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Chapter 4: Problem 137

Use the properties of the gamma function to evaluate the following: (a) \(\Gamma(6)\) (b) \(\Gamma(5 / 2)\) (c) \(\Gamma(9 / 2)\)

Short Answer

Expert verified
\( \Gamma(6) = 120 \), \( \Gamma(5/2) = \frac{3}{4} \sqrt{\pi} \), \( \Gamma(9/2) = \frac{105}{16} \sqrt{\pi} \).

Step by step solution

01

Understanding the Gamma Function

The gamma function, denoted as \( \Gamma(n) \), is defined for positive values as \( \Gamma(n) = (n-1)! \) for natural numbers \(n\). For non-integer values, it is expressed using the property: \( \Gamma(n+1) = n \Gamma(n) \). These properties will help evaluate the given problems.
02

Evaluate \(\Gamma(6)\)

Since \(6\) is a natural number, apply the definition directly: \( \Gamma(6) = (6-1)! = 5! \). Now, compute \(5!\): \[5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\] Thus, \(\Gamma(6) = 120\).
03

Evaluate \(\Gamma(5/2)\)

To evaluate \( \Gamma\left(\frac{5}{2}\right) \), use the recursive property \( \Gamma\left(n+1\right) = n\Gamma\left(n\right) \). Start with:- \[ \Gamma\left(\frac{5}{2}\right) = \frac{3}{2} \Gamma\left(\frac{3}{2}\right) \]- \[ \Gamma\left(\frac{3}{2}\right) = \frac{1}{2} \Gamma\left(\frac{1}{2}\right) \]Using the known value \( \Gamma\left(\frac{1}{2}\right) = \sqrt{\pi} \), we find:- \( \Gamma\left(\frac{3}{2}\right) = \frac{1}{2} \sqrt{\pi} \)- \( \Gamma\left(\frac{5}{2}\right) = \frac{3}{2} \times \frac{1}{2} \sqrt{\pi} = \frac{3}{4} \sqrt{\pi} \)
04

Evaluate \(\Gamma(9/2)\)

Apply the recursive property multiple times to find \( \Gamma\left(\frac{9}{2}\right) \):- \[ \Gamma\left(\frac{9}{2}\right) = \frac{7}{2} \Gamma\left(\frac{7}{2}\right) \]- \[ \Gamma\left(\frac{7}{2}\right) = \frac{5}{2} \Gamma\left(\frac{5}{2}\right) \]We've already calculated \( \Gamma\left(\frac{5}{2}\right) = \frac{3}{4} \sqrt{\pi} \). Substitute to find:- \( \Gamma\left(\frac{7}{2}\right) = \frac{5}{2} \times \frac{3}{4} \sqrt{\pi} = \frac{15}{8} \sqrt{\pi} \)- \( \Gamma\left(\frac{9}{2}\right) = \frac{7}{2} \times \frac{15}{8} \sqrt{\pi} = \frac{105}{16} \sqrt{\pi} \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Properties of Gamma Function
The Gamma function, denoted as \( \Gamma(n) \), plays a crucial role in various areas of mathematics, particularly in complex analysis and calculus. It extends the concept of factorials to real and complex numbers. Here are some key properties you should know:

  • For natural numbers, the Gamma function relates directly to the factorial, where \( \Gamma(n) = (n-1)! \).
  • The function is defined for positive real numbers and complex numbers with a real part greater than zero.
  • It is a smooth and continuous function, despite being derived from the discrete factorial concept.

Knowing these properties helps students approach problems involving the Gamma function with confidence, understanding its broader application in integrals and probability theory.
Factorial and Gamma Function
The relationship between the factorial and the Gamma function is foundational to understanding how Gamma generalizes the factorial concept beyond integers. For any natural number \( n \), the Gamma function is defined as:

\[ \Gamma(n) = (n-1)! \]

This formula allows the calculation of factorials using the continuous Gamma function for positive integers.

Furthermore, the Gamma function handles non-integers. When employed for half-integers such as \( \Gamma(5/2) \), it extends its utility beyond whole numbers. Knowing that \( \Gamma(1/2) = \sqrt{\pi} \) is crucial because it provides a starting point to compute functions like \( \Gamma(5/2) \), illustrating the flexibility of the Gamma function in mathematical problems.
Recursive Property of Gamma Function
The recursive property of the Gamma function is essential for calculating its value at non-integer and fractional points. This property states that for any \( n \),

\[ \Gamma(n+1) = n \Gamma(n) \]

This formula is especially powerful because it helps calculate Gamma values sequentially, building from known values. For example, knowing \( \Gamma(1/2) = \sqrt{\pi} \) provides us a basis to derive \( \Gamma(3/2) \) and further to \( \Gamma(5/2) \), and so on.

When applied, this property allows students to solve complex Gamma functions by simplifying them into known calculations. It acts as a stepping stone, bridging complex equations from what may initially appear as intimidating, into digestible parts. Understanding and utilizing this recursive property is key for mastering problems involving the Gamma function.

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Most popular questions from this chapter

When a bus service reduces fares, a particular trip from New York City to Albany, New York, is very popular. A small bus can carry four passengers. The time between calls for tickets is exponentially distributed with a mean of 30 minutes. Assume that each caller orders one ticket. What is the probability that the bus is filled in less than three hours from the time of the fare reduction?

The total service time of a multistep manufacturing operation has a gamma distribution with mean 18 minutes and standard deviation 6. (a) Determine the parameters \(\lambda\) and \(r\) of the distribution. (b) Assume that each step has the same distribution for service time. What distribution for each step and how many steps produce this gamma distribution of total service time?

The time between the arrival of electronic messages at your computer is exponentially distributed with a mean of two hours. (a) What is the probability that you do not receive a message during a two- hour period? (b) If you have not had a message in the last four hours, what is the probability that you do not receive a message in the next two hours? (c) What is the expected time between your fifth and sixth messages?

Suppose that \(X\) has a lognormal distribution with parameters \(\theta=10\) and \(\omega^{2}=16\). Determine the following: (a) \(P(X<2000)\) (b) \(P(X>1500)\) (c) Value exceeded with probability 0.7

Suppose that \(X\) has a beta distribution with parameters \(\alpha=2.5\) and \(\beta=2.5 .\) Sketch an approximate graph of the probability density function. Is the density symmetric?
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