Complex Analysis 3 | Complex Derivative and Examples [dark version]

TL;DR

Complex differentiability at z0 requires the existence of a limit that produces a direction-independent linear approximation of f near z0.

Briefing Cornell Notes

Briefing

Complex differentiability in the complex plane hinges on a linear approximation that must work in every direction, and the derivative is defined through the coefficient of that linear term. For a function f defined on an open set U ⊂ ℂ, differentiability at a point z0 (with z0 an interior point) is expressed by the existence of a limit that behaves like a slope. Because both inputs and outputs are complex numbers, the usual geometric “graph” intuition from real analysis doesn’t translate cleanly; instead, the key idea is abstract: near z0, f must be well-approximated by a linear function of the form f(z) = f(z0) + (linear coefficient)·Δf(z0), where Δf(z0) captures the limiting linear behavior. That linear coefficient is not just a heuristic—it becomes the derivative once the associated Δ-term is continuous at z0.

Once this framework is set, the derivative f′(z0) is defined as the value produced by the Δ-function when evaluated at z0. Since Δf(z0) must be continuous at z0 for differentiability to hold, the derivative can be computed using the same limit expression used in the slope definition. The result is a complex number: f′(z0) lives in ℂ, not ℝ. This is the practical payoff of the definition: it turns differentiability into a precise requirement that the function’s first-order behavior is linear and direction-independent.

Two examples make the contrast sharp. First, consider a complex-valued linear polynomial f(z) = Mz + c on all of ℂ. Writing it around any point z0 shows that the “Δ” part is constant and equals M, so the linear approximation matches the function exactly at first order. The derivative exists everywhere and equals M regardless of the chosen point. The simplicity here reflects the fact that linear functions already have the required direction-independent behavior.

The second example is the complex conjugation function f(z) = z̄, again defined on all of ℂ. This map reflects points across the real (x) axis: the real part stays fixed while the imaginary part changes sign. The function is continuous, but complex differentiability fails. Computing the derivative at 0 leads to a limit of the form z̄/z as z → 0, and that limit depends on the path taken toward 0. Approaching along the real axis with z = 1/n yields a limit of 1, while approaching along the imaginary axis with z = −i/n yields a limit of −1. Since the candidate limit is not the same from different directions, the limit does not exist, so f′(0) does not exist. The same reasoning applies at every point, implying that complex conjugation is nowhere complex differentiable.

The takeaway is that complex differentiability is a much stronger condition than continuity: even very simple-looking functions can fail to be differentiable because the required linear approximation cannot be made consistent across all directions in the complex plane.

Cornell Notes

Complex differentiability at z0 requires more than a real “slope.” A function f: U → ℂ is complex differentiable at z0 (with U open and z0 an interior point) only if its first-order behavior admits a linear approximation that works uniformly in every direction. The derivative f′(z0) is defined as the coefficient coming from the Δ-term in the linear approximation, and it is computed via the corresponding limit; f′(z0) is a complex number. For f(z)=Mz+c, the Δ-term is constant (equal to M), so f′(z0)=M everywhere. For f(z)=z̄, the limit defining the derivative depends on the path to 0 (1 along the real axis, −1 along the imaginary axis), so the derivative fails to exist anywhere, even though the function is continuous.

Why does complex differentiability require a linear approximation that works in every direction?

In complex analysis, the input z approaches z0 through infinitely many directions in ℂ. Differentiability at z0 means the limit defining the “slope” exists and yields a direction-independent linear first-order term. That’s why the derivative is tied to a linear approximation of the form f(z)=f(z0)+ (linear coefficient)·(Δ-term), with the Δ-term required to behave continuously at z0.

How does the derivative definition connect to the limit that resembles a slope?

The derivative f′(z0) is defined using the same limit structure as the real-variable difference quotient: it is the value produced by the Δ-function evaluated at z0. Because Δf(z0) must be continuous at z0, the derivative can be computed by taking the limit of the appropriate quotient as z→z0. The output is a complex number, not a real one.

What makes f(z)=Mz+c differentiable everywhere, and what is its derivative?

For f(z)=Mz+c, expanding around any point z0 gives f(z)=f(z0)+M(z−z0). The linear part is already exactly M times the increment, so the Δ-term is constant and equals M. Therefore the complex derivative exists for every z0 and equals M.

Why is complex conjugation f(z)=z̄ not complex differentiable, even though it is continuous?

The derivative at 0 would require the limit of z̄/z as z→0. Approaching along different paths gives different results: along the real axis with z=1/n, z̄/z=1; along the imaginary axis with z=−i/n, z̄/z=−1. Since the limit depends on the direction, it does not exist, so the function is not complex differentiable at 0. The same path-dependence argument applies at every point, making it nowhere complex differentiable.

How can path dependence be used as a diagnostic for non-differentiability?

To test differentiability at z0, compute the candidate limit for the derivative. If two sequences (or paths) approaching z0 produce different limiting values, the limit cannot exist. In the conjugation example, real-axis and imaginary-axis approaches yield 1 and −1, respectively, proving non-differentiability.

Review Questions

What conditions must hold for a complex function f to be complex differentiable at a point z0?
Compute the complex derivative of f(z)=Mz+c and justify why it does not depend on z0.
For f(z)=z̄, show how two different approaches to 0 lead to different values of z̄/z, preventing the derivative from existing.

Key Points

1
Complex differentiability at z0 requires the existence of a limit that produces a direction-independent linear approximation of f near z0.
2
The derivative f′(z0) is the coefficient from the Δ-term in the linear approximation and is computed via the corresponding limit.
3
For complex functions, f′(z0) is generally a complex number, not a real slope.
4
Linear complex polynomials f(z)=Mz+c are complex differentiable everywhere, with derivative f′(z0)=M.
5
Complex conjugation f(z)=z̄ is continuous but nowhere complex differentiable because the derivative limit depends on the path to the point.
6
Path dependence (different limits along different sequences approaching z0) is a decisive way to prove a complex derivative does not exist.

Highlights

Complex differentiability is essentially a requirement that the same linear first-order behavior emerges no matter which direction z approaches z0.

For f(z)=Mz+c, the derivative is constant: f′(z0)=M for every z0 in ℂ.

For f(z)=z̄, the derivative candidate z̄/z approaches 1 along the real axis but −1 along the imaginary axis, so the limit fails to exist.

Continuity alone is not enough in complex analysis: a function can be continuous yet nowhere complex differentiable. 