Power Series, Exponentials and Bump Functions

A power series is a polynomial that never stops: $\sum_{n=0}^\infty c_n x^n$. Power series are where the functions of school mathematics, $e^x$, $\log x$, $\sin x$ and $\cos x$, finally get definitions rather than descriptions. In Book 2A we used them on trust, and every property assumed there is proved here, using the uniform convergence of 2B.5 Uniform Convergence and Arzelà–Ascoli.

The chapter then goes in two directions that the rest of the guidebook needs. First, the exponential makes sense for matrices, and $e^{tA}$ solves every linear system of differential equations $x' = Ax$ (2B.10 Ordinary Differential Equations). Second, there are functions that are infinitely differentiable but are not given by their power series. The simplest, $e^{-1/x^2}$, is so flat at $0$ that every derivative vanishes there. From it we build bump functions and smooth cutoffs: smooth functions that are $1$ on one region and $0$ outside a slightly larger one. They are humble objects, and they are indispensable. Every local estimate in geometric analysis, including the one at the heart of Perelman's noncollapsing theorem, multiplies by a cutoff function.

By the end of this chapter you will be able to:

• find the radius of convergence of a power series, and differentiate and integrate one term by term;
• define $\exp$, $\log$, real powers, $\sin$ and $\cos$ from scratch and prove their main properties;
• explain why the real power series of $\frac{1}{1+x^2}$ stops converging at $|x| = 1$;
• compute matrix exponentials and use them to solve $x' = Ax$;
• prove that $e^{-1/x^2}$ is smooth and not analytic, and build smooth bumps and cutoffs with prescribed properties.

6.1Dating by decay

The example uses three facts we haven't yet proved: that a function equal to its own derivative (up to a constant factor) must be an exponential, that $\exp$ has an inverse $\log$, and that $\exp$ turns sums into products, which is why a half-life is the same at every age. All three are proved below.

6.2Power series

At $|x - a| = R$ anything can happen: $\sum x^n$ diverges at both ends of $(-1, 1)$, $\sum x^n/n$ converges at $-1$ but not at $1$, and $\sum x^n/n^2$ converges at both.

Analytic functions are rigid. If an analytic function on an interval vanishes on some small subinterval, it vanishes on the whole interval (Exercise 6.13). This rigidity is why analytic functions are not enough for geometry, and why the smooth non-analytic functions at the end of the chapter are needed.

Abel's theorem. If $\sum c_n$ converges, then $\sum c_n x^n \to \sum c_n$ as $x \to 1^-$: the sum is continuous at the endpoint, from inside, whenever the series converges there. For example $\ln(1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots$ for $|x| < 1$ (integrate $\frac{1}{1+x} = \sum(-x)^n$ term by term), and the alternating harmonic series converges (2A.7 Series), so Abel's theorem gives

Tao proves the theorem by summation by parts; the proof is worth reading once.

6.3The exponential and the logarithm

Since $(1/n!)^{1/n} \to 0$ (for instance by the ratio test), the radius of convergence is infinite, and $\exp$ is defined, continuous and infinitely differentiable on all of $\mathbb{R}$. Differentiating term by term, $\exp' = \exp$, and $\exp(0) = 1$. Everything else follows from these two facts.

Part 1 is the uniqueness of solutions of the simplest differential equation, $f' = cf$. It answers the radiocarbon question: decay at a rate proportional to the amount present must be exponential. Part 2 explains why a half-life makes sense: $N(t + T) = N_0 e^{-\lambda t}e^{-\lambda T}$, so waiting a further time $T$ multiplies what is left by the same factor $e^{-\lambda T}$, whatever $t$ is.

The number $e = \exp(1) = \sum \frac1{n!} = 2.71828\ldots$ agrees with the limit $\lim(1 + \tfrac1n)^n$ from 2A.6 Sequences (Exercise 6.12). By part 2, $\exp(q) = e^q$ for every rational $q$, so $\exp(x)$ is the real power $e^x$ in the sense of 2A.6 Sequences, and we write $e^x$ from now on.

The inverse function theorem of one variable (2A.10 Derivatives) gives $\log'(y) = 1/\exp(\log y) = 1/y$. With $\log 1 = 0$ and the fundamental theorem, $\log y = \int_1^y \frac{dt}{t}$: the logarithm of 2A.11 The Riemann Integral, defined as an area, is the same function. The rules $\log(xy) = \log x + \log y$, $a^{x+y} = a^xa^y$ and $(a^x)^y = a^{xy}$ follow from part 2 of the proposition. (Mathematicians write $\log$ for the natural logarithm, and so does the guidebook from here on; $\ln$ means the same.)

