Appendix A — Background material

A series of standard results used without justification in these notes.

A.1 Taylor expansions

Given a function \(f(x)\) with sufficient derivatives at a point \(X\), the function can be represented as a polynomial using a Taylor series about the point \(X\). There are multiple forms of interest.

The general form with remainder term obscured is

\[ f(x) = f(X) + (X - x) f'(X) + \mathcal{O}\left( (X - x)^2 \right) \, . \tag{A.1}\]

The series expanded as a general polynomial is

\[ f(x) = \sum_{k=0} \frac{(X - x)^k}{k!} f^{(k)}(X) \tag{A.2}\]

where the notation \(f^{(k)}(X)\) corresponds to the \(k^\text{th}\) derivative of \(f\) evaluated at \(X\).

For finite differencing it is useful to restrict to an evenly spaced grid \(x_j = x_0 + j \, \Delta x\). Then, expanding about \(x_i\), we have

\[ f(x_j) = \sum_{k=0} \frac{(j - i)^k \, \Delta x^k}{k!} f^{(k)}(x_i) \, . \tag{A.3}\]

The most useful results, written out explicitly to low orders, are

\[ f(x_{i \pm 1}) = f(x_i) \pm \Delta x \, f'(x_i) + \frac{\Delta x^2}{2} f''(x_i) \pm \frac{\Delta x^3}{6} f'''(x_i) + \mathcal{O}(\Delta x^4)\, . \tag{A.4}\]

A.2 Series expansions

A.2.1 Fourier Series

An \(L_2\) square integrable periodic function of one variable \(x\) defined on \([-\pi, \pi]\) can be represented by the complex Fourier series

\[ f(x) \sim \sum_{n=-\infty}^{\infty} a_n \exp(i n x) \tag{A.5}\]

where the (complex) coefficients \(a_n\) are given by

\[ a_n = \frac{1}{2 \pi} \int_{-\pi}^{\pi} f(x) \exp(-i n x) \, \text{d}x \, . \tag{A.6}\]

When the function depends on more than one variable the series expansion can be performed separately for each, or the dependency on the additional variables can be retained in the coefficient. Typically spatial dependence is expanded in a series and time dependence is retained in the coefficients, for example as

\[ f(x, y, t) \sim \sum_{n_x=-\infty}^{\infty} \sum_{n_y=-\infty}^{\infty} a_n(t) \exp(i n_x x) \exp(i n_y y) \, . \tag{A.7}\]

A.2.2 Eigenfunctions

One key use for series expansions is in differential equations. Each individual mode \(f_n(x) = \exp(i n x)\) is an eigenfunction of both the first and second derivative operators,

\[ \begin{aligned} \partial_x f_n(x) &= i n \exp(i n x) \\ &= i n f_n(x) \, , \\ \partial_{xx} f_n(x) &= -n^2 \exp(i n x) \\ &= -n^2 f_n(x) \, . \end{aligned} \tag{A.8}\]

When problems are linear the action of the differential operator can be studied as an algebraic operation on the individual modes.

This generalizes to more complex problems. Sturm-Liouville theory looks at problems where the spatial differential operator has the form

\[ \mathcal{L}(y) = \frac{\text{d}}{\text{d}x} \left[ p(x) \frac{\text{d} y}{\text{d}x} \right] + q(x) y \, . \tag{A.9}\]

Here \(p, q\) are known functions. The eigenfunctions of this operator obey

\[ \mathcal{L}(y_n) = -\lambda_n w(x) y_n \tag{A.10}\]

where \(w\) is a known weighting function. With these eigenfunctions, an arbitrary function can be represented as

\[ f(x) \sim \sum_{n=1}^\infty a_n y_n(x) \tag{A.11}\]

where the coefficients \(a_n\) can be explicitly computed as

\[ a_n = \int_{-\pi}^{\pi} f(x) y_n(x) w(x) \, \text{d}x \, . \tag{A.12}\]

This is an expansion in terms of orthogonal functions, using that

\[ \int_{-\pi}^{\pi} y_m(x) y_n(x) w(x) \, \text{d}x = \delta_{mn} \, . \tag{A.13}\]

A.2.3 Key examples

In the Fourier Series case \(p(x) = 1\), \(q(x) = 0\), and \(w(x) = 2 \pi\). The eigenvalues are \(\lambda_n = n^2\).

In the Legendre equation case (where the domain is conventionally \(x \in [-1, 1]\)) \(p(x) = 1 - x^2\), \(q(x) = 0\), and \(w(x) = (2 n + 1) / 2\). The eigenvalues are \(\lambda_n = n (n + 1)\).

Spherical harmonics \(Y^m_\ell (\theta, \varphi)\) are eigenfunctions of the covariant second derivative operator on a spherical shell with radius \(r\), so that

\[ r^2 \nabla^2 Y^m_\ell = -\ell (\ell + 1) Y^m_\ell \, . \tag{A.14}\]

Spherical harmonics look like a Fourier mode in \(\varphi\) multiplied by an eigenfunction of the (associated) Legendre equation in \(\theta\).