Complex Differentiability

One Limit, Every Direction

In Complex Power Series, convergence was studied by fixing a complex number and examining the resulting series. We now treat a complex-valued function as a function of a complex variable and ask whether its change near a point has a well-defined linear rate. The crucial feature is that a complex increment can approach zero from any direction in the plane. A complex derivative must give the same limit no matter how that direction varies.

This requirement is stronger than checking a real-variable derivative along the horizontal axis, or even checking derivatives along the two coordinate axes separately. The difference quotient must converge as the increment moves through all nonzero complex values near zero.

Definition (Complex Differentiability at a Point): Let \(U\subseteq\mathbb{C}\) be open, let \(z_0\in U\), and let \(f:U\to\mathbb{C}\). The function \(f\) is complex differentiable at \(z_0\) if the limit \[ \lim_{\substack{h\to0\\h\neq0}}\frac{f(z_0+h)-f(z_0)}{h} \] exists in \(\mathbb{C}\). When it exists, this limit is the complex derivative of \(f\) at \(z_0\), denoted \(f'(z_0)\).

Because \(U\) is open, all sufficiently small increments \(h\) satisfy \(z_0+h\in U\). The limit is a complex limit: \(h\) is not restricted to be real, positive, or to lie on any particular line. This feature distinguishes complex differentiability from the existence of separate directional derivatives.

What the Difference Quotient Gives

The definition can be rewritten as a first-order approximation. If \(f'(z_0)=a\), then the difference quotient approaches \(a\), so its error from \(a\) becomes small as \(h\) approaches zero. Multiplying that error by \(h\) gives the change in the function.

f(z_0+h)=f(z_0)+ah+o(|h|), \qquad a=f'(z_0).

Here \(o(|h|)\) denotes a remainder whose modulus, divided by \(|h|\), tends to zero as \(h\to0\). Thus complex differentiability gives a real-linear approximation to \(f\), but of a specially restricted form: its linear part is multiplication by one complex number \(a\). In real coordinates, multiplication by \(a=\alpha+i\beta\) has matrix

\begin{pmatrix} \alpha&-\beta\\ \beta&\alpha \end{pmatrix}.

Not every real-linear map of the plane has this form. The restriction is exactly what leads to the Cauchy–Riemann equations.

Worked Example: The Derivative of a Quadratic

Let \(f(z)=z^2\), and fix any \(z_0\in\mathbb{C}\). For \(h\neq0\),

\frac{f(z_0+h)-f(z_0)}{h} =\frac{(z_0+h)^2-z_0^2}{h} =\frac{z_0^2+2z_0h+h^2-z_0^2}{h} =2z_0+h.

As \(h\to0\), the final expression tends to \(2z_0\), regardless of the direction of approach. Therefore \(f\) is complex differentiable at every \(z_0\), with \(f'(z_0)=2z_0\). The cancellation and limit here rely on an identity valid for every complex \(h\), not just real increments.

Worked Example: Complex Conjugation Is Not Complex Differentiable

Let \(f(z)=\overline{z}\), and fix any \(z_0\). The difference quotient is

\frac{\overline{z_0+h}-\overline{z_0}}{h} =\frac{\overline{h}}{h}.

For real increments \(h=t\neq0\), this quotient equals \(1\). For purely imaginary increments \(h=it\), where \(t\neq0\), it equals \(-1\), because \(\overline{it}=-it\). The two values cannot approach the same limit. Thus \(f\) is not complex differentiable at any point. The example illustrates why checking only one direction can give a misleading impression of a complex derivative.

Algebra of Complex Derivatives

Once the difference quotient is understood, basic operations on differentiable functions behave as expected. The product rule needs one small point of care: it uses the fact that a complex-differentiable function is continuous at the point in question. That continuity follows directly from the defining limit.

Theorem (Sum and Product Rules for Complex Derivatives): Suppose \(f\) and \(g\) are complex differentiable at \(z_0\). Then \(f+g\) and \(fg\) are complex differentiable at \(z_0\), and \[ (f+g)'(z_0)=f'(z_0)+g'(z_0),\qquad (fg)'(z_0)=f'(z_0)g(z_0)+f(z_0)g'(z_0). \]

Proof. Write \(f_0=f(z_0)\) and \(g_0=g(z_0)\). By the definition of the derivatives, as \(h\to0\) through nonzero complex values, \[ \frac{f(z_0+h)-f_0}{h}\to f'(z_0), \qquad \frac{g(z_0+h)-g_0}{h}\to g'(z_0). \] Adding these quotients proves the sum rule.

For the product rule, first note that the quotient for \(f\) is bounded for all sufficiently small nonzero \(h\), since it converges to \(f'(z_0)\). Hence \[ |f(z_0+h)-f_0| =|h|\left|\frac{f(z_0+h)-f_0}{h}\right|\to0. \] So \(f(z_0+h)\to f_0\). Now split the product difference as follows:

\frac{f(z_0+h)g(z_0+h)-f_0g_0}{h} = f(z_0+h)\frac{g(z_0+h)-g_0}{h} + g_0\frac{f(z_0+h)-f_0}{h}.

The first term tends to \(f_0g'(z_0)\), and the second tends to \(g_0f'(z_0)\), by the limits just established and the algebraic limit laws for complex sequences. Their sum gives the stated product rule. \(\square\)

In particular, complex differentiability implies continuity at the point. This conclusion is a consequence of the definition, not an extra hypothesis. The rules allow derivatives of polynomials to be computed by repeated addition and multiplication once the derivatives of the basic functions are known.

