Fourier Transforms: Solving Homework Statement

[NotStine]

Homework Statement

The problem is related to Fourier transforms. Since my teacher only cared to spend 2 lectures explaining the whole of this topic, I just cannot get my head around some of the principles involved.

The question is as follows:

Use the Fourier Transform formulas with the function f(t) = t3 when 0 ≤ t < ∞ to show that
http://img222.imageshack.us/img222/6450/fst2kd3.png

What on Earth is my starting point?

Homework Equations

f(t) = a0/2 + Sum_n=a^infinity{a_ncos(2.n.pi.t/T) + b_nsin(2.n.pi.t/T)}

The Attempt at a Solution

I'm sorry, like I said I have no clue where to begin from.

1. What is the difference between Fourier series and transform?

2. What would be my starting point for this question? Is this related to Fourier series convergence?

3. What is the "fourier transforms" formula?

I'm afraid my teacher has a habit of making things very complicated, and explaining very little...

 

[Hootenanny]

Fourier Series can be thought of as a discrete Fourier Transform. Do you have a course text that you can refer to? Your starting point would be to determine the Fourier Transform of the given function, but that's going to be very difficult if you don't know what a Fourier Transform is.

For a brief introduction try here: http://mathworld.wolfram.com/FourierTransform.html

 

[NotStine]

After reading the notes, this is how far I got:

f(t) = 2/pi. I_{(0 - inf)}fc_omega.cos_omega.td.omega

However, when I start evaluating the fc_omega integral, I have the limits 0 - infinity and I'm left with

[t³.cos_omega.t / omega] between 0 and infinity. I can't seem to progress from this stage. All the sample questions I have seen so far have integer limits so I don't know how to evaluate this infinite range. Any help?

 

[chaoseverlasting]

You can convert cosw+isinw to eular form which would give you something roughly representing the Fourier transform inside the integral. That could be a starting point.

 

[NotStine]

I just want to confirm if I'm following the right approach.

Step 1: I need to find the sine and cosine transforms of f(t) = t³ when 0 ≤ t < ∞?

Step 2: I then need to rearrange to somehow get the equaton in the picture?

 

[Hootenanny]

Sorry I haven't got back sooner.

Step 1: I need to find the sine and cosine transforms of f(t) = t³ when 0 ≤ t < ∞?

No, you just need to find the bog standard Fourier transform of t³. Note that in this case the Fourier Transform may be defined as;
$$F\left[f\left(t\right);\omega\right] = \int_{0}^\infty f\left(t\right)e^{-i\omega t}dt$$

Step 2: I then need to rearrange to somehow get the equaton in the picture?

Yes. HINT: Euler's formula.

 

[NotStine]

Performing the intergral with integration by parts 3 times, I get the following:

F(t³) = 4 / $$\omega$$⁴

With the full transformation

F(t³) = 2/pi * $$\int_{0}^\infty$$ 4.e^{i.$$\omega$$.t} / $$\omega$$⁴

Which also equals

F(t³) = 8/pi * $$\int_{0}^\infty$$ cos$$\omega$$ + j.sin$$\omega$$ / $$\omega$$⁴

Now using t = 1, I get

1 = 8 / pi * $$\int_{0}^\infty$$ e^i.$$\omega$$ / $$\omega$$⁴

pi / 8 = $$\int_{0}^\infty$$ cos$$\omega$$ + j.sin$$\omega$$ / $$\omega$$⁴

...

Which is not right. I am suppose to get 2*pi / 3. Any ideas where I'm going wrong?