The Fourier series representation of is:Originally Posted by cherry106
,
where:
and
So:
.
Rearranging gives:
RonL
Can someone help me with these two Fourier Series problems please? Also, can you explain how you did the problems because I dont understand how to do Fourier Series ...Thanks in advance!
#1 Use the Fourier Series
f(x) = {0, -pi < x < 0; x^2, 0 < x < pi
to show that
1 - 1/4 + 1/9 - 1/16 +.... = (pi^2)/12
#2 Establish the result in the problem below, where m and n are positive integers. (Hint: sinA sinB = (1/2) [cos(A-B) - cos(A+B)].)
L
∫ sin((n pi x)/L) sin((m pi x)/L) dx = {0, m≠n; L, m=n
-L
Originally Posted by cherry106
Now if the integral on the RHS is over an integer number of cycles of each of the cosines
which sum to the integrand and so is zero.
When the integand on the RHS is one plus a cosine whose period
is a sub-multiple of the interval of integration and so the integral is and when
this is multipled by two gives the intregral on the LHS is
RonL