Q11.8P Expand the integrands of K聽and ... [FREE SOLUTION]

Chapter 11: Q11.8P (page 554)

Expand the integrands of Kand E[see ( 12.3 )] in power series in $k^{2} s i n^{2} 胃$ (assuming small k ), and integrate term by term to find power series approximations for the complete elliptic integrals Kand E.

Short Answer

Expert verified

The value of integral in elliptic integrals are given below.

$K = \frac{蟺}{2} + \frac{{蟺办}^{2}}{2^{3}} + \frac{9 {蟺办}^{4}}{2^{7}} + . . E = \frac{蟺}{2} + \frac{{蟺办}^{2}}{2^{3}} - \frac{3 {蟺办}^{4}}{2^{7}}$

Step by step solution

Given Information

The function is $k^{2} \sin^{2} 胃$ .

Definition of elliptic form.

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

Find the value of K.

The function is $k^{2} \sin^{2} 胃$ .

The formula states that $\frac{1}{\sqrt{1 - x}} = 1 + \frac{x}{2} + \frac{3 x^{2}}{8} + \frac{3 x^{3}}{16} + . . .$ .

$K = {鈭�}_{0}^{\frac{蟺}{2}} d 胃 + {鈭�}_{0}^{\frac{蟺}{2}} \sin^{2} 胃 d 胃 + \frac{3 k^{4}}{8} {鈭�}_{0}^{\frac{蟺}{2}} \sin^{4} 胃 d 胃 + . . K = \frac{蟺}{2} + \frac{{蟺办}^{2}}{2^{3}} + \frac{9 {蟺办}^{4}}{2^{7}} + . .$

Find the value of E.

The function is $k^{2} \sin^{2} 胃$ .

The formula states that $\sqrt{1 + x} = 1 - \frac{x}{2} - \frac{x^{2}}{8} - \frac{x^{3}}{16} + . . .$ .

$E = {鈭�}_{0}^{\frac{蟺}{2}} d 胃 - \frac{k^{2}}{2} {鈭�}_{0}^{\frac{蟺}{2}} \sin^{2} 胃 d 胃 - \frac{k^{4}}{8} {鈭�}_{0}^{\frac{蟺}{2}} \sin^{4} 胃 d 胃 + . . E = \frac{蟺}{2} - \frac{{蟺办}^{2}}{2^{3}} - \frac{3 {蟺办}^{4}}{2^{7}} + . .$

Hence, the value of integral in elliptic integrals are given below.

$K = \frac{蟺}{2} - \frac{{蟺办}^{2}}{2^{3}} - \frac{9 {蟺办}^{4}}{2^{7}} + . . E = \frac{蟺}{2} - \frac{{蟺办}^{2}}{2^{3}} - \frac{3 {蟺办}^{4}}{2^{7}}$

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Expand the integrands of Kand E[see ( 12.3 )] in power series in $k^{2} s i n^{2} 胃$ (assuming small k ), and integrate term by term to find power series approximations for the complete elliptic integrals Kand E.

Short Answer

Step by step solution

Given Information

Definition of elliptic form.

Find the value of K.

Find the value of E.

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91影视

Expand the integrands of Kand E[see ( 12.3 )] in power series ink2sin2胃(assuming small k ), and integrate term by term to find power series approximations for the complete elliptic integrals Kand E.

Short Answer

Step by step solution

Given Information

Definition of elliptic form.

Find the value of K.

Find the value of E.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Oscillations

Fundamentals of Physics

Dynamics

Solid State Physics

Conservation of Energy and Momentum

Magnetism and Electromagnetic Induction

Study anywhere. Anytime. Across all devices.

Expand the integrands of Kand E[see ( 12.3 )] in power series in $k^{2} s i n^{2} 胃$ (assuming small k ), and integrate term by term to find power series approximations for the complete elliptic integrals Kand E.