Q17P Write the integral in equation (... [FREE SOLUTION]

Chapter 11: Q17P (page 559)

Write the integral in equation (12.7) as an elliptic integral and show that (12.8)gives its value. Hints: Write $c o s 胃 = 1 - 2 s i n^{2} (\frac{胃}{2})$ and a similar equation for $c o s 伪$ . Then make the change of variable $x = \frac{\sin (\frac{胃}{2})}{\sin (\frac{伪}{2})}$ .

Short Answer

Expert verified

The elliptic form of the given function is $I = \sqrt{2} {鈭�}_{0}^{1} \frac{1}{\sqrt{1 - u^{2}} \sqrt{1 - s i n^{2} \frac{伪}{2} u^{2}}} d 胃$ , and it is proved that $\sqrt{2} {鈭�}_{0}^{1} \frac{1}{\sqrt{1 - u^{2}} \sqrt{1 - \sin^{2} \frac{伪}{2} u^{2}}} d 胃 = \sqrt{2} K (\sin \frac{伪}{2})$ .

Step by step solution

Given Information

The value of integral from equation (12.7) is ${鈭�}_{0}^{伪} \frac{1}{\sqrt{\cos 胃 - \cos 伪}} d 胃$ .

Definition of elliptic form

The elliptic form of the integral is defined as $K (k) = {鈭�}_{0}^{\frac{蟺}{2}} \frac{1}{\sqrt{1 - k^{2} s i n^{2} 胃}} d 胃$ .

Find the value of Integral

Substitute $\cos 胃 = 1 - 2 \sin^{2} \frac{胃}{2}$ into ${鈭�}_{0}^{伪} \frac{1}{\sqrt{\cos 胃 - \cos 伪}} d 胃$ .

$\frac{1}{\sqrt{2}} {鈭�}_{0}^{伪} \frac{1}{\sqrt{\sin^{2} \frac{伪}{2} - \sin^{2} \frac{胃}{2}}} d 胃$

Factor out $\sin^{2} 胃$ in the equation mentioned above.

$I = \frac{1}{\sqrt{2} \sin \frac{伪}{2}} {鈭�}_{0}^{伪} \frac{1}{\sqrt{1 - \frac{\sin^{2} \frac{胃}{2}}{\sin^{2} \frac{伪}{2}}}} d 胃$

Substitute the values given below in the above equation.

$u = \frac{\sin \frac{胃}{2}}{\sin \frac{伪}{2}} d u = \frac{\cos \frac{胃}{2}}{2 \sin \frac{伪}{2}} d 伪$

The equation becomes as follows:

role="math" localid="1664306147751" $I = \sqrt{2} {鈭�}_{0}^{1} \frac{1}{\sqrt{1 - u^{2}} \sqrt{1 - \sin^{2} \frac{伪}{2} u^{2}}} d 胃$

The formula for the elliptic function is $K (k) = {鈭�}_{0}^{\frac{蟺}{2}} \frac{1}{\sqrt{1 - k^{2} \sin^{2} 胃}} d 胃$ .

Equate the above equation with the value of I, the value of I becomes follows:

$I = \sqrt{2} K (\sin \frac{伪}{2})$

Hence, the statement has been proven.

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91影视

Write the integral in equation (12.7) as an elliptic integral and show that (12.8)gives its value. Hints: Write $c o s 胃 = 1 - 2 s i n^{2} (\frac{胃}{2})$ and a similar equation for $c o s 伪$ . Then make the change of variable $x = \frac{\sin (\frac{胃}{2})}{\sin (\frac{伪}{2})}$ .

Short Answer

Step by step solution

Given Information

Definition of elliptic form

Find the value of Integral

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Most popular questions from this chapter

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Particle Model of Matter

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91影视

Write the integral in equation (12.7) as an elliptic integral and show that (12.8)gives its value. Hints: Write cos胃=1-2sin2(胃2) and a similar equation forcos伪. Then make the change of variablex=sin(胃2)sin(伪2).

Short Answer

Step by step solution

Given Information

Definition of elliptic form

Find the value of Integral

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Particle Model of Matter

Solid State Physics

Force

Wave Optics

Physics of Motion

Medical Physics

Study anywhere. Anytime. Across all devices.

Write the integral in equation (12.7) as an elliptic integral and show that (12.8)gives its value. Hints: Write $c o s 胃 = 1 - 2 s i n^{2} (\frac{胃}{2})$ and a similar equation for $c o s 伪$ . Then make the change of variable $x = \frac{\sin (\frac{胃}{2})}{\sin (\frac{伪}{2})}$ .