Fourier series of complex numbers with diffrent limits of integration?

1. Jan 4, 2012

jose_peeter

Fourier series of complex numbers with diffrent limits of integration?????

Dear all,
i dont know how to simplify a COMPLEX NUMBER fourier series with LIMITS OF INTEGRATION that are not complementary. I MEAN limits LIKE this X to -X being easy to solve and SIMPLIFY but Not X to -Y or anything different. When i say simplify i mean writing the exponents in the form of cos ωt or sin ωt following the euler's identity.

As an example i will FIND the complex fourier series of the following function and find it unable to simplify.

f(t) = 1, 0 < t < 2
0, 2 < t < 4

MY ATTEMPT at the question.

cn = (1/T) * T/2 - (-T/2) ∫ f(t) * e^-(j2npit/T) dt

= (1/4) * 2 - 0∫e^-(j2npit/T) dt

= (1/4) * [ (-2/jnpi)*e^(-jnpit/2) ] 2 - 0

= (-1/2jnpi)*e^(-jnpi) + (1/j2npi)

=here is my problem!. how do i now write this like in the answer

= answer : -∞ to ∞Ʃ (j/2npi)*(cos npi - 1)e^(jnpit/2)

please tell me a trick for any general question when the limits are NON-COMPLEMENTARY to eacb other when using COMPLEX FOURIER SERIES.

thanks,

do you know a way where i can write my handwitten math work and then post.
this is really tedious.
thanks.

Last edited: Jan 4, 2012
2. Jan 4, 2012

LCKurtz

Re: Fourier series of complex numbers with diffrent limits of integration?????

I have a couple of suggestions for you. First is to lose the "size" tag on the fonts. Aside from being annoying, they are hard to read. Also, a little latex will help you with the tedium. It is better to post as you have rather than post an image of your handwriting since it is easier to quote and reply to individual steps.

I didn't check your integrals, but my suggestion would be that you can always use a symmetrical interval when calculating Fourier coefficients since your function is periodic. This is because for any function with period 2P,$$\int_a^{a+2P}f(x)\, dx = \int_{-P}^Pf(x)\, dx$$

3. Jan 4, 2012

Ray Vickson

Re: Fourier series of complex numbers with diffrent limits of integration?????

Why don't you consult available sources? For example, the article http://en.wikipedia.org/wiki/Fourier_series contains the requisite formulas for a general interval later in the article.

RGV

4. Jan 5, 2012

jose_peeter

Re: Fourier series of complex numbers with diffrent limits of integration?????

hi,
i don't understand what they say in wikipedia or what LCKURTZ is saying. Can you please show me IN MATHEMATICS(not words, thank you) by choosing your own suitable inteval for the EXACT example above.

sorry i am demanding a lot of work but, please you should know that i have failed the damn signals module. so please help me pass this time, to revive my GPA.

5. Jan 5, 2012

LCKurtz

Re: Fourier series of complex numbers with diffrent limits of integration?????

Your function is given to be $$f(t) = \left \{\begin{array}{rl} 1,&0<t<2\\ 0,&2<t<4 \end{array}\right.$$ Presumably that is one period, so T = 4. So the function's value from -2 to 0 must be the same as its value from 2 to 4. So you could just as well write the definition of the function like this:$$f(t) = \left \{\begin{array}{rl} 0,&-2<t<0\\ 1,&0<t<2 \end{array}\right.$$ Now you have a symmetric interval, which you have indicated you know how to do.

Last edited: Jan 5, 2012
6. Jan 5, 2012

Ray Vickson

Re: Fourier series of complex numbers with diffrent limits of integration?????

The Wiki article gives you _exactly_ all the formulas you need, in detail. All you need to do is plug your function into the formulas written there. (Look at Section 3.3, titled "Fourier series on a general interval [a,a+τ]".) If you truly want to pass the course you need to put in the work, and struggle if necessary---there is really no other way!

RGV

7. Jan 10, 2012

jose_peeter

Re: Fourier series of complex numbers with diffrent limits of integration?????

I just have one problem, the integral limits cannot be -2 to 2 because f(t) is 0 from -2 to 0. so limits effectively reduce to 0 to 2. this is the original problem i was facing. so how do i do it now? if i assume the f(t) to be 1 from -2 to 2. then it is WRONG.

thanks.

8. Jan 10, 2012

LCKurtz

Re: Fourier series of complex numbers with diffrent limits of integration?????

Just break the interval up:$$c_n=\frac 1 4 \int_{-2}^{2}f(t)e^{-\frac{in\pi t}{2}}\,dt = \frac 1 4 \int_{-2}^{0}0e^{-\frac{in\pi t}{2}}\,dt +\frac 1 4 \int_{0}^{2}1e^{-\frac{in\pi t}{2}}\,dt$$

9. Jan 11, 2012

jose_peeter

Re: Fourier series of complex numbers with diffrent limits of integration?????