Problem 146 - Applied Statistics and Probability for Engineers

Question

Use integration by parts to show that \(\Gamma(r)=(r-1)\) \(\Gamma(r-1).\)

Step-by-Step Solution

Answer

The identity \( \Gamma(r) = (r-1) \Gamma(r-1) \) is verified using integration by parts.

Step 1: Understanding the Problem

We are given the Gamma function, denoted \( \Gamma(r) = \int_0^{\infty} t^{r-1} e^{-t} \, dt \).We need to show that \( \Gamma(r) = (r-1) \Gamma(r-1) \). This identity represents a recursive property of the Gamma function.

Step 2: Recall Integration by Parts Formula

The integration by parts formula is given by \[ \int u \, dv = uv - \int v \, du \].We will choose the functions \( u \) and \( dv \) from the integrand \( t^{r-1} e^{-t} \).

Step 3: Choose Functions for Integration by Parts

Choose \( u = t^{r-1} \) and \( dv = e^{-t} \, dt \).Then, differentiate and integrate to find \( du \) and \( v \):- \( du = (r-1) t^{r-2} \, dt \)- \( v = -e^{-t} \).

Step 4: Apply Integration by Parts

Substituting \( u \), \( du \), \( v \), and \( dv \) into the integration by parts formula:\[\int t^{r-1} e^{-t} \, dt = \left[ -t^{r-1} e^{-t} \right]_0^{\infty} + (r-1) \int t^{r-2} e^{-t} \, dt\].

Step 5: Evaluate Boundary Terms

Evaluate the boundary term:\[ \lim_{t \to \infty} -t^{r-1} e^{-t} = 0 \] because exponential decay dominates polynomial growth, and:\[ \left[ -t^{r-1} e^{-t} \right]_0 = 0 \] because as \( t \to 0 \), \( e^{-t} \to 1 \) leaving \( 0 \).

Step 6: Simplify and Conclude

After evaluating the boundary, we have:\[0 = 0 + (r-1) \int t^{r-2} e^{-t} \, dt\]So, \(\Gamma(r) = (r-1) \Gamma(r-1)\), where \( \Gamma(r-1) = \int t^{r-2} e^{-t} \, dt \).This completes the proof as required.

Key Concepts

Recursive Property

The Gamma function, a complex mathematical construct, has an intriguing recursive property. This recursive nature is expressed in the relationship \( \Gamma(r) = (r-1) \Gamma(r-1) \). Exploring this recursive property allows us to extend the understanding of factorials to non-integer values. The Gamma function generalizes the factorial function, where for natural numbers:

• \( \Gamma(n) = (n-1)! \)
• \( \Gamma(1) = 0! = 1 \)

To leverage this recursive property, we argue that solving \( \Gamma(r) \) can always involve expressing it in terms of \( \Gamma(r-1) \). This insight is invaluable when dealing with complex calculus problems by simplifying them through recursion.

Integration by Parts

Integration by parts is a technique from calculus used to tackle integral expressions like \( \int u \, dv = uv - \int v \, du \). It's especially useful when dealing with products of functions, which is often the case with the Gamma function. To solve \( \Gamma(r) = \int_0^{\infty} t^{r-1} e^{-t} \, dt \) using integration by parts:

• Choose \( u = t^{r-1} \), hence \( du = (r-1) t^{r-2} \, dt \)
• Choose \( dv = e^{-t} \, dt \), leading to \( v = -e^{-t} \)

Applying these choices to the by-parts formula splits the integral into terms that are more manageable, simplifying its computation as each term becomes easier to evaluate. This shows the connection between two consecutive values \( \Gamma(r) \) and \( \Gamma(r-1) \).

Calculus

Calculus is the branch of mathematics that studies continuous change and includes the study of integrals and derivatives. The Gamma function, \( \Gamma(r) \), plays a critical role in advanced calculus, particularly in dealing with problems that extend beyond integer domains. Calculus allows us to handle functions like the Gamma function, which are not easily simplified using basic algebra. It uses integration, an essential concept, that helps verify properties like the recursive property of the Gamma function. By utilizing calculus tools such as integration by parts, mathematicians can unpack complex behavior within functions and solve higher-level problems. This approach transforms seemingly convoluted expressions into manageable forms by iterating simpler operations.

Mathematical Proof

A mathematical proof is a logical argument that verifies the truth of a statement based on established axioms and previous results. Proving \( \Gamma(r) = (r-1) \Gamma(r-1) \) involves:

• Setting up the problem using definitions, such as the definition of the Gamma function
• Applying techniques like integration by parts to manipulate the representation of \( \Gamma(r) \)
• Evaluating boundary terms to handle limits as \( t \to \infty \) or \( t \to 0 \)

The process concludes by showing that both sides of the equation are equal, completing the logical argument. Proofs are foundational in mathematics because they affirm the validity of equations we frequently use without constant re-verification.