Power Series, Fourier Series, and Transforms

A numerical series $\sum_{n=0}^\infty a_n$ is the limit of its partial sums $S_N=\sum_{n=0}^N a_n$ as $N\to\infty$. Before working with series of functions, we must be able to decide whether a numerical series converges.

Ratio test (d'Alembert). If $\displaystyle\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right| = L$, then: if $L<1$ the series converges absolutely; if $L>1$ it diverges; if $L=1$ the test is inconclusive. Works well when $a_n$ contains factorials or exponentials.

Root test (Cauchy). If $\displaystyle\limsup_{n\to\infty}\sqrt[n]{|a_n|} = L$, same conclusions as the ratio test. Often more effective when $a_n$ is an $n$-th power.

Leibniz test for alternating series. A series $\sum_{n=0}^\infty (-1)^n b_n$ with $b_n\geq0$ converges if $b_n$ is decreasing ($b_{n+1}\leq b_n$) and $b_n\to0$. Convergence is conditional (not absolute in general).

Absolute convergence. If $\sum|a_n|$ converges, then $\sum a_n$ converges (but not vice versa). Absolutely convergent series can be rearranged: changing the order of terms does not alter the sum.

Notable series: geometric $\sum r^n$ converges to $1/(1-r)$ for $|r|<1$; harmonic $\sum 1/n$ diverges; generalized harmonic $\sum 1/n^p$ converges for $p>1$, diverges for $p\leq1$.

A power series is a series of the form $\sum_{n=0}^\infty c_n (x-x_0)^n$, where $x$ is a variable. Unlike numerical series, convergence depends on $x$.

Radius of convergence. The Cauchy-Hadamard theorem guarantees the existence of $R\in[0,\infty]$ such that:

• For $|x-x_0| < R$: the series converges absolutely (and uniformly on every compact set).
• For $|x-x_0| > R$: the series diverges.
• For $|x-x_0| = R$: must be checked case by case (at the endpoints).

Hadamard formula: $\displaystyle R = \frac{1}{\limsup_{n\to\infty}\sqrt[n]{|c_n|}}$. When the ratio limit exists, $R = \lim_{n\to\infty} |c_n/c_{n+1}|$.

Fundamental properties. Inside the radius of convergence, the power series represents a function $f(x)$ that is:

• Continuous (indeed $C^\infty$).
• Termwise differentiable: $f'(x) = \sum_{n=1}^\infty n c_n (x-x_0)^{n-1}$, with the same $R$.
• Termwise integrable: $\int f(x),dx = \sum_{n=0}^\infty \frac{c_n}{n+1}(x-x_0)^{n+1} + C$, with the same $R$.

This flexibility makes power series the ideal tool for solving differential equations in series form (Frobenius method).

An infinitely differentiable function $f$ at $x_0$ can be expressed locally by the Taylor series:

For $x_0=0$ it is called the Maclaurin series. Not all $C^\infty$ functions are representable by their Taylor series: the remainder $R_N(x)\to0$ must hold, which is true for analytic functions.

Fundamental expansions (memorize these):

• $\displaystyle e^x = \sum_{n=0}^\infty \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$, $R=\infty$.
• $\displaystyle \sin x = \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$, $R=\infty$.
• $\displaystyle \cos x = \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$, $R=\infty$.
• $\displaystyle \frac{1}{1-x} = \sum_{n=0}^\infty x^n$, $R=1$ (geometric series).
• $\displaystyle \ln(1+x) = \sum_{n=1}^\infty \frac{(-1)^{n-1} x^n}{n} = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots$, $R=1$ (converges also at $x=1$).
• $\displaystyle (1+x)^\alpha = \sum_{n=0}^\infty \binom{\alpha}{n} x^n$ (binomial series), $R=1$, where $\binom{\alpha}{n}=\frac{\alpha(\alpha-1)\cdots(\alpha-n+1)}{n!}$.

Building new expansions. Often one starts from a known expansion and applies substitutions, integrations, or differentiations. Example: from $1/(1-x)=\sum x^n$, integrating termwise gives $-\ln(1-x)=\sum x^n/n$.

While a Taylor series approximates locally, the Fourier series represents a periodic function over an entire interval as a sum of sines and cosines (the "fundamental frequencies").

For a function $f$ with period $2L$ (i.e. $f(x+2L)=f(x)$ for every $x$), the Fourier series is:

The Fourier coefficients are computed via the $L^2$ inner product:

Why it works. The functions ${\cos(n\pi x/L),\sin(n\pi x/L)}$ form an orthogonal basis in $L^2[-L,L]$: the integral of the product of two different functions is zero. The coefficients $a_n,b_n$ are the "coordinates" of $f$ in this basis — exactly like projecting a vector onto basis vectors.

Parsimony and symmetry:

• If $f$ is even ($f(-x)=f(x)$): $b_n=0$, only cosines (cosine series).
• If $f$ is odd ($f(-x)=-f(x)$): $a_n=0$, only sines (sine series).
• The term $a_0/2$ is the average value of $f$ over the interval: $\frac{1}{2L}\int_{-L}^{L} f$.

Complex form. Using $e^{i\theta}=\cos\theta+i\sin\theta$, the series writes elegantly as:

The coefficients $c_n$ are complex: for real $f$, $c_{-n}=\overline{c_n}$. The complex form is more compact and generalizes naturally to the Fourier transform.

