Question

obtain the fourier series in
(-π,π)for f(x)=xcosx​

Answer 1

Step-by-step explanation:

45x cos is your answer.......

Answer 2

Answer:

Step-by-step explanation:

A Fourier series representation of a function consists of two parts. The odd part and the even part.

A generalized definition of a function in terms of Fourier Series is done in the interval of  [0,T] where  T is  π  

To find the Fourier

The expression for the fourier series of a function is given as follows:

f(x)=a0+∑n=1∞(ancos(2πnxT)+bnsin(2πnxT))  where  a0,an  and  bn  

are Euler Coefficients.

a0=2T∫T0f(x)dx  

an=2T∫T0f(x)cos(2πnxT)dx  

bn=2T∫T0f(x)sin(2πnxT)dx  

Now in the given interval  [0,π] ,  f(0+π−x)=−cos(x)=−f(x) is odd.  

So, a0=0  and  an=0.  

bn=4π∫π20cos(x)sin(2nx)dx=2π[cos(2n+1)x2n+1+cos(2n−1)x2n−1]0π2=2π[12n+1+12n−1]=2π[4n4n2−1]  

This gives  f(x)  as

f(x)=8π∑n=1∞(n4n2−1sin(2nx))