Homework 11 collaboration area
Question: I'm having trouble getting HWK 11, Page 499, Problem 3 started.
Answer: You will need to use Euler's identity
$ e^{i\theta}=\cos\theta+i\sin\theta $
and separate the definitions of the complex coefficients into real and imaginary parts. For example,
$ c_n=\frac{1}{2L}\int_{-L}^L f(x)e^{-inx}\,dx= $
$ =\frac{1}{2L}\int_{-L}^L f(x)(\cos(-nx)+i\sin(-nx))\,dx= $
$ =\frac{1}{2L}\int_{-L}^L f(x)(\cos(nx)-i\sin(nx))\,dx= $
$ =\frac{1}{2L}(\int_{-L}^L f(x)(\cos(nx)\,dx - i\int_{-L}^L f(x)\sin(nx)\,dx)= $
$ =\frac{1}{2}(a_n-ib_n). $
Do the same thing for $ c_{-n} $ and combine.
Question: Page 501 #3: (Example 1, really), What is $ C_n $? Is it the same $ C_n $ from the complex fourier series equation? If so, why have we discarded the negative n terms? Example 1 makes some really big algebraic leaps that I'm having trouble following. Can someone explain it more clearly?
Question: Page 506, Prob 15, if:
$ 2a_o=\frac{2\pi^4}{9} $
and
$ (a_n)^2=\frac{(4\pi^2)(cos)^2(nx)}{9} $
I dont understand where the $ \frac{\pi^4}{4} $ comes from? Can anyone point out what I am doing wrong?
Questions: prob 11 on page 512:
Should I still use equation 10 to compute A(w) or should I use equation 12 to compute B(w) since f(x) is odd.
When I find A or B, what should the integral range be? (0 to pi?)
Answer: The function f is only defined for positive x. The Fourier Cosine Integral was cooked up by extending f to the negative real axis in such a way to make it an even function. That made the B(w) integral turn out to be zero.
Hence, you only need to calculate the A(w) integral in the form
A(w)= (2/pi) integral from 0 to infinity ...
Since f(x) is zero after pi, your integral would only really go from 0 to pi.
