Complex Analysis
Analytic Continuation
In 1859 Bernhard Riemann analytically continued $\zeta(s) = \sum n^{-s}$ beyond the half-plane $\mathrm{Re}\,s > 1$ in eight pages. The continuation gives $\zeta(-1) = -1/12$. In 1948 Hendrik Casimir predicted that two conducting plates feel an attractive force proportional to $\zeta(-3)$. In 1997 Steve Lamoreaux at Los Alamos measured that force to within 5%, matching the theoretical value $\hbar c \pi^2 / (240 d^4)$. The divergent series $1 + 2 + 3 + \ldots$, via analytic continuation, became a measurable physical force.
- **Casimir effect (Lamoreaux, 1997):** the force between parallel conducting plates agrees with $\hbar c \pi^2 / (240 d^4)$, a direct use of $\zeta(-3) = 1/120$
- **String theory:** the critical bosonic-string dimension $d = 26$ follows from $\sum_{n=1}^\infty n = -1/12$; analytic continuation gives the right physics
- **Riemann Hypothesis (Clay Millennium Prize):** \$1,000,000 for a proof; $10^{13}$ zeros verified on the critical line
- **Wick rotation in QFT:** continuing $t \to i\tau$ turns the Schrödinger equation into a heat equation - the basis of Path Integral Monte Carlo
Предварительные знания
The Identity Theorem
The Identity Theorem is one of the most striking facts in complex analysis: an analytic function is completely determined by its values on any sequence with a limit point. Analytic functions have no 'local' freedom.
**Identity Theorem:** if f is analytic in a connected domain D, and the zero set of f has a limit point in D, then f ≡ 0 in D. **Corollary:** if f = g on any sequence with a limit point in D, then f ≡ g in D. **Rigidity:** there is no C^∞ bump function that is analytic - analytic functions cannot be 'locally zero' without being globally zero. **Real vs. complex:** a real-analytic function can agree on a set without a limit point and still differ. Complex analyticity is far more rigid.
f and g are entire functions that agree on all points 1/n for n=1,2,3,.... By the Identity Theorem:
Analytic Continuation Along Paths
Analytic continuation extends a function beyond its original domain of definition by overlapping disks of convergence. Different paths of continuation may yield different values - leading to multivalued functions.
**Continuation by overlapping elements:** if f₁ in D₁ and f₂ in D₂ agree on D₁∩D₂ ≠ ∅, then f₂ is the analytic continuation of f₁ to D₂. **Monodromy Theorem:** if f continues analytically along every path in a simply connected domain D, the continuation is single-valued in D. **Multi-valuedness:** in non-simply connected domains (e.g. C\{0}), continuation around a loop may change the value: - ln z: value increases by 2πi after CCW loop around 0 - √z: sign flips after CCW loop around 0
Continuing ln z counterclockwise around the origin by 2π gives:
Riemann Surfaces
A Riemann surface is the geometric object on which a multivalued function becomes single-valued. Rather than cutting the plane, we build a surface where each 'sheet' represents one branch.
**Riemann surface for √z:** - Two sheets glued along the branch cut [0,+∞) - Transition between sheets when crossing the cut - On this surface, √z is a genuine single-valued function **Riemann surface for ln z:** - Infinitely many sheets (one per 2πi multiple) - Helical structure: the 'parking garage' topology **Genus:** topological invariant χ = 2−2g - Sphere (g=0): Riemann sphere for rational functions - Torus (g=1): elliptic curves, doubly periodic functions
The Riemann surface for √z has:
Analytic Continuation of the Zeta Function
The Riemann zeta function ζ(s) = Σn⁻ˢ (Re s > 1) extends analytically to all of C\{1}. The continuation encodes deep arithmetic in its zeros - and ζ(−1) = −1/12 is a precise mathematical result, not a paradox.
**Continuation via functional equation:** ζ(s) = 2ˢπˢ⁻¹ sin(πs/2) Γ(1−s) ζ(1−s) **Values outside the series domain:** - ζ(0) = −1/2 - ζ(−1) = −1/12 (NOT the sum 1+2+3+...) - ζ(−2n) = 0 (trivial zeros) **Riemann Hypothesis:** all nontrivial zeros lie on the critical line Re s = 1/2. **Key point:** ζ(−1) = −1/12 is the value of the analytic continuation, which exists and is finite - not a regularization of a divergent series.
Riemann 1859: an eight-page revolution
In 1859 Bernhard Riemann's eight-page memoir *Über die Anzahl der Primzahlen unter einer gegebenen Grösse*, written for the Berlin Academy, analytically continued $\zeta(s)$ to the whole plane via a keyhole contour and the functional equation $\zeta(s) = 2^s \pi^{s-1} \sin(\pi s/2) \Gamma(1-s) \zeta(1-s)$. In the same paper Riemann stated his hypothesis: every nontrivial zero of $\zeta(s)$ lies on the line $\mathrm{Re}\,s = 1/2$. After 165 years (as of 2024), the first $10^{13}$ zeros have been verified to sit on the critical line, yet no general proof exists. The hypothesis is one of the seven Clay Mathematics Institute Millennium Problems, with a \$1,000,000 prize.
ζ(−1) = −1/12 because:
Key Ideas
- **Identity theorem:** analytic f is determined by values on any sequence with a limit point
- **Continuation:** extend via overlapping discs; path-dependence → multivalued functions
- **Riemann surfaces:** the correct domain where multivalued functions become single-valued
- **ζ(−1)=−1/12:** analytic continuation value; the series Σn diverges, the function does not
Connected Topics
Analytic continuation unifies the whole theory:
- Taylor and Laurent Series — Overlapping discs of convergence are the mechanism of analytic continuation
- Harmonic Functions — Harmonic continuation parallels analytic continuation via the Cauchy-Riemann equations
- Complex Analysis in Physics — Wick rotation t→iτ is analytic continuation in the time variable
Вопросы для размышления
- How does the Identity Theorem explain why no analytic function can be 'supported' on a compact set (unlike smooth bump functions)?
- What happens to analytic continuation when two paths give different values? How does the Riemann surface resolve this?
- Why has the Riemann Hypothesis resisted proof despite numerical verification of trillions of zeros on the critical line?