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.