6.4Complex exponentials and trigonometry

Tao's "digression on complex numbers" is the shortest route to $\sin$ and $\cos$. Complex numbers $z = x + iy$, with $i^2 = -1$ and $|z| = \sqrt{x^2 + y^2}$, form a complete metric space with $d(z, w) = |z - w|$ (it is $\mathbb{R}^2$ with the Euclidean metric). All of Theorem 6.2 works for complex power series, with "interval" replaced by "disc" $|z - a| < R$. So

converges for every complex $z$, and the proof of $e^{z + w} = e^ze^w$ goes through (or multiply the series, using the binomial theorem; Tao's section on multiplication of power series justifies the rearrangement).

Everything about trigonometry follows. Since $\overline{e^{i\theta}} = e^{-i\theta}$, we get $|e^{i\theta}|^2 = e^{i\theta}e^{-i\theta} = 1$, that is, $\cos^2\theta + \sin^2\theta = 1$. Differentiating the series, $\cos' = -\sin$ and $\sin' = \cos$. The addition formulas are the real and imaginary parts of $e^{i(\alpha + \beta)} = e^{i\alpha}e^{i\beta}$. And $\pi$ can be defined as twice the smallest positive zero of $\cos$; one shows it exists (by the intermediate value theorem, since $\cos 0 = 1$ and $\cos 2 < 0$) and that $\cos$ and $\sin$ have period $2\pi$. So $t \mapsto e^{it}$ goes once around the unit circle at unit speed as $t$ runs over $[0, 2\pi)$, which is the connection with angles.

6.4.1Why the series for $\frac{1}{1+x^2}$ stops at $1$

The function $f(x) = \frac{1}{1+x^2}$ is perfectly smooth on the whole real line. Its Taylor series at $0$ is the geometric series

which has radius of convergence $1$ and diverges for $|x| > 1$ (Figure 6.1). Nothing happens to $f$ at $x = \pm 1$, so why should the series stop there?

The answer is in the complex plane. As a function of a complex variable, $\frac{1}{1 + z^2}$ blows up at $z = \pm i$, where $1 + z^2 = 0$. A complex power series converges on a disc, and the disc centred at $0$ can't extend past the nearest point where the function misbehaves. That point, $i$, is at distance $1$. The real interval of convergence is just the disc's shadow on the real line.

This is the seam to complex analysis (Book 5A, optional), where it becomes a theorem: the radius of convergence of the Taylor series of a complex-differentiable function at a point is the distance to the nearest singularity.

6.5The matrix exponential

The exponential series makes sense whenever we can multiply, add and take limits. For a square matrix $A$,

To see that it converges, measure matrices by the operator norm $\|A\| = \sup\{|Av| : |v| \leq 1\}$. It satisfies $\|AB\| \leq \|A\|\,\|B\|$, so $\|A^k/k!\| \leq \|A\|^k/k!$, and the series converges absolutely by the M-test, in the complete space of $n \times n$ matrices ($\mathbb{R}^{n^2}$, any norm, 2B.3 Compactness). Moreover $\|e^A\| \leq e^{\|A\|}$.

The hypothesis $AB = BA$ is essential: for most pairs of matrices $e^{A+B} \neq e^Ae^B$. That failure is the beginning of Lie theory (8A.5 Lie Groups and Group Actions).

6.6Smooth functions that are not analytic

Cauchy gave this example in 1823 to show that a function is not determined by its Taylor series. For analysis it is a gift: it lets us build smooth functions that are exactly zero in one place and positive in another, which no analytic function can do.

6.6.1Bumps and cutoffs

Use the one-sided version: $\psi(x) = e^{-1/x}$ for $x > 0$ and $\psi(x) = 0$ for $x \leq 0$. The same proof shows $\psi$ is smooth, with all derivatives $0$ at $0$. From $\psi$:

• A bump. $\phi(x) = \psi(1 - x^2)$ is smooth, positive on $(-1, 1)$, and zero outside. In $\mathbb{R}^n$, $\phi(|x|)$ is a smooth bump on the unit ball (smooth even at $0$, because $1 - |x|^2$ is a polynomial in the coordinates).
• A smooth step. $S(x) = \dfrac{\psi(x)}{\psi(x) + \psi(1 - x)}$ is smooth (the denominator is never $0$), equal to $0$ for $x \leq 0$, to $1$ for $x \geq 1$, and increasing in between (Exercise 6.14).
• A cutoff. $\chi(x) = S(2 - |x|)$ equals $1$ on $[-1, 1]$, $0$ outside $(-2, 2)$, and is smooth everywhere: near $0$ it is constant, so the corner of $|x|$ doesn't matter.

