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.

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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.

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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

Electromagnetism

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