Problem 53 - Calculus Early Transcendentals: Pearson New International Edition

Question

(Gamma Function) Let \(\Gamma(n)=\int_{0}^{\infty} x^{n-1} e^{-x} d x, n>0 .\) This integral converges by Problems 51 and \(52 .\) Show each of the following (note that the gamma function is defined for every positive real number \(n\) ): (a) \(\Gamma(1)=1\) (b) \(\Gamma(n+1)=n \Gamma(n)\) (c) \(\Gamma(n+1)=n !\), if \(n\) is a positive integer.

Step-by-Step Solution

Answer

(a) \(\Gamma(1) = 1\), (b) \(\Gamma(n+1) = n \Gamma(n)\), (c) \(\Gamma(n+1) = n!\) for positive integer \(n\).

Step 1: Introduction to the Gamma Function

The Gamma function is an important function in mathematics which generalizes the factorial function to real and complex numbers. It is defined as \(\Gamma(n) = \int_{0}^{\infty} x^{n-1} e^{-x} \, dx\) for \(n > 0\). This exercise will demonstrate key properties of the Gamma function.

Step 1: Evaluate \(\Gamma(1)\)

To find \(\Gamma(1)\), substitute \(n = 1\) into the definition: \(\Gamma(1) = \int_{0}^{\infty} x^{1-1} e^{-x} \, dx = \int_{0}^{\infty} e^{-x} \, dx\). The result of this integral is 1, since it represents the area under the curve of \(e^{-x}\) from 0 to \(\infty\). Thus, \(\Gamma(1) = 1\).

Step 2: Prove Recursive Formula \(\Gamma(n+1) = n \Gamma(n)\)

Consider \(\Gamma(n+1) = \int_{0}^{\infty} x^n e^{-x} \, dx\). Use integration by parts where \(u = x^n\) and \(dv = e^{-x}dx\). This gives \(du = nx^{n-1}dx\) and \(v = -e^{-x}\). Applying integration by parts, we get \([ -x^n e^{-x} ]_0^{\infty} + \int_{0}^{\infty} nx^{n-1} e^{-x} \, dx\). The first term evaluates to 0, and the result of the integration by parts is \(n \Gamma(n)\). Thus, \(\Gamma(n+1) = n \Gamma(n)\).

Step 3: Show \(\Gamma(n+1) = n!\) for Positive Integer \(n\)

Using the recursive formula proved in the previous step, \(\Gamma(n+1) = n \Gamma(n)\). Assume \(\Gamma(n) = (n-1)!\), which is true for \(\Gamma(1) = 0! = 1\). Applying the recursive formula repeatedly: \(\Gamma(2) = 1 \cdot \Gamma(1) = 1!\), \(\Gamma(3) = 2 \cdot \Gamma(2) = 2!\), and so on, results in \(\Gamma(n+1) = n!\). By mathematical induction, this holds for any positive integer \(n\).

Key Concepts

Factorial

The concept of factorial is central to the Gamma function's properties. The factorial of a non-negative integer is the product of all positive integers less than or equal to that number. It is typically denoted as

• \(!n\) for a positive integer \(n\), calculated as \(n! = n \times (n-1) \times (n-2) \times ... \times 2 \times 1\).
• Special case: \(0!\) is defined to be 1.

The factorial function is originally defined for natural numbers. However, the Gamma function extends the factorial concept to real and complex numbers. For positive integers,

• \(\Gamma(n+1)\) equals \(n!\), connecting the Gamma function to the factorial function.

Understanding how factorials work is essential when dealing with permutations, combinations, and various mathematical computations. In the context of the Gamma function:

• the factorial appears as a part of the expression \(\Gamma(n+1) = n!\) for integer \(n\).

Recursive Formula

Recursive formulas are equations that express a term in a sequence as a function of preceding terms. The Gamma function satisfies a very important recursive relation:

• \(\Gamma(n+1) = n \Gamma(n)\).

This recursive property builds upon the defined integral form of the Gamma function and allows calculations of the function for higher values based on lower ones. Using this recursive formula:

• If you know \(\Gamma(n)\), you can easily compute \(\Gamma(n+1)\) by multiplying \(\Gamma(n)\) by \(n\).
• For example: \(\Gamma(2) = 1 \cdot \Gamma(1)\), \(\Gamma(3) = 2 \cdot \Gamma(2)\), and so on.

This recursion closely ties the Gamma function to the factorial function, as it mirrors the foundational definition of factorials. This recursive nature simplifies many computations as one can iteratively compute higher values using lower ones.

Integration by Parts

Integration by parts is a fundamental technique in calculus used to integrate products of functions. It's applicable when one function can be easily differentiated and the other easily integrated. It is central to proving the recursive formula of the Gamma function. The rule is derived from the product rule for differentiation and is stated as follows:

• \(\int u \, dv = uv - \int v \, du\).

In proving the recursive formula \(\Gamma(n+1) = n \Gamma(n)\):

• Choose \(u = x^n\) and \(dv = e^{-x} \, dx\).
• Then, \(du = n x^{n-1} \, dx\) and \(v = -e^{-x}\).

Substitute into the integration by parts formula:

• Evaluate the boundary terms at 0 and infinity, both vanishing to 0.
• Simplify the remaining integral to find it equals \(n \Gamma(n)\).

Thus, integration by parts elegantly confirms the recursive property of the Gamma function, solidifying the connection to the factorial.