Euler — The Exponential is a Circle
Leonhard Euler (1707–1783) wrote the Rosetta Stone that translates between growth, rotation, and shape. He gave us $e$, named $i$, popularized $\pi$, and proved they are one thing.
Take compound interest: $(1+1/n)^n\to e$. Take rotation: multiplying by $i$ is $90^\circ$. Take the circle: $x^2+y^2=1$. Euler proved these are the same object seen from three sides.
Three proofs, each a worldview: Taylor: $e^{i\theta}=1+i\theta-\theta^2/2!-i\theta^3/3!+\cdots = (1-\theta^2/2!+\cdots)+i(\theta-\theta^3/3!+\cdots)=\cos\theta+i\sin\theta$. ODE: $f(\theta)=\cos\theta+i\sin\theta$ satisfies $f'=if$, $f(0)=1$, whose unique solution is $e^{i\theta}$. Limit: $e^{i\theta}=\lim_{n\to\infty}(1+i\theta/n)^n$ — $n$ tiny rotations by $\theta/n$ compose to full rotation $\theta$.
Corollary: multiplication by $e^{i\theta}$ rotates the complex plane by $\theta$. The circle group $S^1$ is the exponential image of $i\mathbb{R}$.
Five constants, three operations, one equality. $0$ additive identity, $1$ multiplicative identity, $\pi$ shape, $e$ growth, $i$ rotation. Feynman called it “our jewel”. It is not mystical — it is inevitable once rotation is exponentiation.
The three constants Euler made canonical
$\pi$ — ratio $C/d$. Euler made it analysis: $\zeta(2)=\pi^2/6$, $\prod$, integrals. $e$ — base where $\frac{d}{dx}e^x=e^x$, $e=\sum1/n!$, $e=\lim(1+1/n)^n$. $i$ — $i^2=-1$, geometry of plane, $i=e^{i\pi/2}$.
The unit circle parametrizes as $(\cos\theta,\sin\theta)=e^{i\theta}$. Euler bound analysis to geometry.
Euler's other circles — geometry you were not told
Euler line: In any non-equilateral triangle, orthocenter $H$, centroid $G$, circumcenter $O$ collinear with $OG:GH=1:2$. Vector proof: put $O$ at origin, then $\vec{OH}=\vec{OA}+\vec{OB}+\vec{OC}$ and $\vec{OG}=\frac13(\vec{OA}+\vec{OB}+\vec{OC})$.
Nine-point / Euler circle: Midpoints of sides (3), feet of altitudes (3), midpoints $A\!-\!H$, $B\!-\!H$, $C\!-\!H$ (3) — all one circle. Center $N$ midpoint of $OH$, radius $R/2$. Feuerbach proved it tangent to incircle. Euler discovered 1765.
$O$ circumcenter, $I$ incenter, $R$ circumradius, $r$ inradius. Distance between centers dictated by radii.
Euler tames $\pi$ and invents $\zeta$
Basel problem (1734): $\sum_{n\ge1}1/n^2=\pi^2/6$. Euler factored $\sin x/x=\prod_{n\ge1}(1-x^2/n^2\pi^2)$ and compared $x^2$ coefficient. Generalization $\zeta(2k)$ rational multiple of $\pi^{2k}$ via Bernoulli numbers.
Expand each factor as geometric series $1+p^{-s}+p^{-2s}+\cdots$ (Part II Taylor) and multiply: every $n^{-s}$ appears exactly once — Fundamental Theorem of Arithmetic. Since harmonic series diverges, $\prod_p(1-1/p)^{-1}$ diverges ⇒ $\sum_p1/p$ diverges — Euclid with teeth.
Interactive — $e^{i\theta}$ is rotation
Interactive — Euler line & nine-point circle
Taylor Series & the Question of Convergence
When does local derivative data determine a function globally? The answer is a disk whose edge is drawn by invisible complex singularities — the first hint of Riemann's world.
For $f$ $C^\infty$ at $a$, Taylor polynomial $P_n(x)=\sum_{k=0}^n f^{(k)}(a)/k!(x-a)^k$. The series is $P_\infty$. Remainder $R_n=f-P_n$.
