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48  Complex functions

Differentiability in the complex plane and the Cauchy-Riemann equations

A function \(f\) of a complex variable maps complex numbers to complex numbers. Writing \(z = x + iy\) and separating \(f(z) = u(x,y) + iv(x,y)\), the function is really a pair of real-valued functions of two real variables. The question is: what extra condition on \(u\) and \(v\) makes \(f\) genuinely complex-differentiable? The answer — the Cauchy-Riemann equations — is the key that unlocks the entire subject.

48.1 Complex numbers: brief review

Every complex number \(z = x + iy\) is a point in the Argand plane, with real part \(x\) on the horizontal axis and imaginary part \(y\) on the vertical.

Polar form. \(z = r e^{i\theta}\), where \(r = |z| = \sqrt{x^2+y^2}\) is the modulus and \(\theta = \arg z\) is the argument — the angle from the positive real axis, measured counterclockwise. Euler’s formula \(e^{i\theta} = \cos\theta + i\sin\theta\) connects the polar and rectangular forms.

De Moivre’s theorem. \((re^{i\theta})^n = r^n e^{in\theta}\). Integer powers of complex numbers are most efficiently computed in polar form.

Conjugate. \(\bar{z} = x - iy\). Identities: \(z\bar{z} = |z|^2\); \(\text{Re}(z) = (z+\bar{z})/2\); \(\text{Im}(z) = (z-\bar{z})/(2i)\).

48.2 Complex functions

A complex function \(f: \mathbb{C} \to \mathbb{C}\) maps each \(z = x+iy\) to a complex value \(f(z) = u(x,y) + iv(x,y)\).

Example. \(f(z) = z^2 = (x+iy)^2 = (x^2-y^2) + 2ixy\), so \(u = x^2-y^2\), \(v = 2xy\).

Example. \(f(z) = \bar{z} = x - iy\), so \(u = x\), \(v = -y\). This function will turn out to be nowhere differentiable in the complex sense.

Limits and continuity carry over from real analysis: \(f\) is continuous at \(z_0\) if \(\lim_{z\to z_0}f(z) = f(z_0)\), where the limit must be the same from every direction of approach.

48.3 Complex differentiability

The derivative of \(f\) at \(z_0\) is:

\[f'(z_0) = \lim_{z \to z_0} \frac{f(z) - f(z_0)}{z - z_0}\]

The limit must exist and agree no matter how \(z\) approaches \(z_0\) — a far stronger requirement than real differentiability.

48.3.1 Deriving the Cauchy-Riemann equations

Approach \(z_0 = x_0+iy_0\) along two different directions and demand the limit is the same.

Horizontal approach (\(z = x+iy_0\), \(z-z_0 = \Delta x\)):

With \(\Delta x\) real, the difference quotient separates cleanly into real and imaginary parts: \[\frac{f(z)-f(z_0)}{\Delta x} = \frac{[u(x_0+\Delta x,\,y_0)-u(x_0,y_0)]}{\Delta x} + i\frac{[v(x_0+\Delta x,\,y_0)-v(x_0,y_0)]}{\Delta x}\]

Taking \(\Delta x \to 0\), each fraction is just a partial derivative: \[f'(z_0) = u_x + iv_x\]

Vertical approach (\(z = x_0+iy\), \(z-z_0 = i\Delta y\)):

Now the increment is purely imaginary, so we divide by \(i\Delta y\). Dividing by \(i\) is the same as multiplying by \(-i\), which swaps real and imaginary parts. Carrying through: \[\frac{f(z)-f(z_0)}{i\Delta y} = \frac{1}{i}\cdot\frac{[u(x_0,y_0+\Delta y)-u(x_0,y_0)]+i[v(x_0,y_0+\Delta y)-v(x_0,y_0)]}{\Delta y}\]

Using \(1/i = -i\) and taking \(\Delta y \to 0\): \[f'(z_0) = v_y - iu_y\]

Equating real and imaginary parts:

\[\boxed{u_x = v_y \qquad u_y = -v_x}\]

These are the Cauchy-Riemann (C-R) equations. They are necessary for complex differentiability. If the partial derivatives are continuous in a neighbourhood of \(z_0\), they are also sufficient.

Consequence. For \(f(z) = \bar{z}\): \(u_x = 1\) but \(v_y = -1\). The C-R equations fail everywhere, so \(\bar{z}\) is nowhere differentiable — even though its real and imaginary parts are smooth real functions.

