Finding the Fourier Series of E(t)

In summary, the conversation discusses finding the Fourier series for a given function using the formulas for a0, an, and bn. The speaker is able to find a0, but is stuck at integrating for an and bn due to the presence of n in the cosine function. They mention trying integration by parts and using the double angle formula, but are unsure of how to proceed.
  • #1
Trenthan
54
0

Homework Statement


{ 0 -pi < t < 0
E(t) =
{ sin(t) 0 < t < pi

Find the Fourier series
w = 1, T = 2pi, L = pi
a0 = 1/(2L) integral(-L to L) f(t) dt
an = 1/(L) integral(-L to L) f(t)cos(nwt)dt n = 1,2,3...
bn = 1/(L) integral(-L to L) f(t)sin(nwt)dt n = 1,2,3...

The Attempt at a Solution


now I am able to find a0 but cannot find an. I am stuck at integrating

an = (1/pi)*integral(0 to pi) sin(t)cos(nt) dt ***

because of the "n" in the "cos" i cannot find the integral, first glance i though integration by parts but that just swaps the sin and cos's around. Than I've looked at the double angle rules and since "n" changes i figured i couldn't apply either since n = 1,2,3... etc

Any idea where to go in order to integrate it ?***
bn is the same problem once i figure out how to integrate it, it should work out nicely

Thanks in advance TRENT
 
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  • #2
[tex]\sin x\cos y=\frac12\biggl(\sin(x+y)+\sin(x-y)\biggr)[/tex]
 

FAQ: Finding the Fourier Series of E(t)

How do you find the Fourier series of E(t)?

To find the Fourier series of E(t), you need to first express E(t) as a periodic function with a period T. Then, you can use the Fourier series formula to calculate the coefficients of the series. This involves finding the integrals of E(t) multiplied by sine and cosine functions with different frequencies.

What is the significance of finding the Fourier series of E(t)?

The Fourier series allows us to represent a periodic function, such as E(t), as an infinite sum of sine and cosine functions with different frequencies. This can help us better understand the behavior of the function and make predictions about its values at different points in time.

Can E(t) have a unique Fourier series?

No, E(t) does not have a unique Fourier series. The coefficients of the series can vary depending on the chosen period T and the method used to calculate them. However, as the number of terms in the series increases, the approximation of E(t) becomes more accurate.

How is the convergence of the Fourier series of E(t) determined?

The convergence of the Fourier series of E(t) is determined by the properties of the function E(t) and the chosen period T. If E(t) is a continuous and piecewise smooth function, then the Fourier series will converge to E(t) at points where E(t) is continuous. The rate of convergence is also affected by the smoothness of E(t) and the choice of T.

Can the Fourier series of E(t) be used to solve differential equations?

Yes, the Fourier series of E(t) can be used to solve certain differential equations. This is because the series can help us represent a function as an infinite sum of simpler functions, making it easier to manipulate and solve equations involving E(t). However, not all differential equations can be solved using the Fourier series method.

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