theory, specifically in the Gamma distribution?
4. What is the connection between the Gamma function and the Beta function?
5. How do we evaluate integrals involving the Gamma function?

Tip: The Gamma function plays a crucial role in many areas of mathematics, especially in complex analysis and statistical distributions. When studying it, try visualizing its behavior for different values of $x$ to build intuition!