Chapter 11: Q21P (page 559)

Show that the four answers given in Section 1 for $\int_{0}^{\frac{π}{2}} \frac{dθ}{\sqrt{cosθ}}$ are all correct. Hints: For the beta function result, use(6.4). Then get the gamma function results by using (7.1) and the various Γ function formulas. For the elliptic integral, use the hint of Problem 17 with $α = \frac{π}{2}$ .

Short Answer

Expert verified

The solution is mentioned below.

$I = \sqrt{2} K (\frac{1}{\sqrt{2}}) I = β (\frac{1}{2}, \frac{1}{4}) I = \frac{Γ \frac{1}{2} Γ \frac{1}{4}}{Γ \frac{3}{4}}$

Step by step solution

Given Information

The value of integration is $α = \int_{0}^{\frac{π}{2}} \frac{1}{\sqrt{\cos θ}} d θ$ .

Definition of elliptic form.

The elliptic form of the integral is defined as $K (k) = \int_{0}^{\frac{π}{2}} \frac{1}{\sqrt{1 - k^{2} \sin^{2} θ}} d θ$ .

Find the value of Integral

The value of integration is $α = \int_{0}^{\frac{π}{2}} \frac{1}{\sqrt{\cos θ}} d θ$ .

Substitute $\cos θ = 1 - 2 \sin^{2} \frac{θ}{2}$

The integral becomes as follows

$α = \int_{0}^{\frac{π}{2}} \frac{1}{\sqrt{1 - 2 \sin^{2} \frac{θ}{2}}} d θ$

Substitute the values given below in the above equation.

$x^{2} = 2 \sin^{2} \frac{θ}{2} d x = \frac{\sqrt{2}}{2} \cos \frac{θ}{2} d θ$

The equation becomes as follows.

$I = \sqrt{2} K (\frac{1}{\sqrt{2}}) I = β (\frac{1}{2}, \frac{1}{4}) I = \frac{Γ \frac{1}{2} Γ \frac{1}{4}}{Γ \frac{3}{4}}$

Hence, The solution is mentioned below.

$I = \sqrt{2} K (\frac{1}{\sqrt{2}}) I = β (\frac{1}{2}, \frac{1}{4}) I = \frac{Γ \frac{1}{2} Γ \frac{1}{4}}{Γ \frac{3}{4}} I = \sqrt{2} \int_{0}^{1} \frac{1}{\sqrt{1 - x^{2}} \sqrt{1 - \frac{1}{2 x^{2}}}} d x$

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Short Answer

Step by step solution

Given Information

Definition of elliptic form.

Find the value of Integral

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Show that the four answers given in Section 1 for ∫0π2dθcosθ are all correct. Hints: For the beta function result, use(6.4). Then get the gamma function results by using (7.1) and the various Γ function formulas. For the elliptic integral, use the hint of Problem 17 withα=π2.

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

Atoms and Radioactivity

Nuclear Physics

Electricity

Circular Motion and Gravitation

Translational Dynamics

Quantum Physics

Study anywhere. Anytime. Across all devices.