Gamma function The gamma function is defined by Γ(p) = ∫ from 0 to ∞ of x^(p-1) e^(-x) dx, for p not equal to zero or a negative integer.
a. Use the reduction formula ∫ from 0 to ∞ of x^p e^(-x) dx = p ∫ from 0 to ∞ of x^(p-1) e^(-x) dx for p = 1, 2, 3, ...
to show that Γ(p + 1) = p! (p factorial).

Verified step by step guidance

1. Recall the definition of the Gamma function: \(\Gamma(p) = \int_0^{\infty} x^{p-1} e^{-x} \, dx\), where \(p\) is not zero or a negative integer.
2. Use the given reduction formula: \(\int_0^{\infty} x^p e^{-x} \, dx = p \int_0^{\infty} x^{p-1} e^{-x} \, dx\). Notice that the left integral is \(\Gamma(p+1)\) and the right integral is \(\Gamma(p)\), so rewrite it as \(\Gamma(p+1) = p \Gamma(p)\).
3. Apply this recursive relation repeatedly for positive integers \(p = 1, 2, 3, \ldots\) to express \(\Gamma(p+1)\) in terms of \(\Gamma(1)\): \(\Gamma(p+1) = p \times (p-1) \times (p-2) \times \cdots \times 1 \times \Gamma(1)\).
4. Evaluate \(\Gamma(1)\) by substituting \(p=1\) into the Gamma function definition: \(\Gamma(1) = \int_0^{\infty} x^{0} e^{-x} \, dx = \int_0^{\infty} e^{-x} \, dx\), which is a standard integral.
5. Since \(\Gamma(1) = 1\), conclude that \(\Gamma(p+1) = p!\), where \(p!\) is the factorial of \(p\), completing the proof.

Key Concepts

Gamma Function Definition

The gamma function Γ(p) generalizes the factorial function to real and complex numbers. It is defined as the improper integral Γ(p) = ∫₀^∞ x^(p-1) e^(-x) dx for p > 0 and p not a negative integer. Understanding this integral form is essential to relate the gamma function to factorials.

Reduction Formula for the Gamma Function

The reduction formula ∫₀^∞ x^p e^(-x) dx = p ∫₀^∞ x^(p-1) e^(-x) dx expresses the integral with power p in terms of the integral with power p-1. This recursive relationship is key to proving properties of the gamma function, such as connecting Γ(p+1) to pΓ(p).

Factorial and Its Relation to the Gamma Function

The factorial of a positive integer p, denoted p!, is the product of all positive integers up to p. The gamma function satisfies Γ(p+1) = p!, linking continuous and discrete mathematics. Demonstrating this equality involves using the reduction formula and the base case Γ(1) = 1.