Let f : [a , b] ( ( an integrable function. Prove that:
lim n-> plus infinity S f(x) cos (nx) dx = 0
lim n-> plus infinity S f(x) sin (nx) dx = 0
where S is the integral going from a to b.
Mathematics Homework Solutions
#3287
Real Analysis Problem
We have just finished up integration and are done with a first course in analysis, chapters 1-6 of Rudin. We are also using the Ross and Morrey/Protter book.
Please answer question fully and clearly explaining every step. Any solution short of perfect is useless to me. So if you are not 100% sure whether your answer is right, then please do not answer.
The same problem is also attached as a word document with all the symbols.
******************************************************
Let f: [a,b] --> R an integrable function. Prove that:
i) lim *S* f(x) cos(nx) dx = 0
and
ii) lim *S* f(x) sin(nx) dx = 0.
where lim is n as it approaches plus infinity (it is
not specified so I believe the default when only n
is listed is to plus infinity, I may be mistaken),
*S* is my notation for the integral taken from a to b.
This is a proof regarding limits of integrable functions.
What is this?
By OTA - Overall OTA Rating
Sheng Huang, MSc - 4.6/5
Purchase Cost Now
$2.19 CAD (was ~$23.94)
Included in Download
- Plain text response
- Attached file(s):
- integrable.doc
Why you can trust BrainMass.com
- Your Information is Secure
- Best Online Academic Help Service
- Students find real academic Success
- Real Analysis :Problem Prove a function is integrable over [a,b] - Let f:[a,b] mapped to the Reals be a function that is integrable over [a,b], and let g:[a,b] mapped to the Reals be a function that agrees with f except at two points. Prove g is integrable over [a,b] ...
- Real Anaylsis - Let f: [a,b] be mapped onto the Reals be a function that is integrable over [a,b] and let g: [a,b] be mapped onto the Reals be a function that agrees with f except at finitely many points. Is g integr ...
- Real analysis - Riemann integrable - If f and g are Riemann integrable on [a,b], show that fg is Riemann integrable on [a,b].
- Real Analysis: Criteria for Integrability - Suppose that the function f:[a,b]->R is integrable and there is a postive number m such that f(x) >= m for all x in [a,b]. Show that the reciprocal function 1/f:[a,b]->R is integrable by proving that ...
- Real Analysis : Riemann Integrable Functions - Show that the function h, defined on I by h(x)=x for x rational and h(x)=0 for x irrational, is not Riemann integrable on I.
