Q about Fourier Coefficient Derivation

[Enigma322]

For the calculation of c_n u have to multiply the equation

∑ c_n * e^jnx
by
e-^jmx

what is the reason for this? in my textbook it says nothing about it and on some sites it just said "without justification"
i guess what I am asking is why does this do what we want?
ps: how do u properly make equations here?

 

[blue_leaf77]

For the calculation of c_n u have to multiply the equation

∑ c_n * e^jnx
by
e-^jmx

what is the reason for this? in my textbook it says nothing about it and on some sites it just said "without justification"
i guess what I am asking is why does this do what we want?
ps: how do u properly make equations here?

It's not just multiplication, but you also have to integrate the resulting product function. It's done this way because
$$
\int_{0}^L \exp\left(\frac{2\pi}{L}(m-n)u\right) du \propto \delta_{mn}
$$

 

[Ssnow]

Yes, the elements $\{e^{jnx}\}$ form an orthonormal system on your Hilbert space ...