Once the coefficients are computed, one asks: does the series actually converge to $f(x)$? The answer depends on the regularity of $f$.

Dirichlet theorem (pointwise convergence). If $f$ is bounded and piecewise regular (finitely many jump discontinuities and local extrema in one period), then the Fourier series converges to:

• $f(x)$ at points where $f$ is continuous.
• $\frac{f(x^+)+f(x^-)}{2}$ (the average of left and right limits) at jump discontinuities.

Gibbs phenomenon. Near a jump discontinuity, the Fourier series exhibits oscillations that do not disappear as the number of harmonics increases: the "ringing" amplitude stabilizes at about 9% of the jump, although the affected zone narrows. This is why JPEG images show "artifacts" at sharp edges.

Uniform convergence. If $f$ is continuous and piecewise regular (in particular if $f$ is piecewise $C^1$ and continuous), the Fourier series converges uniformly to $f$ on the whole interval. In this case the convergence is much stronger: no oscillations, no Gibbs.

Parseval identity. For a function $f$ in $L^2[-L,L]$:

Physically: the total energy of the signal is the sum of the energies of its harmonics. It is the infinite-dimensional analogue of the Pythagorean theorem.

The Fourier transform extends the idea of the Fourier series to non-periodic functions: instead of summing over discrete frequencies $n\pi/L$, it integrates over all continuous frequencies $\xi$:

The transform $\hat f(\xi)$ is a complex function that tells "how much" frequency $\xi$ is present in the signal $f(x)$. The inverse ("synthesis") is:

Fundamental properties:

• Linearity: $\widehat{\alpha f+\beta g} = \alpha\hat f + \beta\hat g$.
• Differentiation: $\widehat{f'}(\xi) = 2\pi i\xi,\hat f(\xi)$ — differentiation becomes multiplication by $2\pi i\xi$, turning ODEs into algebraic equations.
• Convolution: $\widehat{f*g}(\xi) = \hat f(\xi),\hat g(\xi)$, where $(f*g)(x)=\int f(t)g(x-t),dt$. Convolution — an expensive operation — becomes a simple product in Fourier space.
• Modulation: $\widehat{f(x)e^{2\pi i\xi_0 x}}(\xi) = \hat f(\xi-\xi_0)$.
• Frequency response: for a linear time-invariant system, the output is $\hat y(\xi)=H(\xi)\hat u(\xi)$, where $H$ is the transfer function.

The Fourier transform is the universal tool for signal processing: audio compression (MP3), images (JPEG), communications (OFDM), solving partial differential equations (heat equation, wave equation).

The Laplace transform is the "cousin" of the Fourier transform, suited to functions defined on $[0,\infty[$ (causal functions). It is the primary tool for solving initial value problems.

Key properties for ODEs. The transform of a derivative eliminates the derivative and introduces initial conditions:

Thanks to these properties, a linear constant-coefficient ODE becomes an algebraic equation in $F(s)$, easily solved.

Table of fundamental transforms:

• $\mathcal{L}{1} = 1/s$
• $\mathcal{L}{t^n} = n!/s^{n+1}$
• $\mathcal{L}{e^{at}} = 1/(s-a)$
• $\mathcal{L}{\sin(\omega t)} = \omega/(s^2+\omega^2)$
• $\mathcal{L}{\cos(\omega t)} = s/(s^2+\omega^2)$
• $\mathcal{L}{t\sin(\omega t)} = 2\omega s/(s^2+\omega^2)^2$
• $\mathcal{L}{e^{at}\sin(\omega t)} = \omega/[(s-a)^2+\omega^2]$
• $\mathcal{L}{\delta(t-t_0)} = e^{-s t_0}$ (Dirac impulse)

Inverse transform. Given $F(s)$, find $f(t)$ using the table in reverse, often with the help of partial fractions to decompose $F(s)$ into simple terms.

The Laplace transform is the most direct method for solving linear Cauchy problems. The procedure is always the same:

General algorithm:

1. Apply $\mathcal{L}$ to both sides of the ODE, using the derivative properties and initial conditions.
2. Solve the algebraic equation for $Y(s)$.
3. Invert $Y(s)$ to obtain $y(t)$, using the table and partial fractions.

RLC circuit. The charge $q(t)$ in a series RLC circuit satisfies $L q'' + R q' + q/C = V(t)$. Transforming with Laplace yields an algebraic equation in $Q(s)$ that includes the initial conditions $q(0)$ and $i(0)=q'(0)$. Inverting gives the complete evolution of the circuit.

Final value theorem. If the limit exists, $\lim_{t\to\infty} f(t) = \lim_{s\to0} sF(s)$. Useful for finding steady-state behavior without inverting: for a stable system, it tells where it settles.

Transfer function. For a linear system with input $u(t)$ and output $y(t)$, define $H(s)=Y(s)/U(s)$ (zero initial conditions). $H(s)$ contains all the system dynamics. The poles of $H(s)$ determine stability: poles with negative real part → stable system.

The combination of Fourier and Laplace provides the tools to analyze any linear system: Fourier for frequency response (sinusoidal steady-state), Laplace for the complete response (transient + steady-state).