Fourier Series Integration by Parts Solution

In summary, the conversation discusses solving the integral of -t*e^{-j2*pi*nt} using integration by parts and the resulting expression being different from the expected answer of 1/(j2*pi*n). The conversation also touches on evaluating the complex Fourier series using Euler's rules.
  • #1
reece
9
0

Homework Statement


Solve [tex]\frac{1}{1}[/tex] [tex]\int^{0}_{-1}[/tex] -t e[tex]^{-j2\pi*nt}[/tex]dt

Homework Equations


So I use integration by parts
u = -t and dv = e[tex]^{-j2\pi*nt}[/tex] , du= -1 and v = [tex]\frac{1}{-j2\pi*n}[/tex]e[tex]^{-j2\pi*nt}[/tex]

The Attempt at a Solution


after integration by parts I get:
[tex]\frac{e^-j2\pi*nt}{j2pi*n}[/tex] + [tex]\frac{1}{(j2pi*n)^{2}}[/tex] [1 - e[tex]^{-j2\pi*nt}[/tex]]

So basically I am suppose to end up with [tex]\frac{1}{j2pi*n}[/tex] but it doesn't work out like it should.
Any help would be good thanks.
 
Last edited:
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  • #2
First, you certainly do not get anything involving "t" when you have evaluated at t= 0 and -1! Did you mean
[tex]\frac{e^{2\pi i n}}{2\pi i n}+ \frac{1- e^{2\pi i n}}{(2\pi i n)^2}[/tex]

What is [itex]e^{-2\pi i nt}[/itex]?
(Sorry, but I just can't force my self to use "j" instead of "i"!)
 
  • #3
woops my bad, those t's arent spose to be in there. But the rest is right.

And yes it is what you typed out.

Im leading to think I shouldn't get the same answer as what I got using the Trig-Complex Relationship equation. Which is why I am getting something different.

The exponential part is what I integrate with to evaluate the complex Fourier series. See the equation. -t is the function.
 
  • #4
Once again, what is
[tex]e^{-2\pi i n}[/tex]?
 
  • #5
In trig FS, you evaluate each component a0, an, bn. Well the exponential replaces that I think.

im guessing you use Euler's Rules (i think) to evaluate it which says

sin theta = 1 / 2i [ e ^ i*theta - e ^ -i*theta]
cos theta = 1 /2 [ e ^ i*theta + e ^ -i*theta]

Hopefully that answered that.
 

1. What is a Complex Fourier Series?

A Complex Fourier Series is a representation of a periodic function as a sum of complex exponential functions. It can be used to analyze the frequency components of a signal and is widely used in fields such as signal processing and physics.

2. How is a Complex Fourier Series different from a Fourier Series?

A Complex Fourier Series includes both real and imaginary terms, while a Fourier Series only includes real terms. This allows for a more accurate representation of periodic functions with complex components.

3. What is the formula for a Complex Fourier Series?

The formula for a Complex Fourier Series is:

f(x) = a0 + ∑n=1 (ancos(nx) + bnsin(nx))

where a0, an, and bn are coefficients determined by the function's period and its complex components.

4. What is the significance of the coefficients in a Complex Fourier Series?

The coefficients in a Complex Fourier Series represent the amplitude and phase of the various frequency components of a periodic function. By analyzing these coefficients, we can gain insight into the behavior and properties of the function.

5. How is a Complex Fourier Series applied in real-world scenarios?

Complex Fourier Series are commonly used in fields such as signal processing, electrical engineering, and physics to analyze and manipulate periodic signals. They are also used in image and sound processing to compress and decompress data. Additionally, they are used in mathematical research to study the properties of functions and their Fourier coefficients.

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