The Cauchy–Riemann Equations

Write \(z=x+iy\) and express \(f\) in real and imaginary parts as \(f(z)=u(x,y)+iv(x,y)\), where \(u\) and \(v\) are real-valued. If the complex derivative exists, increments along the real and imaginary axes must yield the same limit. These two tests force relations among the first partial derivatives.

Theorem (Cauchy–Riemann Equations as Necessary Conditions): Suppose \(f=u+iv\) is complex differentiable at \(z_0=x_0+iy_0\). Then the real partial derivatives of \(u\) and \(v\) with respect to \(x\) and \(y\) exist at \((x_0,y_0)\) and satisfy \[ u_x(x_0,y_0)=v_y(x_0,y_0), \qquad u_y(x_0,y_0)=-v_x(x_0,y_0). \] Moreover, \[ f'(z_0)=u_x(x_0,y_0)+i\,v_x(x_0,y_0). \]

Proof. Let \(a=f'(z_0)\). First take real increments \(h=t\), with real \(t\neq0\). The difference quotient becomes

\frac{f(x_0+t+iy_0)-f(x_0+iy_0)}{t} = \frac{u(x_0+t,y_0)-u(x_0,y_0)}{t} +i\frac{v(x_0+t,y_0)-v(x_0,y_0)}{t}.

Its limit exists and equals \(a\). Taking real and imaginary parts shows that \(u_x(x_0,y_0)\) and \(v_x(x_0,y_0)\) exist, and that \(a=u_x(x_0,y_0)+iv_x(x_0,y_0)\).

Next take \(h=it\), again with real \(t\neq0\). Dividing the change in \(u+iv\) by \(it\) gives

\frac{f(x_0+i(y_0+t))-f(x_0+iy_0)}{it} = \frac{v(x_0,y_0+t)-v(x_0,y_0)}{t} -i\frac{u(x_0,y_0+t)-u(x_0,y_0)}{t}.

This quotient must also tend to \(a\). Its real and imaginary parts show that \(v_y(x_0,y_0)\) and \(u_y(x_0,y_0)\) exist and that \(a=v_y(x_0,y_0)-i\,u_y(x_0,y_0)\). Equating real and imaginary parts of the two expressions for \(a\) yields \(u_x=v_y\) and \(v_x=-u_y\), as claimed. \(\square\)

These equations are necessary because complex differentiability requires a single limit along both coordinate directions. The proof does not establish the converse: matching derivatives along the two axes does not by itself control the difference quotient along every direction.

Worked Example: A Function Differentiable at One Point Only

Consider \(f(z)=|z|^2\). At \(z_0=0\), for every nonzero \(h\),

\frac{f(h)-f(0)}{h} =\frac{|h|^2}{h} =\overline{h}\longrightarrow0.

Thus \(f'(0)=0\). At a nonzero point \(z_0\), expanding the squared modulus gives

\frac{|z_0+h|^2-|z_0|^2}{h} =\overline{z_0}+z_0\frac{\overline{h}}{h}+\overline{h}.

For real \(h=t\), this tends to \(\overline{z_0}+z_0=2\operatorname{Re}(z_0)\). For purely imaginary \(h=it\), it tends to \(\overline{z_0}-z_0=-2i\operatorname{Im}(z_0)\). If these limits were equal, their real and imaginary parts would both be zero, forcing \(\operatorname{Re}(z_0)=\operatorname{Im}(z_0)=0\). That contradicts \(z_0\neq0\). Therefore \(f\) is complex differentiable at zero and nowhere else.

Why the Cauchy–Riemann Equations Are Not Enough

A common pitfall is to find that the Cauchy–Riemann equations hold at a point and conclude that the complex derivative exists there. The theorem proves only necessity. The following example has the required partial derivatives and satisfies the equations at the origin, but its difference quotient has no limit.

Worked Example: Cauchy–Riemann Equations Without a Complex Derivative

Define \(f(0)=0\) and, for \(z\neq0\), define \(f(z)=\overline{z}^{\,2}/z\). Along the real axis, \(f(t)=t\); along the imaginary axis, direct substitution gives

f(it)=\frac{(-it)^2}{it} =\frac{-t^2}{it} =it.

Consequently, at the origin \(u_x=1\), \(v_x=0\), \(u_y=0\), and \(v_y=1\). The Cauchy–Riemann equations hold there. But the difference quotient along the real axis is \(f(t)/t=1\). Along \(h=t(1+i)\), where \(t\neq0\),

\frac{f(h)}{h} = \frac{\overline{h}^{\,2}}{h^2} = \frac{t^2(1-i)^2}{t^2(1+i)^2} = \frac{-2it^2}{2it^2} =-1.

Since these two approaches give different values, the difference quotient has no limit at zero. Thus \(f\) is not complex differentiable there despite satisfying the Cauchy–Riemann equations.

The practical distinction is important: complex differentiability controls every small complex increment, while the Cauchy–Riemann equations record relations among partial derivatives at a point. Additional hypotheses are needed to turn those relations into a sufficient test. The next tutorial develops the broader notion of holomorphic functions and the consequences of complex differentiability on open sets.

Check Your Understanding

Use the definition and results in this tutorial to answer the following questions.

Why must the increment in the definition of a complex derivative be allowed to approach zero from every direction?
For \(f(z)=z^2\), simplify the difference quotient at an arbitrary \(z_0\) and identify the derivative.
Explain how taking real and purely imaginary increments leads to the two Cauchy–Riemann equations.
Why does the existence of a complex derivative imply continuity at the point?
What feature of the example \(f(0)=0\), \(f(z)=\overline{z}^{\,2}/z\) for \(z\neq0\), shows that the Cauchy–Riemann equations are not sufficient by themselves?

What You'll Learn