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Chapter 11: Q12.5P (page 559)
The value of integral in elliptic form is .
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Sketch or computer plot a graph of the function .
Write the integral in equation (12.7) as an elliptic integral and show that (12.8)gives its value. Hints: Write and a similar equation for. Then make the change of variable.
Prove that, for positive integral n:
Without computer or tables, but just using facts you know, sketch a quick rough graph of the function from -2to 3. Hint:This is easy; don鈥檛 make a big job of it. From Section 3, you know the values of the data-custom-editor="chemistry" function at the positive integers in terms of factorials. From Problem 1, you can easily find and plot the function at , . (Approximateas a little less than 2.) From (4.1) and the discussion following it, you know that the function tends to plus or minus infinity at 0 and the negative integers, and you know the intervals where it is positive or negative. After sketching your graph, make a computer plot of the 螕 function from -5to 5and compare your sketch.
Show that the four answers given in Section 1 for 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 螕 function formulas. For the elliptic integral, use the hint of Problem 17 with.
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