Complex Analysis
Picard's Theorem and Exceptional Values
What happens to a function at a point where it behaves infinitely chaotically? Picard's theory gives a shocking answer: near an essential singularity, a function visits almost every point of the complex plane infinitely many times!
- **Quantum mechanics:** essential singularities appear in wave functions during scattering analysis - their behavior determines the energy spectrum
- **Chaos theory:** dynamical systems with essential singularities exhibit behavior closely described by Picard's theory
- **Complex dynamics:** Julia and Mandelbrot sets are connected to points where iterates behave like essential singularities
Предварительные знания
Essential Singularities
e^z hits every complex value except 0 - infinitely often in any neighborhood of the origin. This is Picard's Great Theorem for essential singularities. The Mandelbrot set boundary exploits exactly this chaotic behavior.
Let z₀ be an isolated singularity of f(z). It is **removable** if the Laurent series has no negative powers; a **pole of order m** if the principal part is finite; **essential** if the principal part contains infinitely many terms.
The function f(z) = sin(1/z) has at z = 0:
Casorati-Weierstrass Theorem
**Casorati-Weierstrass theorem**: if z₀ is an essential singularity of f(z), then in any punctured neighborhood of z₀ the image of f is dense in ℂ. In other words, f comes arbitrarily close to every complex number near an essential singularity.
Near an essential singularity the function "goes wild": it takes values arbitrarily close to any point in the complex plane. The image of any punctured neighborhood is everywhere dense in ℂ.
By the Casorati-Weierstrass theorem, the image of a punctured neighborhood of an essential singularity:
Little Picard Theorem
**Little Picard theorem**: a non-constant entire function takes every complex value, with at most one exception. This strengthens Liouville's theorem: where Liouville says bounded entire functions are constant, Picard says non-constant entire functions are nearly surjective.
The single value that an entire function may omit is called its **exceptional** or **lacunary value**. Classic example: e^z never takes the value 0. If a function omits two values, it must be constant.
The little Picard theorem states that a non-constant entire function takes all complex values except:
Great Picard Theorem (Picard-Sokhotski)
**Great Picard theorem (Picard-Sokhotski)**: in any punctured neighborhood of an essential singularity z₀, the function f(z) takes every complex value, with at most one exception, infinitely many times. This is the strongest possible statement about behavior near an essential singularity.
Casorati-Weierstrass: image is dense in ℂ (qualitative). Little Picard: entire functions are nearly surjective globally. Great Picard: near an essential singularity, the function takes almost every value infinitely often-dramatically stronger than density.
The great Picard theorem states that near an essential singularity, the function takes each non-exceptional value:
Key Ideas
- **Essential singularity**: isolated singular point with infinitely many negative-power terms in the Laurent series
- **Casorati-Weierstrass**: the image of any punctured neighborhood of an essential singularity is dense in ℂ
- **Little Picard theorem**: a non-constant entire function takes all values with at most one exception
- **Great Picard theorem**: near an essential singularity, the function takes every value (except possibly one) infinitely many times
Related Topics
Picard's theory builds on Laurent series and the theory of singularities, and connects to entire functions:
- Taylor and Laurent Series — The Laurent series classifies singular points by the form of the principal part
- Residue Theory — Residues are defined by the coefficient a₋₁ of the Laurent series near a singular point
- Entire and Meromorphic Functions — The little Picard theorem is the key global result about the behavior of entire functions
Вопросы для размышления
- Why are essential singularities "worse" than poles? What qualitatively different behavior distinguishes them?
- The function e^z misses the value 0. Can an entire function miss both 0 and 1 simultaneously?
- How does Picard's theorem relate to Liouville's theorem? Is one a consequence of the other?