Finding Fourier Expansion of f(t): Part b & c

In summary, the Fourier expansion of one period of f(t)=1+t absolute value of t<1 is 1+2/pi Sigma(0 to infinity) ((-1^(n+1))/n)sinnpit. For part b, you can set t equal to t_0 and use the property of sine to simplify the expression. For part C, you can substitute the given values into the Fourier series to find the sum.
  • #1
zyphriss2
18
0
Find the Fourier expansion of one period of f(t)=1+t absolute value of t<1

I found this to be 1+2/pi Sigma(0 to infinity) ((-1^(n+1))/n)sinnpit by just the standard methods of the a0 an and bn formuals, which I know is correct

Now the parts I am having problems with is part b and c which our teacher has not covered much at all and I cannot find any help online.

Part b. Use the Fourier expansion of f to find the sum of the Series
Sigma(0toinfinity)(-1^n)/(2n+1)

Part C. If f denotes the function defined on (-infinity to infinity) by the Fourier series of f, find F(1)+F(-5)-3F(0)
 
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  • #2
Just to make things a bit more clear on the notation side, I take it your expansion is:

[tex]1+\frac{2}{\pi}\sum_{n=0}^{\infty}\frac{(-1)^{n+1}}{n} \sin{n\pi t}[/tex]

As for question b:

Notice that if you set t equal to [tex]t_0 \equiv \frac{1}{2}[/tex], then

[tex]\sin{n\pi t_0} = \sin{\frac{n\pi}{2}}[/tex]

which is zero whenever n is even and [tex](-1)^{(n-1)/2[/tex] whenever n is odd... Do you see where I'm going with this?

part C:
I take it by F(0) you just mean f(0)? Have you tried filling in these numbers into the Fourier series?
 

FAQ: Finding Fourier Expansion of f(t): Part b & c

What is a Fourier expansion?

A Fourier expansion is a mathematical representation of a periodic function as a sum of sinusoidal functions with different frequencies. It allows us to break down a complex function into simpler components, making it easier to analyze and understand.

How is a Fourier expansion calculated?

A Fourier expansion is calculated using the Fourier series, which is a mathematical tool that decomposes a function into its constituent frequencies. This involves finding the coefficients of each sinusoidal function in the expansion through integration.

What is the purpose of finding a Fourier expansion?

The main purpose of finding a Fourier expansion is to understand the behavior of a periodic function and to simplify complex functions. It is also used in many fields, such as signal processing, image analysis, and data compression, to name a few.

What are the applications of a Fourier expansion?

A Fourier expansion has various applications in different fields such as engineering, physics, and mathematics. It is used in analyzing signals and systems, solving differential equations, and representing periodic functions in a more manageable form. It is also used in solving heat transfer and vibration problems in engineering.

Are there any limitations to using a Fourier expansion?

Yes, there are limitations to using a Fourier expansion. It is only applicable to periodic functions, and the function must be well-behaved for the expansion to converge. Additionally, discontinuities or sharp corners in a function can cause errors in the expansion.

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