Problem 95
Question
Show that \(\Gamma\left(\frac{7}{2}\right)=\frac{15}{8} \sqrt{\pi}\).
Step-by-Step Solution
Verified Answer
\(\Gamma\left(\frac{7}{2}\right) = \frac{15}{8} \sqrt{\pi}\)
1Step 1: Use the gamma function's offset property
Using the property \(\Gamma \left( z+1 \right) = z \cdot \Gamma \left(z\right)\), we can express \(\Gamma\left(\frac{7}{2}\right)\) as \( \Gamma\left(\frac{5}{2}+1\right) = \frac{5}{2} \cdot \Gamma \left(\frac{5}{2}\right) \). Now we repeat and use this property until we reach \(\Gamma\left(\frac{1}{2}\right)\), we will get: \(= \frac{5}{2} \cdot \frac{3}{2} \cdot \Gamma \left(\frac{3}{2}\right) = \frac{5}{2} \cdot \frac{3}{2} \cdot \frac{1}{2} \cdot \Gamma \left(\frac{1}{2}\right)\)
2Step 2: Substitute for the known value of the gamma function
As we know that \(\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}\), by substituting this in the equation we get: \(= \frac{5}{2} \cdot \frac{3}{2} \cdot \frac{1}{2} \cdot \sqrt{\pi} = \frac{15}{8} \sqrt{\pi}\)
Key Concepts
Special FunctionsMathematical ProofsProperties of Gamma Function
Special Functions
Special functions play a significant role in mathematics, providing solutions to various problems across different fields. They are not elementary functions like polynomials or trigonometric functions, instead they have unique definitions and properties. These functions emerge in complex mathematical calculations and their study helps solve equations that would otherwise be difficult or impossible to tackle with elementary methods.
The Gamma Function, denoted \(\Gamma(z)\), is a prime example of a special function. It is often used in probability and statistics, particularly in the context of factorials for non-integer values. In essence, the gamma function is an extension of the factorial function, with \(\Gamma(n) = (n-1)!\) for integers. The uniqueness of the gamma function comes from its ability to handle complex numbers, which allows it to be utilized in advanced mathematical fields.
Understanding special functions like the gamma function can dramatically enhance one's ability to solve complex mathematical problems. This connection to factorials for non-integers often appears in various proofs and computations, making the gamma function incredibly valuable.
The Gamma Function, denoted \(\Gamma(z)\), is a prime example of a special function. It is often used in probability and statistics, particularly in the context of factorials for non-integer values. In essence, the gamma function is an extension of the factorial function, with \(\Gamma(n) = (n-1)!\) for integers. The uniqueness of the gamma function comes from its ability to handle complex numbers, which allows it to be utilized in advanced mathematical fields.
Understanding special functions like the gamma function can dramatically enhance one's ability to solve complex mathematical problems. This connection to factorials for non-integers often appears in various proofs and computations, making the gamma function incredibly valuable.
Mathematical Proofs
Mathematical proofs are fundamental to validating the correctness of mathematical statements. They involve a logical sequence of steps that demonstrate the truth of a proposition or formula. In the context of the gamma function, proofs often rely on known properties and values to simplify and solve expressions.
The problem we are considering demonstrates the use of a proof involving the gamma function. To prove \(\Gamma\left(\frac{7}{2}\right)=\frac{15}{8} \sqrt{\pi}\), a series of transformations using the offset property of the gamma function is invaluable. This property, \(\Gamma(z+1) = z \cdot \Gamma(z)\), simplifies complex expressions by breaking them down step-by-step; it allows us to connect higher arguments of the gamma function to lower, previously known solutions.
The problem we are considering demonstrates the use of a proof involving the gamma function. To prove \(\Gamma\left(\frac{7}{2}\right)=\frac{15}{8} \sqrt{\pi}\), a series of transformations using the offset property of the gamma function is invaluable. This property, \(\Gamma(z+1) = z \cdot \Gamma(z)\), simplifies complex expressions by breaking them down step-by-step; it allows us to connect higher arguments of the gamma function to lower, previously known solutions.
- Initiate with finding \(\Gamma\left(\frac{7}{2}\right)\) using the offset property.
- Reduce it step-by-step to reach \(\Gamma\left(\frac{1}{2}\right)\), aiding the substitution of known values.
- This approach relies on logically ordered substitutions and calculations, which build from established values and properties.
Properties of Gamma Function
The properties of gamma function are crucial for its applications and the simplification of complex equations. Understanding these properties enables mathematicians to extend factorials to non-integer values, solve challenging equations, and explore new areas like complex analysis.
Among its essential properties is the aforementioned offset property, \(\Gamma(z+1) = z \cdot \Gamma(z)\), which allows recursive calculation of gamma values starting from known base points such as \(\Gamma\left(\frac{1}{2}\right) = \sqrt{\pi}\). This property is instrumental in deriving values such as \(\Gamma\left(\frac{7}{2}\right)\) by successively reducing the problem to simpler terms.
Among its essential properties is the aforementioned offset property, \(\Gamma(z+1) = z \cdot \Gamma(z)\), which allows recursive calculation of gamma values starting from known base points such as \(\Gamma\left(\frac{1}{2}\right) = \sqrt{\pi}\). This property is instrumental in deriving values such as \(\Gamma\left(\frac{7}{2}\right)\) by successively reducing the problem to simpler terms.
- Recursion: It enables calculations of non-integer "factorials" through repeated application.
- Normalization: \(\Gamma\left(1\right) = 1\), analogously to \(0! = 1\).
- Reflection Formula: Helps in connecting the values of \(\Gamma(z)\) and \(\Gamma(1-z)\).
Other exercises in this chapter
Problem 93
Show that a gamma pdf has the unique mode \(\frac{r-1}{\lambda}\); that is, show that the function \(f_{Y}(y)=\frac{\alpha^{\prime}}{\Gamma(r)} y^{\prime-1} e^{
View solution Problem 94
Prove that \(\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}\). (Hint: Consider \(E\left(Z^{2}\right)\), where \(Z\) is a standard normal random variable.)
View solution Problem 96
If the random variable \(Y\) has the gamma pdf with integer parameter \(r\) and arbitrary \(\lambda>0\), show that $$ E\left(Y^{m}\right)=\frac{(m+r-1) !}{(r-1)
View solution Problem 92
Demonstrate that \(\lambda\) plays the role of a scale parameter by showing that if \(Y\) is gamma with parameters \(r\) and \(\lambda\), then \(\lambda Y\) is
View solution