From: hrubin@odds.stat.purdue.edu (Herman Rubin) Subject: Re: How did Euler do this? Date: 2 Jul 1999 12:19:39 -0500 Newsgroups: sci.math Keywords: Tricks for evaluating integrals In article <377C2046.E5973ABA@earthlink.net>, Robert wrote: >Isn't this technique known as the Cesaro limit of the integral? Cesaro limit is for sums; the analog for integral would be to consider \int_0^T (T-t) sin(t) dt/t / T. That this works is seen because the expression is \int_0^T sin(t) dt/t - (1 - cos(T)). The integral with the exponential in it corresponds to the Abel limit, which is always at least as powerful. However, this does not become that much easier. On the other hand, the method of evaluating integrals by adding a parameter and differentiating with respect to that parameter, and possibly continuing the process and doing more manipulations, is quite powerful and was much used in the 18th century. It can be combined with integration by parts (used to evaluate the integral occurring in the derivative) and many other tricks. >James Van Buskirk wrote: >> Steven E. Landsburg wrote in message <7ldrnf$9c34@biko.cc.rochester.edu>... >> >The question is: How did Euler evaluate the following integral?: >> > oo >> > / sin t Pi >> > | _____ dt = --- >> > / t 2 >> > 0 >> Let >> oo >> / sin t e^(-s t) >> f(s) = | ______________ dt >> / t >> 0 >> Then f'(s) = -1/(s^2+1) >> We want to find f(0), of course. We know that f(oo) = 0, and we >> have >> oo >> / -Pi >> f(oo)-f(0) = | f'(s) ds = --- >> / 2 >> 0 -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399 hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558