Three remainder forms
Lagrange, Cauchy, integral. If $|f^{(n+1)}|\le M$ then $|R_n|\le M|x-a|^{n+1}/(n+1)!\to0$ because factorial beats exponential. Hence $e^x,\sin x,\cos x$ have $R=\infty$.
Radius — Hadamard
Converges absolutely $|x-a|<R$, diverges $>R$. Example $1/(1-x)=\sum x^n$, $|x|<1$; $\ln(1+x)=\sum(-1)^{n+1}x^n/n$, $|x|<1$ plus $x=1$; $(1+x)^\alpha$ binomial $R=1$ unless $\alpha\in\mathbb N$.
Key: $f(x)=1/(1+x^2)$ has $R=1$ at $0$ not because of real trouble, but poles at $\pm i$. Radius = distance to nearest singularity in $\mathbb C$. This is bridge Euler→Riemann.
When Taylor fails
$f(x)=e^{-1/x^2}$, $f(0)=0$ is $C^\infty$ with all $f^{(n)}(0)=0$. Maclaurin series $0\neq f$. Smooth $\neq$ analytic. Cantor function gives worse.
Interactive — Taylor approximator
Cantor — How Many Points Does Convergence Need?
Fourier series $\sum c_n e^{int}$ are Taylor series on the unit circle. Asking when they converge uniquely forced Cantor to invent set theory.
Countable vs uncountable
$\mathbb N,\mathbb Z,\mathbb Q$ countable $|\mathbb N|=\aleph_0$. $\mathbb R$ not. Diagonal: assume enumeration $r_1,r_2,\dots$ of $[0,1]$, build $d$ differing at $n$-th digit from $r_n$. Then $d$ not in list. So $|\mathbb R|>|\mathbb N|$, $|\mathcal P(\mathbb N)|=2^{\aleph_0}$.
Cantor set $C$
Closed, perfect, nowhere dense, totally disconnected, measure $0$ (removed $\sum2^n/3^{n+1}=1$), uncountable: $x\in C\iff x=\sum a_n3^{-n}$, $a_n\in\{0,2\}$ bijection to binary, $|\,C\,|=2^{\aleph_0}$, self-similar $C=\frac13C\cup(\frac23+\frac13C)$, $\dim_H=\log2/\log3$.
Cantor function — Devil's staircase
Continuous, monotone, $f'=0$ a.e. (on complement of $C$), yet $f(0)=0,f(1)=1$. Climbs purely on measure-zero set. Constant on each removed middle third — shows why Taylor integral remainder needs absolute continuity.
Riemann — The Function Beyond Its Series
Euler's product $\sum n^{-s}=\prod_p(1-p^{-s})^{-1}$ converges for $\Re(s)>1$. Riemann continued it past its wall to the whole plane — and found a line.
For $\Re(s)>1$, $\zeta(s)=\sum n^{-s}$. Euler product is geometric series (Taylor) + Fundamental Theorem of Arithmetic. At $s=1$ harmonic diverges.
Analytic continuation — Weierstrass chain of Taylor discs
$\sum x^n$ defines $1/(1-x)$ only $|x|<1$, but function exists elsewhere. Same for $\zeta$. Alternating $\eta(s)=\sum(-1)^{n-1}n^{-s}$ converges $\Re(s)>0$, and $\zeta=\eta/(1-2^{1-s})$ extends to $\Re(s)>0$ minus pole. Riemann's functional equation:
Mirror $s\leftrightarrow1-s$ across $\Re=1/2$. Trivial zeros at $-2,-4,\dots$ from $\sin$ factor. Other zeros in critical strip $0\le\Re\le1$, paired $ \rho,1-\rho,\bar\rho,1-\bar\rho$.
Every nontrivial zero has $\Re(\rho)=1/2$.
Very probable, Riemann wrote, after fleeting futile attempts. 167 years, $10^{13}$ zeros verified on line, 0 off, and a one-million-dollar Clay prize.
Why zeros matter — explicit formula
$\psi(x)=\sum_{p^k\le x}\ln p$ weighted prime count. Primes = smooth $x$ + wave $x^{\rho}/\rho=x^{\beta}e^{i\gamma\ln x}/\rho$ per zero. $\Im\rho=\gamma$ frequency, $\Re\rho=\beta$ amplitude $x^{\beta}$. If all $\beta=1/2$, error $\pi(x)-\mathrm{Li}(x)=O(\sqrt{x}\log x)$ — smallest symmetry allows. One zero off line = rogue wave, persistent bias.