When the C-R equations hold, the derivative can be written in any of the equivalent forms: \[f'(z) = u_x + iv_x = v_y - iu_y\]

48.4 Analytic functions

\(f\) is analytic at \(z_0\) if it is differentiable in some open disk centred at \(z_0\). It is analytic in a domain \(D\) if differentiable at every point of \(D\). A function analytic on all of \(\mathbb{C}\) is called entire.

Being differentiable at an isolated point does not make a function analytic — analyticity requires differentiability in an open neighbourhood. Exercise 4 shows this with \(f(z) = |z|^2\), which is differentiable only at the origin and analytic nowhere.

48.4.1 Harmonic functions

If \(f = u + iv\) is analytic, both \(u\) and \(v\) satisfy Laplace’s equation: \[\nabla^2 u = u_{xx} + u_{yy} = 0, \qquad \nabla^2 v = 0\]

Proof. From the C-R equations: \(u_{xx} = (u_x)_x = (v_y)_x = v_{yx}\) and \(u_{yy} = (u_y)_y = (-v_x)_y = -v_{xy}\). Since mixed partials are equal, \(u_{xx} + u_{yy} = 0\).

A function satisfying Laplace’s equation is harmonic. So: the real and imaginary parts of any analytic function are harmonic.

Conversely, given a harmonic \(u\) on a simply connected domain — one with no holes, so that any closed loop inside it can be continuously shrunk to a point — there exists a harmonic conjugate \(v\) (unique up to a constant) such that \(f = u + iv\) is analytic. Finding \(v\) from \(u\) via the C-R equations is a standard technique.

48.5 Elementary analytic functions

48.5.1 Exponential

\[e^z = e^x(\cos y + i\sin y), \qquad \frac{d}{dz}e^z = e^z\]

Entire. Periodic with period \(2\pi i\): \(e^{z+2\pi i} = e^z\).

48.5.2 Trigonometric

\[\cos z = \frac{e^{iz}+e^{-iz}}{2}, \qquad \sin z = \frac{e^{iz}-e^{-iz}}{2i}\]

Both entire. The identities \(\cos^2 z + \sin^2 z = 1\) and all standard trig identities hold. For purely imaginary argument: \(\cos(iy) = \cosh y\), \(\sin(iy) = i\sinh y\) — the complex trig and hyperbolic functions are closely related.

48.5.3 Logarithm

\[\log z = \ln|z| + i\arg z = \ln r + i\theta\]

Multivalued: \(\arg z\) is defined only up to multiples of \(2\pi\). The principal value \(\text{Log}\,z = \ln|z| + i\,\text{Arg}\,z\) uses \(\text{Arg}\,z \in (-\pi, \pi]\) and is analytic everywhere except on the branch cut along the negative real axis — the line where \(\text{Arg}\,z\) jumps discontinuously by \(2\pi\) as you cross it. Without the cut, \(\text{Log}\,z\) cannot be made single-valued and continuous.

48.5.4 Complex powers

\[z^\alpha = e^{\alpha \log z}\]

Multivalued in general. Integer powers are single-valued and entire; non-integer powers require a branch cut.

48.6 Conformal mappings: preview

At any point \(z_0\) where \(f\) is analytic and \(f'(z_0) \neq 0\), the map \(f\) is conformal (Chapter 4): it preserves the angle between any two curves crossing at \(z_0\) (in magnitude and orientation). The scale factor is \(|f'(z_0)|\) and the rotation angle is \(\arg f'(z_0)\).

Conformal mappings transform solutions of Laplace’s equation from one domain into solutions in another — the basis for all analytic potential theory. Chapter 4 develops this fully.

48.7 Exercises

48.7.1 Exercise 1: Cauchy-Riemann equations for \(f(z) = z^2\)

Verify that \(f(z) = z^2\) satisfies the Cauchy-Riemann equations everywhere and compute \(f'(z)\).

48.7.2 Exercise 2: Finding a harmonic conjugate

Show that \(u = x^3 - 3xy^2\) is harmonic, find its harmonic conjugate \(v\), and identify the analytic function \(f = u + iv\).

48.7.3 Exercise 3: \(e^z\) is entire

Verify the Cauchy-Riemann equations for \(f(z) = e^z\) and confirm \(\frac{d}{dz}e^z = e^z\).

48.7.4 Exercise 4: Where is \(f(z) = |z|^2\) differentiable?

48.7.5 Exercise 5: Powers via De Moivre

Compute \((1+i)^8\) and find all fourth roots of \(-16\).