Chapter 11: Q7P (page 542)

In the Table of Laplace Transforms (end of Chapter 8, page 469), verify $Γ$ the function results for L5 and L6. Also show that $L (1 / \sqrt{t}) = \sqrt{π / p}$ .

Short Answer

Expert verified

The following are proved:

(a)The Gamma function $Γ$ results for $L 5, L 6$ .

(b) $L (1 / \sqrt{t}) = \sqrt{π / p}$

Step by step solution

Given Information

The value of a common Gamma function of $\frac{1}{2}$ .

$Γ (\frac{1}{2}) = \sqrt{π}$

Definition of a Gamma function

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

Begin with the definition of the Gamma function.

Start with the definition of a Gamma Function.

$Γ (p) = \int_{0}^{\infty} x^{p - 1} e^{- x} d x, p > 0$

Make the following substitutions in this definition.

$\begin{array}{l} u = p t \\ t = \frac{u}{p} \\ d t = \frac{d u}{p} \end{array}$

Simplify after substitutions.

$\begin{matrix} \int_{0}^{\infty} {(\frac{u}{p})}^{k} e^{- u} \frac{d u}{p} = \frac{1}{p^{k + 1}} \int_{0}^{\infty} u^{k} e^{- u} d u \\ = \frac{1}{p^{k + 1}} Γ (k + 1) \end{matrix}$

Repeat the process for different substitutions

Make the following substitutions in the definition.

$\begin{array}{l} u = (a + p) t \\ t = \frac{u}{a + p} \\ d t = \frac{d u}{(a + p)} \end{array}$

Simplify after making the substitutions.

$\begin{matrix} \int_{0}^{\infty} {(\frac{u}{a + p})}^{k} e^{- u} \frac{d u}{a + p} = \frac{1}{{(a + p)}^{k + 1}} \int_{0}^{\infty} u^{k} e^{- u} d u \\ = \frac{Γ (k + 1)}{{(a + p)}^{k + 1}} \end{matrix}$

Repeat the process again for another substitution

Make the following substitutions in the definition.

$\begin{array}{l} u = p t \\ t = \frac{u}{p} \\ d t = \frac{d u}{p} \end{array}$

Simplify after making the substitutions.

$\begin{matrix} \int_{0}^{\infty} {(\frac{u}{p})}^{- 1 / 2} e^{- u} \frac{d u}{p} = \int_{0}^{\infty} u^{- 1 / 2} e^{- u} d u \\ = \frac{Γ (\frac{1}{2})}{p^{1 / 2}} \\ = \sqrt{\frac{π}{p}} \end{matrix}$

Hence the given statement is proved.

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In the Table of Laplace Transforms (end of Chapter 8, page 469), verify $Γ$ the function results for L5 and L6. Also show that $L (1 / \sqrt{t}) = \sqrt{π / p}$ .

Short Answer

Step by step solution

Given Information

Definition of a Gamma function

Begin with the definition of the Gamma function.

Repeat the process for different substitutions

Repeat the process again for another substitution

One App. One Place for Learning.

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In the Table of Laplace Transforms (end of Chapter 8, page 469), verifyΓthe function results for L5 and L6. Also show thatL(1/t)=π/p.

Short Answer

Step by step solution

Given Information

Definition of a Gamma function

Begin with the definition of the Gamma function.

Repeat the process for different substitutions

Repeat the process again for another substitution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Particle Model of Matter

Fundamentals of Physics

Kinematics Physics

Electricity and Magnetism

Fluids

Wave Optics

Study anywhere. Anytime. Across all devices.

In the Table of Laplace Transforms (end of Chapter 8, page 469), verify $Γ$ the function results for L5 and L6. Also show that $L (1 / \sqrt{t}) = \sqrt{π / p}$ .