Chapter 11: Q1P (page 542)

Using (5.3) with (3.4) and (4.1), find $Γ (3 / 2)$ , $Γ (- 1 / 2)$ , and $Γ (- 3 / 2)$ in terms of $\sqrt{π}$ .

Short Answer

Expert verified

The solution of the given gamma functions are as follows:

$\begin{array}{l} Γ (\frac{3}{2}) = \frac{\sqrt{π}}{3} \\ Γ (\frac{- 1}{2}) = - 2 \sqrt{π} \\ Γ (\frac{- 3}{2}) = \frac{4}{3} \sqrt{π} \end{array}$

Step by step solution

Given information

The given Gamma functions are $Γ (\frac{3}{2}), Γ (- \frac{1}{2})$ and $Γ (- \frac{3}{2})$ . Answers need to be expressed in terms of $\sqrt{π}$ .

Definition of a Gamma function

Gamma function is defined as $Γ (p) = \int_{0}^{\infty} x^{p - 1} e^{- x} d x, p > 0$ .

Write the relevant relations for Gamma functions

Write the relations of a Gamma function.

$\begin{matrix} Γ (\frac{1}{2}) = \sqrt{π} \\ Γ (x) = \frac{1}{x} Γ (x + 1) \\ Γ (x + 1) = p Γ (x) \end{matrix}$

Substitute appropriate values and find the value of Gamma functions

Substitute $\frac{3}{2}$ with into $Γ (x) = \frac{1}{x} Γ (x + 1)$ and use the relations of gamma functions.

$\begin{matrix} Γ (\frac{3}{2}) = \frac{1}{2} Γ (\frac{1}{2}) \\ = \frac{\sqrt{π}}{2} \end{matrix}$

Repeat the process by making different substitutions.

Substitute $x$ with $\frac{- 1}{2}$ in the equation.

$\begin{matrix} Γ (x) = \frac{1}{x} Γ (x + 1) \\ Γ (\frac{- 1}{2}) = \frac{1}{\frac{- 1}{2}} Γ (\frac{1}{2}) \\ = - 2 \sqrt{π} \end{matrix}$

Similarly,Substitute x with $\frac{- 3}{2}$ in the equation.

$\begin{matrix} Γ (x) = \frac{1}{x} Γ (x + 1) \\ Γ (\frac{- 3}{2}) = \frac{1}{\frac{- 3}{2}} Γ (\frac{- 1}{2}) \\ = \frac{4}{3} \sqrt{π} \end{matrix}$

Thus, the required gamma functions are , $Γ (\frac{1}{2}) = \frac{\sqrt{π}}{3}$ , $Γ (\frac{- 1}{2}) = - 2 \sqrt{π}$ and $Γ (\frac{- 3}{2}) = \frac{4}{3} \sqrt{π}$ .

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Using (5.3) with (3.4) and (4.1), find $Γ (3 / 2)$ , $Γ (- 1 / 2)$ , and $Γ (- 3 / 2)$ in terms of $\sqrt{π}$ .

Short Answer

Step by step solution

Given information

Definition of a Gamma function

Write the relevant relations for Gamma functions

Substitute appropriate values and find the value of Gamma functions

Repeat the process by making different substitutions.

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Using (5.3) with (3.4) and (4.1), find Γ(3/2),Γ(-1/2), andΓ(-3/2)in terms ofπ.

Short Answer

Step by step solution

Given information

Definition of a Gamma function

Write the relevant relations for Gamma functions

Substitute appropriate values and find the value of Gamma functions

Repeat the process by making different substitutions.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Atoms and Radioactivity

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Scientific Method Physics

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Study anywhere. Anytime. Across all devices.

Using (5.3) with (3.4) and (4.1), find $Γ (3 / 2)$ , $Γ (- 1 / 2)$ , and $Γ (- 3 / 2)$ in terms of $\sqrt{π}$ .