By Hilary D. Brewster
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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.