Evidence & equivalences
| Year | Result | Meaning |
|---|---|---|
| 1896 | Hadamard, de la Vallée Poussin | No zeros on Re=1 ⇒ PNT |
| 1914 | Hardy | ∞ many zeros on line |
| 1942–89 | Selberg, Levinson, Conrey | >40% zeros on line |
| 1973 | Montgomery–Dyson | Pair correlation = GUE eigenvalues |
| 2004 | Gourdon | 10¹³ zeros on line |
| 1974 | Deligne | Weil analogue RH theorem |
- Mertens: RH ⇔ $M(x)=\sum_{n\le x}\mu(n)=O(x^{1/2+\epsilon})$
- Robin: RH ⇔ $\sigma(n)<e^\gamma n\ln\ln n$ for $n>5040$
- Nyman–Beurling: RH ⇔ dilated fractional parts dense in $L^2$
- Li: RH ⇔ $\lambda_n\ge0$ where $\lambda_n$ are Taylor coefficients of $\ln\xi(s)$ at $s=1$ — Taylor to the last
GNU Octave Lab
Self-contained scripts — Euler, Taylor, Cantor, Riemann. No toolboxes, core plot only.
Euler
% euler_formula.m - e^{i theta} = cos theta + i sin theta
theta = linspace(0,2*pi,1000);
z = exp(1i*theta);
figure; plot(cos(theta),sin(theta),'k--'); hold on;
plot(real(z), imag(z),'b-','LineWidth',2); axis equal; grid on;
title('Unit circle = image of iR under exp');
% Verify identity
fprintf('e^{i pi}+1 = %.2e + %.2ei\n', real(exp(1i*pi)+1), imag(exp(1i*pi)+1));
% Euler line and nine-point for triangle [0 0],[4 0],[1 3]
A=[0 0]; B=[4 0]; C=[1 3];
D=2*(A(1)*(B(2)-C(2))+B(1)*(C(2)-A(2))+C(1)*(A(2)-B(2)));
O=[(A(1)^2+A(2)^2)*(B(2)-C(2))+(B(1)^2+B(2)^2)*(C(2)-A(2))+(C(1)^2+C(2)^2)*(A(2)-B(2));
(A(1)^2+A(2)^2)*(C(1)-B(1))+(B(1)^2+B(2)^2)*(A(1)-C(1))+(C(1)^2+C(2)^2)*(B(1)-A(1))]'/D;
H=A+B+C-2*O; Nc=(O+H)/2; R=norm(A-O);
th=linspace(0,2*pi,300);
figure; plot([A(1) B(1) C(1) A(1)],[A(2) B(2) C(2) A(2)],'k-','LineWidth',2); hold on;
plot(O(1)+R*cos(th), O(2)+R*sin(th),'b--'); plot(Nc(1)+R/2*cos(th), Nc(2)+R/2*sin(th),'r-','LineWidth',2);
plot([O(1) H(1)],[O(2) H(2)],'m-'); axis equal; grid on;
title('Euler line O-G-H and nine-point circle R/2');
% euler_product.m - zeta(2)=pi^2/6 via Euler product
s=2; N=100000; n=1:N; z_sum=sum(n.^(-s));
fprintf('sum 1/n^2 N=%d %.10f true pi^2/6 %.10f err %.2e\n',N,z_sum,pi^2/6,abs(z_sum-pi^2/6));
plist=[2 3 5 7 11 13 17 19 23 29 31 37 41 43 47]; % avoid shadowing primes()
z_prod=1; for p=plist, z_prod*=1/(1-p^(-s)); end
fprintf('Euler product %d primes %.10f\n',length(plist),z_prod);
Taylor
% taylor_convergence.m - remainder decay vs radius wall
clear; close all;
x=3; N=0:25;
% e^x error at x=3, entire function R=inf
err_exp=[]; for n=N, s=0; for k=0:n, s+=x^k/factorial(k); end; err_exp(end+1)=abs(exp(x)-s); end
figure; semilogy(N,err_exp,'o-'); grid on; xlabel('N'); ylabel('|R_N|'); title('e^x at x=3, R=inf, factorial wins');
% 1/(1-x) outside radius: x=1.2 diverges
x=1.2; partial=[]; s=0; for n=0:20, s+=x^n; partial(end+1)=s; end
figure; plot(0:20,partial,'o-'); grid on; title('1/(1-x) at x=1.2 outside R=1 diverges');
Cantor
% cantor_set.m
1; % script marker: without this, a leading `function` makes Octave
% treat the whole file as a function file and the demo never runs.
