Limit, continuity and the complex derivative
The definition of the derivative is the same as in real calculus, but now the increment can arrive from infinitely many directions and the quotient has to give the same result in all of them. That single requirement is what makes complex variable so different.
01A limit with infinitely many paths
On the real line, approaching a point allows only two paths: from the left and from the right. That is why the two one-sided limits are enough. In the complex plane, \( z \) can approach \( z_0 \) along infinitely many directions and also along spiral curves. The limit exists only if all those paths give the same number.
The change from interval to disk is what makes everything that follows more demanding. Checking a complex limit by trying two or three directions proves nothing: to show that it does not exist it is enough to find two paths that give different results, but to assert that it does exist you have to bound the modulus.
The transfer function of a circuit, \( H(s) \), is a complex function. The behavior near a pole —what happens as \( s \) approaches the resonant frequency— is literally a limit problem in the plane. And all the machinery that comes later —Cauchy, residues, Nyquist— rests on those functions being differentiable in the complex sense.
02Continuity
\( f \) is continuous at \( z_0 \) if the limit exists and equals the value of the function: \( \lim_{z \to z_0} f(z) = f(z_0) \). Since \( f = u + jv \), this is equivalent to both real functions \( u(x,y) \) and \( v(x,y) \) being continuous at \( (x_0, y_0) \). Nothing new, then: polynomials, the exponential, sine and cosine are continuous on the whole plane, and quotients are continuous except where the denominator vanishes.
The argument \( \operatorname{Arg} z \), taken on the principal branch, is not continuous across the negative real axis: from above it tends to \( +\pi \) and from below to \( -\pi \). That 360° jump is exactly what a network analyzer shows when the phase “wraps,” and that is why measurement software includes an unwrapping function.
03The derivative, and why it demands so much
The definition is identical to the real one:
But now \( \Delta z \) is a complex number that can tend to zero from any direction, and the quotient has to give the same value in all of them. That is the fundamental difference from real calculus: it is not the same definition with different symbols, it is a far more restrictive condition.
Let us write \( \Delta z = r\,e^{j\theta} \) and keep the direction \( \theta \) fixed while \( r \to 0 \). Each \( \theta \) gives one value of the quotient. If, as \( r \) shrinks, all those values gather at a single point, the derivative exists; if they stay spread out, it does not.
On the left, the point \( z_0 \) and the circle of radius \( |\Delta z| \) along which the increment approaches. On the right, the value of the quotient \( \Delta w / \Delta z \) for all directions at once. Shrink the radius and see whether that curve contracts to a point.
04The conjugate: the textbook counterexample
Take \( f(z) = \bar{z} \), which flips the sign of the imaginary part. It is continuous on the whole plane and its formula could not be simpler. The difference quotient comes out by hand:
Approaching along the real axis (\( \theta = 0 \)) the quotient is \( 1 \); along the imaginary axis (\( \theta = 90^\circ \)) it is \( -1 \). Two paths, two results: the conjugate is not differentiable at any point of the plane, despite being continuous at all of them. In the lab it shows up as a circle of radius one that never shrinks.
| Function | Difference quotient | Differentiable? |
|---|---|---|
| \( z^2 \) | \( 2z_0 + \Delta z \) | Yes, on the whole plane |
| \( 1/z \) | \( \dfrac{-1}{z_0(z_0 + \Delta z)} \) | Yes, except at \( z = 0 \) |
| \( \bar{z} \) | \( e^{-2j\theta} \) | At no point |
| \( |z|^2 \) | \( \bar{z}_0 + \overline{\Delta z} + z_0\,e^{-2j\theta} \) | Only at \( z = 0 \) |
| \( \operatorname{Re}(z) \) | \( \cos\theta\;e^{-j\theta} \) | At no point |
The pattern repeats: as soon as \( \bar{z} \), or \( |z| \), or the real or imaginary part alone appears, the derivative falls apart. The deeper reason is given by the next topic, with the Cauchy-Riemann equations: those functions mix \( z \) with \( \bar{z} \), and a differentiable function has to depend on \( z \) alone.
The case of \( |z|^2 \) is interesting because it fails almost everywhere in the plane but is saved at the origin: there \( z_0 = 0 \) wipes out the term that depended on the direction and what is left is \( \overline{\Delta z} \), which does tend to zero. It is differentiable at an isolated point, and that is not enough for anything: it has to be differentiable on an entire neighborhood.
05Differentiable on a neighborhood: analytic function
A function is analytic —or holomorphic— at a point if it is differentiable at that point and on a whole disk around it. The distinction matters: \( |z|^2 \) is differentiable at the origin but is analytic nowhere, because there is no disk on which it is.
