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Chapter 11: Q11.1P (page 553)
Use the term 1/(12p)in (11.5) to show that the error in Stirling’s formula (11.1) is < 10%for p > 1; < 1%for p > 10; < 0.1%for p > 100; < 0.01%for p > 1000.
If p > 1000 the error will be lesser than 0.01%.
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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 Γ function formulas. For the elliptic integral, use the hint of Problem 17 with.
Use the recursion relation (3.4), and if needed, equation (3.2) to simplify:
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.
Computer plot graphs of sn u, cn u, and dn u, for several values of k, say, for example, .Also plot 3D graphs of sn, cn, and dn as functions of u and k.
Verify equations (9.2), (9.3), and (9.4). Hint: In (9.2a) , you want to write in terms of an error function. Make the change of variable t = u √2 t = u (√2)in the integral. Warning: Don’t forget to adjust the limits; when .
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