Series and Transform Methods

Let $y''+p(x)\,y'+q(x)\,y=0$ be a second-order linear homogeneous equation. A point $x_0$ is an ordinary point of the equation if $p$ and $q$ are analytic at $x_0$ and a singular point otherwise. Near an ordinary point every solution is analytic and is found as a power series with unknown coefficients,

$$y(x)=\sum_{n=0}^{\infty}a_n\left(x-x_0\right)^n,$$

by differentiating term by term, substituting into the equation and equating the coefficient of each power of $x-x_0$ to zero, which gives a recurrence relation that determines $a_2,a_3,\ldots$ from the free coefficients $a_0=y\left(x_0\right)$ and $a_1=y'\left(x_0\right).$ The radius of convergence of the series is at least the distance from $x_0$ to the nearest singular point of the equation in the complex plane. For example, the Airy equation $y''-x\,y=0$ has only ordinary points and the recurrence $\left(n+2\right)\left(n+1\right)a_{n+2}=a_{n-1},$ whose solutions converge on all of $\mathbb{R}.$

A singular point $x_0$ is a regular singular point if $\left(x-x_0\right)p(x)$ and $\left(x-x_0\right)^2q(x)$ are analytic at $x_0,$ and an irregular singular point otherwise. Near a regular singular point the Frobenius method seeks solutions of the form

$$y(x)=\left(x-x_0\right)^r\sum_{n=0}^{\infty}a_n\left(x-x_0\right)^n,\qquad a_0\neq 0,$$

where the exponent $r$ is a root of the indicial equation $r\left(r-1\right)+p_0r+q_0=0$ with $p_0=\lim_{x\rightarrow x_0}\left(x-x_0\right)p(x)$ and $q_0=\lim_{x\rightarrow x_0}\left(x-x_0\right)^2q(x).$ The larger root always gives a solution of this form; if the roots differ by an integer the second solution may contain a term $\log_e\left(x-x_0\right)$ times the first. Bessel’s equation $x^2y''+x\,y'+\left(x^2-\nu^2\right)y=0$ has a regular singular point at $0$ with indicial roots $\pm\nu,$ and Legendre’s equation $\left(1-x^2\right)y''-2x\,y'+\ell\left(\ell+1\right)y=0$ has regular singular points at $\pm 1$ and, when $\ell\in\mathbb{N},$ a polynomial solution of degree $\ell,$ the Legendre polynomial $P_\ell.$

Let $f:\left[0,\infty\right)\rightarrow\mathbb{R}$ be integrable on every bounded interval and of exponential order, meaning that there exist $M,a\geq 0$ with $\lvert f(t)\rvert\leq M\,e^{at}$ for all $t\geq 0.$ The Laplace transform of $f$ is the function of $s$ given by the improper integral

$$\mathcal{L}\left\{f\right\}(s)=F(s):=\int_0^{\infty}e^{-st}f(t)\,dt,$$

which converges absolutely for $s>a$ by comparison with $\int_0^{\infty}M\,e^{-\left(s-a\right)t}\,dt.$ The transform is linear and Lerch’s theorem states that two continuous functions with the same transform are equal, so $f$ is the inverse Laplace transform $\mathcal{L}^{-1}\left\{F\right\}$ of $F.$ Some common transforms, valid for $s$ larger than the stated constant, are:

$$\mathcal{L}\left\{1\right\}=\frac{1}{s},\qquad\mathcal{L}\left\{t^n\right\}=\frac{n!}{s^{n+1}},\qquad\mathcal{L}\left\{e^{at}\right\}=\frac{1}{s-a}$$ $$\mathcal{L}\left\{\cos\omega t\right\}=\frac{s}{s^2+\omega^2},\qquad\mathcal{L}\left\{\sin\omega t\right\}=\frac{\omega}{s^2+\omega^2}.$$

Let $f$ be $C^1$ with $f$ and $f'$ of exponential order. Integration by parts gives the derivative rule, which turns differentiation into multiplication by $s$ and builds in the initial values,

$$\mathcal{L}\left\{f'\right\}(s)=s\,F(s)-f(0),\qquad\mathcal{L}\left\{f''\right\}(s)=s^2F(s)-s\,f(0)-f'(0),$$

and in general $\mathcal{L}\left\{f^{(n)}\right\}(s)=s^nF(s)-\sum_{k=0}^{n-1}s^{n-1-k}f^{(k)}(0).$ Let $a\geq 0$ and $H$ be the Heaviside step function, which is $0$ for $t<0$ and $1$ for $t\geq 0.$ The shift theorems state that multiplying $f$ by an exponential shifts its transform and that delaying $f$ multiplies its transform by an exponential.

$$\mathcal{L}\left\{e^{at}f(t)\right\}(s)=F\left(s-a\right),\qquad\mathcal{L}\left\{H\left(t-a\right)f\left(t-a\right)\right\}(s)=e^{-as}F(s)$$

Also $\mathcal{L}\left\{t\,f(t)\right\}=-F'(s)$ and $\mathcal{L}\left\{\int_0^tf(\tau)\,d\tau\right\}=\frac{F(s)}{s}.$ The convolution of $f$ and $g$ on $\left[0,\infty\right)$ is

$$\left(f*g\right)(t):=\int_0^tf(\tau)\,g\left(t-\tau\right)d\tau,$$

which is commutative, associative and bilinear, and the convolution theorem states that the transform of a convolution is the product of the transforms, $\mathcal{L}\left\{f*g\right\}=F\,G.$

