
Re: improper integrals
Posted:
Dec 4, 2005 10:31 AM


SusanP wrote:
>> Invent a continuous function f: R> R whose improper >> integral is 0, but which is unbounded as x> oo >> and x> +oo. (I know the function is far from monotone)
The World Wide Wade wrote:
> Here's an explicit formula for such a function: > > f(x) = x*[(1 + (cos(x))^2)/2]^(x^6). > > Although I think building your spikes "by hand" is the > easiest way to go for this problem.
I'm revisiting this thread because I came across two references that might be useful to archive here. The relevant pages of both references are (currently) available on the internet.
1. Édouard Goursat and Earle Raymond Hedrick, A FIRST COURSE IN MATHEMATICAL ANALYSIS, Volume I, Ginn and Company, 1904.
Pages 182183 have a discussion of sin(x^2) and x / [1 + (x^6)*(sin x)^2] that is relevant to the original poster's question.
http://name.umdl.umich.edu/ABA9351.0001.001
2. Richard Courant and Fritz John, INTRODUCTION TO CALCULUS AND ANALYSIS, Volume 1, Interscience Publishers, 1965. [Reprinted by WileyIEEE in 1988 in paperback.]
Page 311 [= pp. 253254 in the 1988 WileyIEEE paperback version] has the following remark and continues with a proof of it.
"In fact, an improper integral can exist even when the integrand is unbounded, as is shown by the example
integral[u=0 to u=infinity] of 2u * cos(u^4)."
http://books.google.com/books?as_epq=Fresnel+integrals+show http://books.google.com/books?as_epq=different+and+lie+between
Dave L. Renfro

