Q4P Prove the given system of equati... [FREE SOLUTION]

Chapter 12: Q4P (page 597)

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

Short Answer

Expert verified

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

$K_{1 / 2} (x) = 鈭� (\frac{蟺}{2 (x)} e^{(- x)} .$

And

$I_{1 / 2} (x) = 鈭� (\frac{2}{蟺 (x)} s i n h (x)$

Compare the given equation with standard equation as follows:

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

Simplify further as follows:

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

Simplify further as follows:

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

Hence, it proved.

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91影视

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

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

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91影视

Prove the given system of equations I1/2(x)=鈭�(2蟺(x)sinh(x)andK1/2(x)=鈭�(蟺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

Physical Quantities And Units

Physics of Motion

Torque and Rotational Motion

Solid State Physics

Measurements

Magnetism and Electromagnetic Induction

Study anywhere. Anytime. Across all devices.

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