Key Concepts of Cauchy-Riemann Equations

Why This Matters

The Cauchy-Riemann equations are the gateway to everything that makes complex analysis powerful and beautiful. When you're tested on complex differentiability, analyticity, or conformal mappings, you're really being tested on whether you understand how these two deceptively simple partial differential equations constrain the relationship between the real and imaginary parts of a complex function. Master this, and concepts like harmonic functions, power series representations, and angle-preserving transformations suddenly click into place.

Here's the key insight: complex differentiability is far more restrictive than real differentiability. The Cauchy-Riemann equations encode exactly what that restriction looks like. Don't just memorize the formulas—understand that they're telling you the real and imaginary parts of an analytic function can't vary independently. Every concept below connects back to this fundamental constraint, so know what principle each item illustrates, not just the definitions.

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The Core Equations and Their Forms

The Cauchy-Riemann equations express the precise relationship that partial derivatives must satisfy for complex differentiability to exist.

Definition of Cauchy-Riemann Equations

• Two coupled partial differential equations—for $f(z) = u(x, y) + iv(x, y)$, these equations link how $u$ and $v$ change with respect to $x$ and $y$
• The equations themselves: $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$, encoding the constraint that makes complex differentiation unique
• Foundation for analyticity—every theorem about analytic functions ultimately traces back to these two relationships

Polar Form of Cauchy-Riemann Equations

• Coordinate transformation—when $z = re^{i\theta}$, the equations become $\frac{\partial u}{\partial r} = \frac{1}{r}\frac{\partial v}{\partial \theta}$ and $\frac{\partial u}{\partial \theta} = -r\frac{\partial v}{\partial r}$
• Essential for radial symmetry—functions like $f(z) = z^n$ or $f(z) = \log z$ are far easier to analyze in polar form
• The $r$ factors matter—notice how they appear differently in each equation, reflecting how distances scale in polar coordinates

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Differentiability and Analyticity Conditions

Complex differentiability is extraordinarily demanding—satisfying the Cauchy-Riemann equations is necessary, but the story doesn't end there.

Necessary Conditions for Complex Differentiability

• Must satisfy Cauchy-Riemann equations at the point—if $f$ is complex differentiable at $z_0$, then the equations hold there; no exceptions
• Continuity required nearby—the function must be continuous in some neighborhood around the point, not just at the point itself
• Failure means non-differentiable—if either equation fails at a point, complex differentiability is impossible there

Sufficient Conditions for Complex Differentiability

• Cauchy-Riemann plus continuous partials—if the equations hold and the partial derivatives $\frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}, \frac{\partial v}{\partial x}, \frac{\partial v}{\partial y}$ are continuous, then $f$ is complex differentiable
• Differentiability implies analyticity—once differentiable in a region, the function is automatically infinitely differentiable there
• Power series representation follows—analytic functions can be expressed as convergent power series, a stunning consequence of complex differentiability

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The Real-Imaginary Relationship

The Cauchy-Riemann equations reveal that $u$ and $v$ are not independent—they're locked together in a mathematical dance.

Relationship Between Real and Imaginary Parts

• Interconnected variation—changes in $u(x,y)$ completely determine how $v(x,y)$ must change, and vice versa
• Harmonic conjugates—given $u$, the equations let you reconstruct $v$ (up to a constant); these paired functions are called harmonic conjugates
• Consistency in the complex plane—this coupling ensures the function's behavior doesn't depend on the direction of approach in the limit definition

Connection to Harmonic Functions

• Both $u$ and $v$ are harmonic—they satisfy Laplace's equation $\frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} = 0$, a consequence of applying Cauchy-Riemann twice
• Second-order constraint—the Cauchy-Riemann equations are first-order, but they imply this second-order condition on each component
• Physical applications—harmonic functions model steady-state heat distribution, electrostatic potential, and incompressible fluid flow

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Geometric and Visual Interpretations

The Cauchy-Riemann equations have elegant geometric meanings that help you visualize what analytic functions do.

Orthogonality of Level Curves

• Gradients are perpendicular—$\nabla u$ and $\nabla v$ are orthogonal wherever the Cauchy-Riemann equations hold
• Level curves intersect at right angles—curves where $u = \text{constant}$ cross curves where $v = \text{constant}$ at $90°$
• Visualization tool—this orthogonal grid structure helps you sketch the behavior of analytic functions geometrically

Role in Conformal Mappings

• Angle preservation—functions satisfying Cauchy-Riemann equations preserve angles between curves at points where $f'(z) \neq 0$
• Local shape preservation—infinitesimally small figures maintain their shape under the mapping, though they may be scaled and rotated
• Applications everywhere—conformal maps are used in fluid dynamics, aerodynamics, electrostatics, and cartography

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Testing for Analyticity

The Cauchy-Riemann equations give you a practical algorithm for determining whether a function is analytic.

Applications in Determining Analyticity

• Systematic test—compute all four partial derivatives, check both equations, verify continuity of partials
• Region-by-region analysis—a function might be analytic in some regions but not others; the equations tell you exactly where
• Foundation for proofs—when asked to prove a function is analytic, Cauchy-Riemann verification is typically your first move

Examples: Functions That Do and Don't Satisfy the Equations

• $f(z) = z^2$ is analytic—with $u = x^2 - y^2$ and $v = 2xy$, both equations check out everywhere
• $f(z) = \bar{z}$ is nowhere analytic—with $u = x$ and $v = -y$, we get $\frac{\partial u}{\partial x} = 1 \neq -1 = \frac{\partial v}{\partial y}$
• $f(z) = |z|^2$ is differentiable only at the origin—a classic example showing differentiability at a single point without analyticity

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Quick Reference Table

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Self-Check Questions

1. If a function satisfies the Cauchy-Riemann equations at a point but has discontinuous partial derivatives there, can you conclude the function is complex differentiable? Why or why not?

2. Compare $f(z) = e^z$ and $f(z) = e^{\bar{z}}$: which satisfies the Cauchy-Riemann equations, and what does this tell you about their analyticity?

3. Given that $u(x, y) = x^2 - y^2$ is the real part of an analytic function, how would you use the Cauchy-Riemann equations to find the imaginary part $v(x, y)$?

4. Explain why the level curves $u = c_1$ and $v = c_2$ intersect at right angles for an analytic function. Which form of the Cauchy-Riemann equations makes this clearest?

5. FRQ-style: A student claims that $f(z) = |z|^2$ is analytic because it's differentiable at the origin. Identify the flaw in this reasoning and explain the distinction between complex differentiability at a point versus analyticity.