Rescaling gives cutoffs adapted to any scale: $\chi(x/\rho)$ is $1$ on the ball of radius $\rho$ and $0$ outside radius $2\rho$, and its derivatives scale like $|\chi'(x/\rho)/\rho| \leq C/\rho$ and $C/\rho^2$ for the second derivative (Exercise 6.17). That scaling is the whole art of localising an estimate: the cost of cutting off at radius $\rho$ is a factor $1/\rho$ per derivative.

6.7History

Newton found the binomial series and power series for sine and cosine in the 1660s; Euler, in his Introductio in analysin infinitorum (1748), made the exponential function central and wrote down $e^{ix} = \cos x + i\sin x$. Cauchy's 1821 Cours d'analyse gave the radius of convergence via what is now the root test, and Hadamard rediscovered the formula $1/\limsup|c_n|^{1/n}$ in 1888. Abel's theorem dates from 1826. Cauchy's flat function appeared in 1823. The matrix exponential is implicit in 19th-century work on linear differential equations and in Sophus Lie's theory of continuous groups (1870s onwards).

6.8Exercises

Solution

(a) $1$. (b) $3$, since $(2^n + 3^n)^{1/n} \to 3$. (c) $0$. (d) $\infty$ (this is $\cosh x$). (e) $1$: the coefficients are $1$ at perfect squares and $0$ elsewhere, so $\limsup|c_n|^{1/n} = 1$.

Solution

$S$ is open by definition and contains $(c, d)$. If $a \in I$ is a limit of points $s_k \in S$, then each $f^{(n)}$ is continuous and vanishes at the $s_k$, so $f^{(n)}(a) = 0$ for all $n$. Near $a$, $f$ equals its Taylor series at $a$, which is $0$; so $a \in S$. Thus $S$ is clopen in the connected interval $I$, and $S = I$. A bump vanishes on an interval but not everywhere, so it can't be analytic.

Hint

For monotonicity, write $S = \frac{1}{1 + \psi(1-x)/\psi(x)}$ on $(0, 1)$ and check that $\psi(1 - x)/\psi(x) = \exp\big(\frac1x - \frac1{1-x}\big)$ is decreasing.

Solution

(a) $\mathrm{diag}(e^{2t}, e^{-t})$: growth along the first axis, decay along the second (a saddle). (b) $A^2 = 0$, so $e^{tA} = I + tA = \begin{pmatrix}1 & t\\0 & 1\end{pmatrix}$: shear, growing only linearly. (c) $A = aI + bJ$ with $aI$ and $bJ$ commuting, so $e^{tA} = e^{at}\begin{pmatrix}\cos bt & -\sin bt\\ \sin bt & \cos bt\end{pmatrix}$: spirals, outward if $a > 0$, inward if $a < 0$. Finally, $e^Ae^B = \begin{pmatrix}1&1\\0&1\end{pmatrix}\begin{pmatrix}1&0\\1&1\end{pmatrix} = \begin{pmatrix}2 & 1\\1 & 1\end{pmatrix}$, while $e^{A+B} = \cosh 1\,I + \sinh 1\,(A + B) = \begin{pmatrix}\cosh 1 & \sinh 1\\ \sinh 1 & \cosh 1\end{pmatrix}$.

Solution

(a) $\chi_\rho(x) = 1$ when $|x|/\rho \leq 1$ and $0$ when $|x|/\rho \geq 2$. The function is smooth: near $0$ it is identically $1$, and away from $0$, $|x|$ is smooth. By the chain rule, $\partial_i\chi_\rho = \chi'(|x|/\rho)\frac{x_i}{\rho|x|}$, bounded by $C_1/\rho$; the second derivatives involve $\chi''/\rho^2$ and $\chi'/(\rho|x|)$, and the latter is only non-zero for $|x| \geq \rho$, so it is at most $C_1/\rho^2$. (b) $\nabla(\chi_\rho u) = \chi_\rho\nabla u + u\nabla\chi_\rho$, and $|a + b|^2 \leq 2|a|^2 + 2|b|^2$, with $0 \leq \chi_\rho \leq 1$ supported in $B(0, 2\rho)$.