Finding Fourier Series of f(x)=√(x2) -pi/2<x<pi/2

• Evilavatar2
In summary: The function you are trying to Fourier series is an even function, but you wrote it like an odd function. You need to use the half-period formula. The coefficient on the half-period formula is 2/pi.
Evilavatar2

Homework Statement

Find the Fourier series of the function
f(x) =√(x2) -pi/2<x<pi/2 , with period pi

The Attempt at a Solution

I have tried attempting the question, but couldn't get the answer. uploaded my attempted qns with the picture attached

Your work looks solid at first glance...maybe an algebraic error, I'll go back and look at the details soon.
Remember, your final form should be something like:
## f(x) = a_0 + \sum_{n=1}^\infty a_n \cos 2n x ##

You were right to use the half-period formula and notice it was an even function. You did not write it like one f(x) = |x|, not f(x) = x.
For ##a_0##, it looks like you did 1/2pi, instead of 1/pi for the full period of pi. On the half-period formula, you want to double the result to get the full period, so you should end up with 2/pi as your coefficient on that integral
## a_0 = \frac2\pi \int_0^{\pi/2} f(x) \, dx .##
For your ##a_n ## terms, your coefficient was incorrect as well, since it should be (L was correctly identified as pi/2).
##a_n = \frac2L \int_0^{L} f(x)\cos\left(\frac{nx}{L} \right)\, dx .##
Other than those coefficients, your integration by parts looks to be done correctly from what I can make out.
**edit** you missed a negative sign in the sine integral. Integral of sin(x) dx = - cos x. **end of edit**
Let me know if you are still having trouble.

Last edited:
Thanks a lot for helping. I have decided to redo the whole question and corrected the ao. But i can't find the mistake that you pointed out for the integral sin(x)dx= -cos x. Also my answer is still incorrect.

Attachments

• WhatsApp Image 2017-01-21 at 12.48.34 AM.jpeg
34.7 KB · Views: 436
I must have lost track of the negative sign somewhere--your new solutions seem to be in the right neighborhood.
Look at the coefficient on your ##a_n## integral. It is the same as the one you used on your ##a_0## integral. It should be twice as big.

Evilavatar2

1. What is a Fourier series?

A Fourier series is a mathematical representation of a periodic function as an infinite sum of sines and cosines. It is used to decompose a complicated function into simpler trigonometric functions, making it easier to analyze and understand.

2. How do you find the Fourier series of a function?

To find the Fourier series of a function, you need to follow a series of steps. First, you need to determine the period of the function. Then, you can use the Fourier series formula to find the coefficients of the sines and cosines. Finally, you can combine the coefficients with the trigonometric functions to form the Fourier series.