let $\phi_n=(e^{inx})$ for $integer \ n$ I have verified that each elements of this set is orthogonal to eachother on $[0,2\pi]$ But while trying to normalize I ended up with the following problem:
To normalize I need to have $||\phi_n||=1$ for each $integer$ $n$
Set :, so we can find the constant $\alpha$ $$||\phi_n||^2 = <\phi_n,\phi_n> = \alpha\int_0^{2\pi}e^{2inx}dx=1$$
Solving we end up with :
$$\alpha\frac{e^{4\pi in}-1}{2in}=1$$ I can't solve for $\alpha$ here since $$e^{4\pi in}=1$$ by euler's idendity. What have I done wrong?
You need to take a complex conjugate of the first function when you take the inner product
$$||\phi_n||^2 = <\phi_n,\phi_n> = \alpha\int_0^{2\pi}e^{-inx}e^{inx}dx=\alpha \int_0^{2\pi}dx=1$$ which will give you a way to solve for the normalization constant.