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Topic: improper integrals
Replies: 42   Last Post: Dec 4, 2005 10:31 AM

 Messages: [ Previous | Next ]
 Dave L. Renfro Posts: 4,792 Registered: 12/3/04
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)

> 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 182-183 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 Wiley-IEEE in 1988 in paperback.]

Page 311 [= pp. 253-254 in the 1988 Wiley-IEEE 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)."

Dave L. Renfro

Date Subject Author
11/19/05 SusanP
11/19/05 Robert Low
11/19/05 Dave L. Renfro
11/19/05 SusanP
11/19/05 quasi
11/19/05 SusanP
11/19/05 Dave L. Renfro
11/19/05 SusanP
11/19/05 quasi
11/19/05 SusanP
11/19/05 quasi
11/19/05 SusanP
11/19/05 C6L1V@shaw.ca
11/19/05 quasi
11/20/05 SusanP
11/20/05 quasi
11/20/05 quasi
11/20/05 SusanP
11/20/05 quasi
11/20/05 SusanP
11/20/05 quasi
11/20/05 quasi
11/20/05 SusanP
11/20/05 quasi
11/20/05 C6L1V@shaw.ca
11/20/05 SusanP
11/20/05 quasi
11/20/05 SusanP
12/4/05 Dave L. Renfro
11/20/05 quasi
11/21/05 SusanP
11/21/05 quasi