广义Jordan引理

THE EXTENDED JORDAN’S LEMMA

通过证明三个简单的引理，将复变函数积分的Ｊｏｒｄａｎ引理推广，使之能在更宽的范围内成立。原先的Ｊｏｒｄａｎ引理要求：被积函数ｆ（ｚ）除在上半平面有有限个孤立奇点外，处处解析并且当‖ｚ‖→∞时一致趋于零。我们发现Ｊｏｒｄａｎ引理可在较宽的条件下使用：被积函数ｆ（ｚ）除在上半平面有有限个孤立奇点外，处处解析并且对于ｐ＞０，当‖ｚ‖→∞时则Ｊｏｒｄａｎ引理就可以使用。

Generally,in the one complex variable integration,Jardan’s lemma is used only when the integrand f(z)tends uniformly tozero in arg z(0≤arg z≤π)as ‖z‖→∞. We find that Jordan’s lemmacan be used for a wider range than that mentioned above. The extendedJordan’s lemma can be described as follows. Let f(z)be analytic in the upper half of the z plane(Im z≥0),with the exception of a finite number of isolated singularities, and itonly has first order singularities in the real axis,and for p>0,if so Jordan’s lemma can also be used. where res(z_i )is the residue of f(z)e￣(ipx) at the polez_i in the upper half of the z plane. [