For real functions, a function can be differentiable once and not twice. For complex functions that is impossible: if \( f \) is analytic on a domain, it has derivatives of all orders and equals its Taylor series. Being differentiable once on a neighborhood, in the complex plane, forces everything else. From that rigidity come Cauchy’s theorems and the calculus of residues.
The differentiation rules, on the other hand, are the same as ever and are proved the same way: sum, product, quotient and chain rule. And so are the familiar derivatives:
Functions that are analytic on the whole plane are called entire: polynomials, \( e^z \), \( \operatorname{sin} z \) and \( \cos z \). A rational function \( H(s) = N(s)/D(s) \) —that is, any transfer function of a circuit with lumped elements— is analytic on the whole plane except at the zeros of the denominator, which are the poles. That sentence, now said in the vocabulary of complex variable, is what enables the stability analysis done in Control.
06In the lab
For \( f(z) = \bar{z}/z \), compute the limit as \( z \to 0 \) approaching along the real axis, along the imaginary axis and along the 45° line. Check that they give 1, −1 and −j respectively, and conclude that the limit does not exist. Repeat with \( f(z) = z^2/|z|^2 \).
In a spreadsheet, evaluate the difference quotient of \( f(z) = e^z \) at \( z_0 = 1 + j \) for \( |\Delta z| = 10^{-1}, 10^{-2}, 10^{-3} \) and eight directions. Verify that the 24 values converge to \( e^{1+j} \approx 1.469 + j\,2.287 \), and that the error falls in proportion to \( |\Delta z| \).
With a generator and an oscilloscope, measure the phase shift of an RC low-pass filter between 10 Hz and 100 kHz, and plot it. Identify where the instrument jumps from −180° to +180° and correct it by hand. It is the discontinuity of \( \operatorname{Arg} z \), seen with a test lead in your hand.
07Common mistakes
- Trying two directions and declaring that the limit exists. That works to disprove it, not to assert it: there are functions that agree along all straight lines and fail along a parabola.
- Confusing continuous with differentiable. Differentiable implies continuous, never the other way around; \( \bar{z} \) is the textbook example.
- Believing that a simple formula guarantees differentiability. \( \bar{z} \), \( |z| \) and \( \operatorname{Re}(z) \) are elementary and are differentiable at no point.
- Saying “analytic” when there is only differentiability at one point. Analytic requires an entire neighborhood.
- Forgetting that \( \operatorname{Arg} z \) has a branch cut. Differentiating across the negative real axis is not valid with the principal branch.
08Self-assessment
Why aren’t one-sided limits enough, as they are for real functions?
Because \( z \to z_0 \) allows infinitely many directions and also curves. The condition is over a disk, not over an interval.
What is \( \Delta w/\Delta z \) for \( f(z) = \bar{z} \) if the increment goes in at 45°?
\( e^{-2j\,45^\circ} = e^{-j90^\circ} = -j \). Along the real axis it would give 1: two paths, two values, no derivative.
\( |z|^2 \) is differentiable at \( z = 0 \). Is it analytic there?
No. Analytic requires differentiability on an entire neighborhood, and in any disk around the origin there are points where it fails.
Where is \( H(s) = \dfrac{1}{s^2 + 2s + 5} \) analytic?
On the whole plane except at \( s = -1 \pm j2 \), the zeros of the denominator. They are the poles of the circuit, and there the function is not defined.
If \( f \) is analytic on a domain, how many times can it be differentiated?
Infinitely many. It is a property exclusive to the complex case: being differentiable once on a neighborhood forces all the derivatives to exist.
What do you see in the lab when a function is not differentiable?
That the trace of the difference quotient does not contract as the radius shrinks: a curve of fixed size remains, because each direction gives a different value.
09Further reading
- Ruel V. Churchill and James W. Brown. Complex Variables and Applications. 7th ed. (Spanish edition, “Variable compleja y aplicaciones”), McGraw-Hill, 2004. The chapters on limits, continuity and derivatives, with the careful construction of the counterexamples.
- A. David Wunsch. Complex Variables with Applications. 2nd ed. (Spanish edition, “Variable compleja con aplicaciones”), Addison-Wesley Iberoamericana, 1997. Written for engineering: every concept arrives with a circuit or a field behind it.
- Murray R. Spiegel. Complex Variables. Schaum’s Outline Series (Spanish edition, “Variable compleja”), McGraw-Hill, 2011. Solved exercises on limits along different paths, which is where you learn to disprove them.