A. Fourier Series

Here we recall, in the most general form, how a function is expanded into a series in an orthogonal system of functions — we use this expansion everywhere we solve boundary value problems by separation of variables. Suppose in the domain $G$ there is given a system of functions $\varphi_1(M), \varphi_2(M), \ldots$, pairwise orthogonal with the weight function $\rho(M)$ $> 0$, that is

The function $f(M)$ is called expandable in this system if it can be represented by a convergent series

The coefficients $c_n$ (they are called Fourier coefficients) are found thanks to orthogonality. Let us multiply both sides of (A.2$\displaystyle f(M)$ $\displaystyle = \sum_{n=1}^{\infty} c_n \cdot \varphi_n(M).$) by $\rho(M) \cdot \varphi_m(M)$ and integrate over the domain $G$. Due to orthogonality (A.1$\displaystyle \iiint_G \rho(M) \cdot \varphi_n(M) \cdot \varphi_m(M) \,dG$ $\displaystyle = 0, \quad n$ $\displaystyle \neq m.$) only one term survives on the right-hand side — the one with $n$ $= m$:

whence the expansion coefficients are determined uniquely:

The denominator in formula (A.4$\displaystyle c_n$ $\displaystyle = \frac{\displaystyle \iiint_G \rho(M) \cdot f(M) \cdot \varphi_n(M) \,dG}{\displaystyle \iiint_G \rho(M) \cdot \varphi_n^2(M) \,dG}.$) is the squared norm of the function $\varphi_n$:

Formula (A.4$\displaystyle c_n$ $\displaystyle = \frac{\displaystyle \iiint_G \rho(M) \cdot f(M) \cdot \varphi_n(M) \,dG}{\displaystyle \iiint_G \rho(M) \cdot \varphi_n^2(M) \,dG}.$) is universal: whatever the orthogonal system, the expansion coefficients in it are computed in the same way. The weight $\rho(M)$ is determined by the coordinate system in which the variables separate: it is the factor with which the coordinate enters the volume element (the Jacobian) and with respect to which orthogonality holds for the system $\{\varphi_n\}$. For the main systems it equals: