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Chapter 11: Q7P (page 542)
In the Table of Laplace Transforms (end of Chapter 8, page 469), verifythe function results for L5 and L6. Also show that.
The following are proved:
(a)The Gamma function results for .
(b)
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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.
In the pendulum problem, is an approximate solution when the amplitude 伪 is small enough for the motion to be considered simple harmonic. Show that the corresponding exact solution when 伪 is not small is is the modulus of the elliptic function. Show that this reduces to the simple harmonic motion solution for small amplitude 伪
Express the following integrals as functions, and then, by , in terms of functions. When possible, use function formulas to write an exact answer in terms of , etc. Compare your answers with computer results and reconcile any discrepancies. .
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