By George Pólya; Leon Bowden
'Mathematics, taught and discovered adequately, improves the brain and implants reliable conduct of thought.' This guideline underlies all of Professor Pólya's works on instructing and problem-solving. This ebook captures a few of Pólya's pleasure and imaginative and prescient. In it he offers enlightenment for all those that have ever puzzled how the legislation of nature have been labored out mathematically. The virtue of the current publication is the tension at the background of yes common chapters of technological know-how; those could be a resource of pleasure and deeper knowing of arithmetic even for newcomers who've little, or maybe no, wisdom of physics
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Example text
Let/(x):;:. 0 for x E M in some neighborhood of a and { E. g(x) = f(x, 0) has a singularity at some 0 :s; x :s; a. If this singularity is of power or logarithmic type then we say that the integral F (e) has a weak singularity. This Definition is obviously extended for unbounded intervals. Let F(e) = r f(x,e)dx where a> 0 and e > 0 is a small parameter. HerefE CX'([a, 1) x (0, eol) is a smooth function for a :s; x:s; 1,0 < e :s; eo, with some eo. Let the integral converge for e > 0 and diverge for e o. If the function g(x) f (x, 0) is of power or = = Mathematical Basics 42 logarithmic order at x singularity. Definition: Let f M X S ~ lR be a function of two variables and a be a limit point of M and {n} be an asymptotic sequence as x ~ a. Let for any xed YES the function f is expanded in an asymptotic series 00 f(x,y) ~ Lan(Y)n(x), (x~a,xEM). n=O This asymptotic expansion is called uniform with respect to the parameter YES, if the relation 00 RN(x,y) = f(x,y)- L an (y) n (x) = O(N(x», (x ~ a, x E M) n=O is valid uniformly with respect to YES. Theorem: A uniform asymptotic expansion can be integrated with respect to the parameter term by term.