Question

The Gamma Function \(\Gamma(n)\) is defined by \(\Gamma(n)=\int_{0}^{\infty} x^{n-1} e^{-x} d x, \quad n>0\) (a) Find \(\Gamma(1), \Gamma(2),\) and \(\Gamma(3)\). (b) Use integration by parts to show that \(\Gamma(n+1)=n \Gamma(n)\). (c) Write \(\Gamma(n)\) using factorial notation where \(n\) is a positive integer.

Short answer

The evaluated gamma functions are \(\Gamma(1) = 1\), \(\Gamma(2) = 1\), and \(\Gamma(3) = 2\). The property \(\Gamma(n+1)=n\Gamma(n)\) was demonstrated using integration by parts. Lastly, for positive integer \(n\), \(\Gamma(n) = (n-1)!\).

Step-by-step solution

Find Specific Values of the Gamma Function

First, it is necessary to evaluate \(\Gamma(1)\), \(\Gamma(2)\), and \(\Gamma(3)\) using the provided integral. This might require performing integration multiple times. For \(\Gamma(1)\), simply substitute \(n=1\) into the definition: \(\Gamma(1) = \int_{0}^{\infty}x^{1-1}e^{-x}dx = \int_{0}^{\infty}e^{-x}dx\). Calculating this integral yields \(\Gamma(1) = 1\). Repeat the process for \(\Gamma(2)\) and \(\Gamma(3)\).

Demonstrate the Recurrence Property Using Integration by Parts

The next step is to prove the property \(\Gamma(n+1)=n\Gamma(n)\) using the method of integration by parts. Start with the integral formula for \(\Gamma(n+1)\) and then apply the integration by parts formula \(\int udv = uv - \int vdu\). It might be helpful to let \(u = x^n\) and \(dv = e^{-x}dx\), then proceed from there. Make sure to get \(\Gamma(n)\) on one side and everything else on the other side.

Express the Gamma Function in Terms of Factorials

Finally, rewrite the gamma function \(\Gamma(n)\) for positive integers \(n\) in terms of factorials. This can be done by observing the previous results and finding a pattern. The definition of factorial, \(n!\), is the product of all positive integers less than or equal to \(n\). Notice the structure of \(\Gamma(n) = (n-1)\Gamma(n-1)\) and think about how it aligns with the definition of factorial.

Key concepts

Integration By Parts

The concept of integration by parts is a critical tool in calculus, particularly useful when dealing with products of functions whose integrals are known. Loosely speaking, it's the integration counterpart to the product rule used in differentiation. When applied strategically, it allows us to break down complex integrals into simpler parts that are easier to tackle.

The formula for integration by parts can be stated as \[ \text{If } u = f(x), dv = g(x)dx, \text{then } \frac{du}{dx} = f'(x), v = \text{the integral of } g(x)dx, \text{and the integration by parts is defined as} \int u dv = uv - \text{the integral of }v du. \] This method often requires choosing 'u' and 'dv' in a way that simplifies the resulting integral, and sometimes it must be repeated several times or combined with other techniques to achieve the final solution.

Factorial Notation

Factorial notation is a mathematical expression that greatly simplifies the description of multiplying a series of descending natural numbers. The notation for the factorial of a non-negative integer 'n' is 'n!'. It is defined as the product of all positive integers less than or equal to 'n'.

Mathematically, it is expressed as \[ n! = n \times (n-1) \times (n-2) \times ... \times 2 \times 1. \] Factorial notation is particularly useful in permutations, combinations, and other areas of combinatorics. It's also seen in the coefficients of a binomial expansion and plays a key role in probability and various mathematical series.

• The factorial of 0 is defined to be 1 (0! = 1).
• Factorials only apply to non-negative integers.

In the context of the Gamma function, factorial notation becomes relevant when working with positive integer inputs, which is reflected in \( \Gamma(n+1) = n \Gamma(n) \) and in turn suggests a profound relationship between factorials and the Gamma function.

Indefinite Integrals

Indefinite integrals, though seemingly less concrete than definite integrals, are of fundamental importance in calculus. They represent a family of functions with an indefinite number of members rather than a single numerical value. When dealing with an indefinite integral, you're essentially finding the most general form of the antiderivative of a function.

Given \( f(x) \) whose integral is to be found, we write \[ \text{the indefinite integral of } f(x) dx = F(x) + C, \] where \( F(x) \) is the antiderivative of \( f(x) \) and \( C \) is the constant of integration. This constant of integration represents an infinite number of possible constant values, reflecting the fact that there are infinitely many antiderivatives for a given function, each varying by just a constant.

Mastering the process of finding indefinite integrals is essential for solving a broad range of problems in calculus, including those related to the Gamma function, where indefinite integration allows for the expression of complex recursive relationships inherent in functions like the factorials and others.