An initial value problem for a linear equation with constant coefficients is solved by transforming both sides, which by the derivative rule gives an algebraic equation for $Y=\mathcal{L}\left\{y\right\},$ solving for $Y$ and inverting, usually by partial fractions and the table of transforms. For $a\,y''+b\,y'+c\,y=g(t)$ with $y(0)=y'(0)=0$ this gives

$$Y(s)=\frac{G(s)}{a\,s^2+b\,s+c},$$

where $\frac{1}{a\,s^2+b\,s+c}$ is the transfer function of the equation and its inverse transform $h$ is the impulse response, so that $y=h*g$ by the convolution theorem; this is variation of parameters for constant coefficients. The method handles discontinuous forcing through the Heaviside function and impulsive forcing through the Dirac delta $\delta,$ which is not a function but the limit of unit-area pulses of shrinking width, defined by $\int_{-\infty}^{\infty}\delta\left(t-a\right)f(t)\,dt=f(a)$ for continuous $f,$ so that $\mathcal{L}\left\{\delta\left(t-a\right)\right\}=e^{-as}$ and the response to $\delta$ is $h.$

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be periodic with period $2\pi$ and Riemann integrable on $\left[-\pi,\pi\right].$ The functions $1,\cos nx,\sin nx$ for $n\in\mathbb{N}$ are orthogonal with respect to the inner product $\langle f,g\rangle:=\int_{-\pi}^{\pi}f(x)\,g(x)\,dx,$ with $\langle\cos nx,\cos nx\rangle=\langle\sin nx,\sin nx\rangle=\pi$ for $n\geq 1$ and $\langle 1,1\rangle=2\pi.$ The Fourier coefficients of $f$ are its inner products with these functions, normalised,

$$a_n:=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos nx\,dx,\qquad b_n:=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\sin nx\,dx,$$

and the Fourier series of $f$ is the series of functions with these coefficients, whose partial sums $S_N$ are trigonometric polynomials.

$$f(x)\sim\frac{a_0}{2}+\sum_{n=1}^{\infty}\left(a_n\cos nx+b_n\sin nx\right)$$

By Euler’s formula the series is equally $\sum_{n=-\infty}^{\infty}c_ne^{inx}$ with complex Fourier coefficients $c_n:=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)\,e^{-inx}\,dx,$ so that $c_{\pm n}=\frac{1}{2}\left(a_n\mp ib_n\right).$ For a function of period $2L$ the same formulae hold with $x$ replaced by $\frac{\pi x}{L}$ and integrals over $\left[-L,L\right]$ divided by $L$ in place of $\pi.$ If $f$ is even then $b_n=0$ and the series is a cosine series; if $f$ is odd then $a_n=0$ and it is a sine series. A function on $\left[0,L\right]$ is expanded in a cosine or sine series by extending it to an even or odd function of period $2L.$

By orthogonality, the partial sum $S_N$ is the trigonometric polynomial of degree $N$ closest to $f$ in the norm $\lVert f\rVert_2:=\sqrt{\langle f,f\rangle},$ since $\lVert f-T\rVert_2^2=\lVert f-S_N\rVert_2^2+\lVert S_N-T\rVert_2^2$ for every trigonometric polynomial $T$ of degree $N,$ and Bessel’s inequality $\frac{a_0^2}{2}+\sum_{n=1}^{N}\left(a_n^2+b_n^2\right)\leq\frac{1}{\pi}\lVert f\rVert_2^2$ follows. The Fourier series converges in mean square to $f,$ meaning that $\lVert f-S_N\rVert_2\rightarrow 0,$ so Bessel’s inequality becomes Parseval’s identity.

$$\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)^2\,dx=\frac{a_0^2}{2}+\sum_{n=1}^{\infty}\left(a_n^2+b_n^2\right)$$

In particular the Riemann–Lebesgue lemma holds: $a_n,b_n\rightarrow 0$ as $n\rightarrow\infty.$ Mean-square convergence is the density of the trigonometric polynomials in the continuous periodic functions, from the Stone–Weierstrass theorem, extended to integrable functions.

Pointwise convergence requires more regularity. The one-sided limits of $f$ at $x$ are the limits $\lim_{t\rightarrow x^-}f(t)$ and $\lim_{t\rightarrow x^+}f(t)$ of $f$ restricted to $t<x$ and to $t>x$ respectively. A function is piecewise $C^1$ on $\left[-\pi,\pi\right]$ if it and its derivative are continuous except at finitely many points, at each of which both have one-sided limits. The Dirichlet convergence theorem states that if $f$ is piecewise $C^1$ then its Fourier series converges at every $x$ to the average of the one-sided limits of $f,$

$$\lim_{N\rightarrow\infty}S_N(x)=\frac{1}{2}\left(\lim_{t\rightarrow x^-}f(t)+\lim_{t\rightarrow x^+}f(t)\right),$$

which is $f(x)$ wherever $f$ is continuous, and that the convergence is uniform on every closed interval on which $f$ is continuous. If $f$ is continuous and piecewise $C^1$ the convergence is uniform on $\mathbb{R}$ and the series may be differentiated term by term wherever $f'$ is itself piecewise $C^1,$ while the series of any piecewise $C^1$ function may be integrated term by term. Near a jump the partial sums overshoot the limit by about $9\%$ of the jump however large $N$ is, the Gibbs phenomenon, which is why convergence is not uniform there. The smoother $f$ is, the faster its coefficients decay: if $f\in C^k$ is periodic then $a_n,b_n$ decay faster than $n^{-k}.$ For example, the odd function equal to $x$ on $\left(-\pi,\pi\right)$ has $b_n=\frac{2\left(-1\right)^{n+1}}{n},$ and Parseval’s identity gives $\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}.$