Chapter 12: Q4P (page 597)

Prove the given system of equations $I_{1 / 2} (x) = \sqrt (\frac{2}{π (x)} s i n h (x)$ and $K_{1 / 2} (x) = \sqrt (\frac{π}{2 (x)} e^{(- x)} .$

Short Answer

Expert verified

The given system of equations $K_{1 / 2} (x) = \sqrt (\frac{π}{2 (x)} e^{(- x)}$ and $I_{1 / 2} (x) = \sqrt (\frac{2}{π (x)} s i n h (x)$ are proved.

Step by step solution

Concept of recursion relation:

The recursion relation is given as follows:

$\frac{d}{d x} (x^{- p} J_{p} (x)) = - x^{- 1} J_{p} + 1 (z)$

Prove the given system of equations 1/2(x)=√(2π(x)sinh(x) and K1/2(x)=√(π2(x)e(-x).

Consider the given equations as below.

$K_{1 / 2} (x) = \sqrt (\frac{π}{2 (x)} e^{(- x)} .$

And

$I_{1 / 2} (x) = \sqrt (\frac{2}{π (x)} s i n h (x)$

Compare the given equation with standard equation as follows:

$K (x) = {(i)}^{- p} J_{P} (i x) = \frac{π}{2} {(i)}^{p + 1} H_{P} (i x) = \frac{π}{2} {(i)}^{p + 1} [J_{P} (i x) + i N_{P} (I X)]$

Simplify further as follows:

role="math" localid="1659266495025" $l_{1 l 2} (x) = i^{(- \frac{1}{2})} l_{1 l 2} (i x) = i^{(- \frac{1}{2})} \sqrt{\frac{2}{π i (x)}} \sin (i x) = \sqrt{\frac{2}{π i (x)}} \sin (i x) k_{\frac{1}{2}} (x) = \frac{π}{2} {(i)}^{\frac{3}{2}} (\sqrt{\frac{2}{π i (x)}} \sin (x) - (i) {(i)}^{- \frac{1}{2}} \sqrt{\frac{2}{π i (x)}} \cos i (x))$

Simplify further as follows:

$k_{\frac{1}{2}} (x) = \sqrt{\frac{π}{2 (x)}} (\cos h (x) - \sin h (x)) = \sqrt{\frac{π}{2 (x)}} {(e)}^{- x} J_{\frac{1}{2}} (i x) + i Y_{\frac{1}{2}} (i x)) k_{\frac{1}{2}} (x) = \sqrt{\frac{π}{2 (x)}} {(e)}^{- x}$

Hence, it 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.

Prove the given system of equations $I_{1 / 2} (x) = \sqrt (\frac{2}{π (x)} s i n h (x)$ and $K_{1 / 2} (x) = \sqrt (\frac{π}{2 (x)} e^{(- x)} .$

Short Answer

Step by step solution

Concept of recursion relation:

Prove the given system of equations 1/2(x)=√(2π(x)sinh(x) and K1/2(x)=√(π2(x)e(-x).

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Rotational Dynamics

Force

Modern Physics

Fields in Physics

Space Physics

Translational Dynamics

Study anywhere. Anytime. Across all devices.

Prove the given system of equations I1/2(x)=√(2π(x)sinh(x)andK1/2(x)=√(π2(x)e(-x).

Short Answer

Step by step solution

Concept of recursion relation:

Prove the given system of equations 1/2(x)=√(2π(x)sinh(x) and K1/2(x)=√(π2(x)e(-x).

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Rotational Dynamics

Force

Modern Physics

Fields in Physics

Space Physics

Translational Dynamics

Study anywhere. Anytime. Across all devices.

Prove the given system of equations $I_{1 / 2} (x) = \sqrt (\frac{2}{π (x)} s i n h (x)$ and $K_{1 / 2} (x) = \sqrt (\frac{π}{2 (x)} e^{(- x)} .$