# How to solve this Real Analysis?

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

-
hint: rewrite this as $u\,e^{ix}+v\,e^{-ix}$ – Raymond Manzoni Sep 1 '12 at 18:13
$\sin(\theta) = \frac{1}{2i}(e^{i\theta}-e^{-i\theta})$ 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 1 '12 at 18:14
Could you please show a few step so that I able to continue – Garett Sep 1 '12 at 18:33
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 1 '12 at 18:42
or expand $\sin(x+a)$ and rewrite $\sin(x)$ and $\cos(x)$ in exponential form... – Raymond Manzoni Sep 1 '12 at 18:51

$\sin(x+\frac{\pi}{4}) = \cos(\frac{\pi}{4})\sin(x)+ \cos(\frac{\pi}{4})\cos(x)=\frac{1}{\sqrt 2} \sin(x) + \frac{1}{\sqrt 2} \cos(x)\,.$

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

$$\sin(x+\frac{\pi}{4}) = \left( \frac{\sqrt {2}}{4}i+\frac{\sqrt {2}}{4} \right) {{\rm e}^{-ix}}+\left( -\frac{\sqrt {2}}{4}i+\frac{\sqrt {2}}{4} \right) {{\rm e}^{ix}}$$

-
No this is not the fourier Series.What you do is the expansion only – Garett Sep 1 '12 at 18:31
In a somewhat connected vein, see math.stackexchange.com/questions/71593/… – James S. Cook Sep 1 '12 at 18:45
@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 1 '12 at 18:47
Thank you Mhenni Benghorbal. I think now I can understand something on it – Garett Sep 3 '12 at 16:44

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.

-
Thanks James for your expalantion. I'm able to do understand now – Garett Sep 3 '12 at 16:44