function C=cantor_levels(n)
C=[0 1];
for k=1:n
newC=[];
for i=1:size(C,1)
l=C(i,1); r=C(i,2); m=(r-l)/3;
newC=[newC; l l+m; r-m r];
end
C=newC;
end
end
n=5; C=cantor_levels(n);
figure; hold on;
for i=1:size(C,1), plot(C(i,:),[0 0],'k-','LineWidth',4); end
axis([0 1 -0.2 0.2]); title(sprintf('Cantor C_%d: %d intervals length (2/3)^%d=%.4f',n,2^n,n,(2/3)^n));
% cantor_function.m - Devil's staircase
1; % script marker (see cantor_set.m)
function y=cantor_func(x,depth)
y=zeros(size(x));
for idx=1:numel(x)
xi=x(idx); yv=0;
for i=1:depth
xi*=3; d=min(floor(xi+1e-12),2); xi-=d; % clamp: x=1 edge gives digit 3
if d==0,
elseif d==1, yv+=0.5^i; break;
else yv+=0.5^i;
end
end
y(idx)=yv;
end
end
x=linspace(0,1,2000); y=cantor_func(x,10);
figure; plot(x,y,'LineWidth',1.5); grid on; title('Cantor function f''=0 a.e.');
Riemann
% zeta_euler_prime.m
1; % script marker: functions below, script code after
function z=zeta_sum(s,N), n=1:N; z=sum(n.^(-s)); end
function z=zeta_euler(s,plist), z=1; for p=plist, z/=1-p^(-s); end; end
function c=prime_pi(n), c=sum(isprime(1:n)); end
plist=[2 3 5 7 11 13 17 19 23 29]; % avoid shadowing built-in primes()
fprintf('zeta(2) sum %.6f product %.6f pi^2/6 %.6f\n',zeta_sum(2,10000),zeta_euler(2,plist),pi^2/6);
% Prime pi vs Li (offset logarithmic integral: li(2) ~ 1.04516)
x=1000; c=prime_pi(x); t=linspace(2,x,20000); Li=trapz(t,1./log(t))+1.04516378;
fprintf('pi(%d)=%d Li~%.1f err %.1f\n',x,c,Li,Li-c);
% zeta_zeros.m - plot critical strip and first zeros
zeros_imag=[14.13472514,21.02203964,25.01085758,30.42487612,32.93506159,37.58617816];
figure; hold on;
plot([0 0],[0 40],'k--'); plot([1 1],[0 40],'k--'); plot([0.5 0.5],[0 40],'r-','LineWidth',1.5);
for k=1:length(zeros_imag), plot(0.5,zeros_imag(k),'bo','MarkerFaceColor','b'); end
xlabel('Re(s)'); ylabel('Im(s)'); title('Critical strip, zeros on Re=1/2 (RH)'); axis([-0.5 1.5 0 40]); grid on;
% Functional eq symmetry: if rho zero, so is 1-rho and conj
From a shell: octave --persist euler_formula.m, or type the script name at the Octave prompt. Files that define local functions (cantor_set.m, cantor_function.m, zeta_euler_prime.m) begin with the 1; marker so Octave parses them as scripts rather than function files — omit it and the demo code after the functions silently never runs. Everything uses core plotting only; no Forge packages required. All six scripts verified on Octave 8.4.
One Story, Four Voices
Euler gave us the exponential as a circle. Taylor asked when a circle of convergence can be trusted to represent a function. Cantor, forced by Fourier series — Taylor on the circle — to examine the continuum, found infinities beyond counting and a set of measure zero that is as large as the line itself. Riemann stood on all three: he continued Euler's product past its natural wall — a Taylor re-centering chain — into the complex plane Cantor made rigorous, and found a single line on which the music of the primes depends.
A series, a set, a line, and the exponential that binds them. The infinite, told four times, is one story.