Unknown method (for me) to obtain the value of a variable

This are notes from a NN. This is one of many steps to solve differential equations with separate variables. In this step NN does following:

$$g_1(y) = \sum_{n=1}^\infty A_n \sin\left(\frac{n\cdot\pi}{b}y\right)$$

And then, to obtain $A_n$ he does:

$$A_n = \frac{2}{b}\int_0^b g_1(y)\cdot \sin\left(\frac{n\pi}{b}y\right)dx$$

Why? Or what is the name of this method?

Thx

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What does NN mean? – JavaMan Mar 29 '11 at 1:54
NN for No Name. Just someone – user8817 Mar 29 '11 at 1:58
I'm slightly confused about your notation. Why does $g_1(y)$ not have a $y$-term on the RHS? Also, is $\delta x$ supposed to be $dx$? – JavaMan Mar 29 '11 at 2:12
This is the "Fourier trick." – Eric O. Korman Mar 29 '11 at 2:24
auch! sorry ... i will fix it. The 'x' should be a 'y' and de $\delta$ should be a $dx$ – user8817 Mar 29 '11 at 2:24

This is a (slightly garbled) Fourier sine series. Here $g_1(x)$ is supposed to be a (sufficiently nice, e.g. piecewise smooth) function defined on the interval $[0, b]$ with $g_1(0) = g_1(b) = 0$. We then expand it in the series $g_1(x) = \sum_{n=1}^\infty a_n \sin(\frac{n \pi}{b} x)$ where $a_n = \frac{2}{b} \int_0^b g_1(x) \sin(\frac{n \pi}{b} x) \, dx$.

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Yess! Seems you are right! Thx a lot all of you for your time. I'm not a registered user, and because of that i can not mark the answer as 'accepted', if some admin can do it, it would be nice. – user8817 Mar 29 '11 at 2:30