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Chapter 11: Q17P (page 540)

Express as a function

Short Answer

Expert verified

The solution to this problem is

Step by step solution

01

Given Information

The given information is the gamma function to be used in the solution.

The given integral is

02

Definition of Gamma function

To extend the factorial to any real number x>0 (whether or not the real number is a whole number), the gamma function is defined as for .

03

Find the solution

Use substitution method to solve the integral.

Let , then we have,

The given integral is now in the form of .

Express the integral in gamma function, and then evaluate the gamma function and integral numerically.

The Gamma function is for . Then solve for the given function as follow:

Hence, the solution to this problem is

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Most popular questions from this chapter

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

In Problem 4 to 13, identify each of the integral as an elliptic (see Example 1 and 2). Learn the notation of your computer program (see Problem 3) and then evaluate the integral by computer.

13. $\int_{\frac{- 1}{2}}^{\frac{3}{4}} \frac{\sqrt{9 - 4 t^{2}}}{\sqrt{1 - t^{2}}} d t$

Prove that $B (p, q) = B (q, p)$ . Hint:Put $x = 1 - y$ in Equation (6.1).

Use Stirling’s formula to evaluate $\underset{n \to \infty}{l i m} \frac{{(n!)}^{\frac{1}{n}}}{n}$ .

In Problem 4 to 13, identify each of the integral as an elliptic (see Example 1 and 2). Learn the notation of your computer program (see Problem 3) and then evaluate the integral by computer.

8. $\int_{0}^{\frac{\sqrt{3}}{2}} \frac{\sqrt{49 - 4 t^{2}}}{\sqrt{1 - t^{2}}} d t$

See all solutions

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