Chapter 12: Q9P (page 590)

To show that, $\sqrt{ττx /} 2 J_{1 / 2} (x) = \sin x$ .

Short Answer

Expert verified

Hence, $\sqrt{\frac{πx}{2}} J_{\frac{1}{2}} (x) = \sin x$ is proved.

Step by step solution

Concept of the Infinite series:

Infinite series:

$J_{p} (x) = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{⌜ (n + 1) ⌜ (n + 1 + p)} {(\frac{x}{2})}^{2 n + p}$

Calculation of the function ττx/2J1/2(x) :

Considering the provided statement to show that the provided infinite series for $J_{p} (x)$ converges for all the values of x by the ratio test.

The infinite series is,

$\sqrt{\frac{π x}{2}} J_{\frac{1}{2}} (x) = \sin x$

Substituting the series for $\frac{1}{2}$ for p in the equation (1) as follows:

$\sqrt{\frac{πx}{2}} J_{\frac{1}{2}} (x) = \sqrt{\frac{πx}{2}} \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{⌜ (n + 1) ⌜ (n + 1 + \frac{1}{2})} {(\frac{x}{2})}^{2 π + \frac{1}{2}} = \sqrt{\frac{πx}{2}} \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{n! (n + \frac{1}{2}) (n + \frac{1}{2} - 1) . . . . . . . . . . \frac{1}{2} ⌜ (\frac{1}{2}) 2^{π} \sqrt{2}} x^{2 n} = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{(2 n + 1)!} x^{2 n + 1} = \sin x$

Hence, $\sqrt{\frac{π x}{2}} J_{\frac{1}{2}} (x) = \sin x is proved$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

To show that, $\sqrt{ττx /} 2 J_{1 / 2} (x) = \sin x$ .

Short Answer

Step by step solution

Concept of the Infinite series:

Calculation of the function ττx/2J1/2(x) :

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Oscillations

Fields in Physics

Kinematics Physics

Modern Physics

Atoms and Radioactivity

Geometrical and Physical Optics

Study anywhere. Anytime. Across all devices.

To show that, ττx/2J1/2(x)=sinx.

Short Answer

Step by step solution

Concept of the Infinite series:

Calculation of the function ττx/2J1/2(x) :

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Oscillations

Fields in Physics

Kinematics Physics

Modern Physics

Atoms and Radioactivity

Geometrical and Physical Optics

Study anywhere. Anytime. Across all devices.

To show that, $\sqrt{ττx /} 2 J_{1 / 2} (x) = \sin x$ .