Key Properties of Analytic Functions

Why This Matters

Analytic functions are the heart of complex analysis—they're the "well-behaved" functions that make the entire subject work. You're being tested on your ability to recognize what makes a function analytic, how analyticity constrains behavior, and what happens when analyticity breaks down. These properties aren't isolated facts; they form an interconnected web where differentiability implies infinite differentiability, which implies power series representation, which implies the stunning rigidity captured by results like the Identity Theorem.

The key insight is that complex differentiability is far more restrictive than real differentiability. This restriction gives analytic functions remarkable properties: their values are determined by their behavior on boundaries, they can't have isolated "bumps," and knowing a function on even a tiny set can determine it everywhere. Don't just memorize definitions—understand why each property follows from complex differentiability and how these properties connect to each other.

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Foundations: What Makes a Function Analytic

The defining characteristic of analytic functions is complex differentiability, but this simple requirement has profound consequences. A single complex derivative existing in a neighborhood forces the function to be infinitely differentiable and representable as a convergent power series.

Definition of Analytic Functions

• Complex differentiability in a neighborhood—a function is analytic at a point if it has a complex derivative at every point in some open disk around that point
• Power series representation follows automatically: every analytic function equals its Taylor series $f(z) = \sum_{n=0}^{\infty} a_n (z - z_0)^n$ within its radius of convergence
• Infinite differentiability is guaranteed—unlike real analysis, where $C^1$ doesn't imply $C^2$, complex differentiability once means differentiability forever

Cauchy-Riemann Equations

• Necessary and sufficient conditions for analyticity (when partials are continuous): if $f(z) = u(x,y) + iv(x,y)$, then $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$
• Geometric meaning—these equations ensure the derivative is the same regardless of the direction from which you approach, coupling the real and imaginary parts
• Practical test for checking analyticity: verify both equations hold and that the partial derivatives are continuous

Power Series Representations

• Local representation $f(z) = \sum_{n=0}^{\infty} a_n (z - z_0)^n$ holds for any analytic function, with coefficients given by $a_n = \frac{f^{(n)}(z_0)}{n!}$
• Radius of convergence $R$ defines the largest disk centered at $z_0$ where the series converges—the series diverges outside this disk and converges absolutely inside
• Computational power—series can be added, multiplied, differentiated, and integrated term-by-term within the convergence disk

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Rigidity: Why Analytic Functions Are Uniquely Determined

One of the most striking features of analytic functions is their rigidity—knowing a function on a small piece determines it everywhere. This is completely unlike real analysis, where you can modify a smooth function on any interval without affecting it elsewhere.

Identity Theorem

• Uniqueness from accumulation—if two analytic functions agree on any set with a limit point in their domain, they're identical throughout the connected domain
• Zero set structure follows: the zeros of a non-constant analytic function are isolated, meaning they can't cluster anywhere in the domain
• Proof strategy tool—to show two analytic functions are equal everywhere, you only need to show they agree on a convergent sequence

Maximum Modulus Principle

• Boundary behavior dominates—if $f$ is analytic and non-constant on a domain, then $|f(z)|$ achieves its maximum only on the boundary, never in the interior
• No local maxima inside—an analytic function's modulus cannot have any interior "peaks," reflecting how analytic functions average over circles
• Fundamental applications include proving Liouville's theorem and establishing bounds on analytic functions

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Harmonic Connections: Real Parts of Analytic Functions

The Cauchy-Riemann equations force the real and imaginary parts of analytic functions to satisfy Laplace's equation. This links complex analysis to potential theory, physics, and PDEs.

Harmonic Functions

• Laplace's equation $\nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$ is satisfied by both the real part $u$ and imaginary part $v$ of any analytic function
• Mean value property—the value at any point equals the average over any circle centered there, explaining why harmonic functions have no interior extrema
• Harmonic conjugates—given a harmonic $u$, there exists a harmonic $v$ (unique up to a constant) such that $u + iv$ is analytic

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Breaking Points: Singularities and Their Classification

Singularities are where analyticity fails—but how it fails matters enormously. The classification of singularities determines the local and global behavior of functions.

Singularities and Their Classification

• Three types—removable singularities (function bounded nearby, can be "filled in"), poles (function blows up like $(z-z_0)^{-n}$), and essential singularities (chaotic behavior, Picard's theorem applies)
• Laurent series diagnosis—the principal part $\sum_{n=1}^{\infty} b_n (z-z_0)^{-n}$ determines the type: no terms means removable, finitely many means pole, infinitely many means essential
• Pole order $n$ equals the smallest power in the principal part and determines the residue calculation method

Residue Theorem

• Contour integral evaluation—$\oint_C f(z)\,dz = 2\pi i \sum \text{Res}(f, z_k)$ where the sum is over all singularities inside the contour $C$
• Residue calculation at a simple pole: $\text{Res}(f, z_0) = \lim_{z \to z_0} (z - z_0)f(z)$; for higher-order poles, use derivatives or Laurent expansion
• Real integral applications—the residue theorem transforms difficult real integrals into algebraic residue calculations, one of the most powerful computational tools in analysis

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Extensions and Transformations

Analytic functions can be extended beyond their original domains and used to transform regions while preserving geometric structure.

Analytic Continuation

• Domain extension—given an analytic function on region $D_1$, find an analytic function on larger region $D_2 \supset D_1$ that agrees on $D_1$
• Uniqueness guaranteed by the Identity Theorem: if a continuation exists, it's unique, so "the" analytic continuation is well-defined
• Classic example—the Riemann zeta function $\zeta(s)$ starts as a series for $\text{Re}(s) > 1$ but continues to all of $\mathbb{C}$ except $s = 1$

Conformal Mappings

• Angle preservation—analytic functions with $f'(z) \neq 0$ preserve angles between curves, making them "shape-preserving" at infinitesimal scales
• Local bijectivity near points where $f'(z) \neq 0$; the mapping magnifies by $|f'(z)|$ and rotates by $\arg(f'(z))$
• Physical applications in fluid dynamics (streamlines), electrostatics (equipotentials), and cartography (map projections)

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Quick Reference Table

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Self-Check Questions

1. If $f$ and $g$ are both analytic on a connected domain and $f(1/n) = g(1/n)$ for all positive integers $n$, what can you conclude? Which theorem applies?

2. Compare and contrast how the Maximum Modulus Principle and the mean value property for harmonic functions both reflect the "no interior extrema" phenomenon.

3. A function has Laurent series $f(z) = \frac{1}{z^2} + \frac{1}{z} + 1 + z + z^2 + \cdots$ near $z = 0$. Classify the singularity and find the residue.

4. Why does complex differentiability imply infinite differentiability, while real differentiability does not? What role do the Cauchy-Riemann equations play?

5. Given a harmonic function $u(x,y) = x^2 - y^2$, explain how you would find its harmonic conjugate $v$ and verify that $f = u + iv$ is analytic.