Given that $f(x)= \sin(x + \pi/4)$ is periodic with period $2\pi$. Find the complex Fourier series.
It's quite a moderate tough question. Can someone help me out. Thanks in advance
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hint: rewrite this as $u,e^{ix}+v,e^{-ix}$ – Raymond Manzoni Sep 01 '12 at 18:13
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$\sin(\theta) = \frac{1}{2i}(e^{i\theta}-e^{-i\theta})$ if you're looking for a sum of imaginary exponentials then this is probably a good way to look at it. – James S. Cook Sep 01 '12 at 18:14
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Could you please show a few step so that I able to continue – Garett Sep 01 '12 at 18:33
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use James' method with $\theta=x+\pi/4$ and put the $i\pi/4$ terms out of the $e^{\pm ix}$ exponentials... – Raymond Manzoni Sep 01 '12 at 18:42
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or expand $\sin(x+a)$ and rewrite $\sin(x)$ and $\cos(x)$ in exponential form... – Raymond Manzoni Sep 01 '12 at 18:51
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Note: $\sin(x+\pi/4) = \frac{1}{2i}(e^{i(x+\pi/4)}-e^{-i(x+\pi/4)})$. Furthermore,
$$ e^{i(x+\pi/4)} = e^{ix}e^{i\pi/4} $$
Now, use $e^{i\theta} =\cos(\theta)+i\sin(\theta)$ to find the explicit value of the coeffient $e^{i\pi/4}$. I leave the rest to you.
James S. Cook
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Here is your Fourier series,
$ \sin(x+\frac{\pi}{4}) = \cos(\frac{\pi}{4})\sin(x)+ \cos(\frac{\pi}{4})\cos(x)=\frac{1}{\sqrt 2} \sin(x) + \frac{1}{\sqrt 2} \cos(x)\,.$
Note that, $\sin(x+\frac{\pi}{4})$ is orthogonal to $\{ 1, \sin(mx),\cos(mx)\} \,, m=2,3,\dots $ on the $[0,2\pi]$.
Use the identities $ \sin(x) = \frac{ {\rm e}^{ix} - {\rm e}^{-ix} }{2i} $ and $ \cos(x) = \frac{ {\rm e}^{ix} + {\rm e}^{-ix} }{2} $ to get it in the complex form
$$ \sin(x+\frac{\pi}{4}) = \left( \frac{\sqrt {2}}{4}i+\frac{\sqrt {2}}{4} \right) {{\rm e}^{-ix}}+\left( -\frac{\sqrt {2}}{4}i+\frac{\sqrt {2}}{4} \right) {{\rm e}^{ix}} $$
Mhenni Benghorbal
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In a somewhat connected vein, see http://math.stackexchange.com/questions/71593/deriving-fourier-series-using-complex-numbers-introduction?rq=1 – James S. Cook Sep 01 '12 at 18:45
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1@Garett: This is the Fourier series. You can use $ {\rm e}^{ix} =\cos(x)+i\sin(x) $ to change to the complex form. – Mhenni Benghorbal Sep 01 '12 at 18:47
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Thank you Mhenni Benghorbal. I think now I can understand something on it – Garett Sep 03 '12 at 16:44