Let \(L : \mathbb {C}\to \mathbb {C}\) be a \(\mathbb {R}\)-linear map represented in the basis \(1,\mathfrak {i}\) by the matrix \(M = \left[ \begin{array}{cc} a & b \\ c & d \end{array} \right] \in \mathbb {R}^{2 \times 2}\). Then the following are equivalent:

• \(L\) is \(\mathbb {C}\)-linear;

• \(b = - c\) and \(a = d\).

If a function \(f \colon A \to \mathbb {C}\) has complex derivative \(f'(z_0) = \lambda \in \mathbb {C}\) at a point \(z_0 \in A\), then we can write a linear approximation

where the error term \(\epsilon \) is small near \(z_0\) in the sense that \(\lim _{z \to z_0} \frac{\epsilon (z)}{|z-z_0|} = 0\).

If a function \(f \colon A \to \mathbb {C}\) has a complex derivative \(f'(z_0)\) at a point \(z_0 \in A\), then it is continuous at \(z_0\).

Let \(f \colon A \to \mathbb {C}\) be a function defined on a set \(A \subset \mathbb {C}\), and let \(u \colon A \to \mathbb {R}\) and \(v \colon A \to \mathbb {R}\) be its real and imaginary parts, viewed as real-valued functions of two real variables, \(u(x,y) = \Re \mathfrak {e}\big( f (x + \mathfrak {i}y) \big)\) and \(v(x,y) = \Im \mathfrak {m}\big( f (x + \mathfrak {i}y) \big)\), so that \(f = u + \mathfrak {i}\, v\). If \(f\) has a complex derivative \(f'(z_0)\) at an interior point \(z_0 = x_0 + \mathfrak {i}y_0 \in A\), then \(u\) and \(v\) are differentiable at \((x_0,y_0)\) and their partial derivatives satisfy the Cauchy-Riemann equations

(These equations are equivalent to the differential \(\mathrm{d}f (x_0, y_0) \colon \mathbb {R}^2 \to \mathbb {R}^2\) being \(\mathbb {C}\)-linear when we identify \(\mathbb {R}^2 = \mathbb {C}\).)

We can then write the derivative at \(z_0\) in any of the following ways:

If two functions \(f, g \colon A \to \mathbb {C}\) have complex derivatives \(f'(z_0), g'(z_0)\) at a point \(z_0 \in A\), then the sum function \(f + g\) has a complex derivative at \(z_0\) given by

If a function \(f \colon A \to \mathbb {C}\) is has a complex derivative \(f'(z_0)\) at a point \(z_0 \in A\) and \(c \in \mathbb {C}\) is a complex number, then the function \(c f\) has complex derivative

at \(z_0\).

If two functions \(f, g \colon A \to \mathbb {C}\) have complex derivatives \(f'(z_0), g'(z_0)\) at a point \(z_0 \in A\), then the product function \(f g\) has complex derivative

at \(z_0\).

If two functions \(f, g \colon A \to \mathbb {C}\) have complex derivatives \(f'(z_0), g'(z_0)\) at a point \(z_0 \in A\) and \(g(z_0) \ne 0\), then the quotient function \(f / g\) has complex derivative

at \(z_0\).

If \(f \colon A \to B \subset \mathbb {C}\) is differentiable at \(z_0 \in A\) and \(g \colon B \to \mathbb {C}\) is differentiable at \(f(z_0) \in B\), then the composition \(g \circ f \colon A \to \mathbb {C}\) is differentiable at \(z_0\), with derivative

Suppose that \(f\) is a complex-valued function defined on a subset of the complex plane, which has a nonzero complex derivative \(f'(z_0) \ne 0\) at a point \(z_0\) and which has a local inverse function near \(z_0\) in the sense that there are open sets \(U, V \subset \mathbb {C}\) with \(z_0 \in U\) and \(f(z_0) \in V\), and the restriction of \(f\) to \(U\) is continuous \(U \to V\) with a continuous inverse. Then the local inverse function \(f^{-1} \colon V \to U\) has complex derivative at \(w_0 := f(z_0)\) given by

Every function \(f \colon U \to \mathbb {C}\) which is analytic on an open set \(U \subset \mathbb {C}\) is also continuous on \(U\).

Every polynomial function \(p \colon \mathbb {C}\to \mathbb {C}\) is analytic.

Every rational function \(f \colon U \to \mathbb {C}\) is analytic on its domain of definition \(U \subset \mathbb {C}\).

The complex exponential function \(\exp \colon \mathbb {C}\to \mathbb {C}\) is analytic. Its (complex) derivative at \(z \in \mathbb {C}\) is \(\exp '(z) = \exp (z)\).

The principal branch of the \(n\)th root function \(z \mapsto \sqrt[n]{z}\) is analytic on its domain \(\mathbb {C}\setminus (-\infty ,0]\).

(Different branch choices can be made to obtain analyticity on other domains, but for \(n \ge 2\), no branch of \(\sqrt[n]{z}\) can be made analytic on all of \(\mathbb {C}\).)

Suppose that \(f \colon D \to \mathbb {C}\) is a analytic function on a connected open subset \(D \subset \mathbb {C}\) of the complex plane such that \(f'(z) = 0\) for all \(z \in D\). Then \(f\) is a constant function.

Suppose that \(f \colon U \to \mathbb {C}\) is a analytic function on an open subset \(U \subset \mathbb {C}\) of the complex plane. Let \(u,v \colon U \to \mathbb {R}\) denote the real and imaginary parts of \(f\) defined by \(u(x,y) = \Re \mathfrak {e}\big( f(x + \mathfrak {i}y)\big)\) and \(v(x,y) = \Im \mathfrak {m}\big( f(x + \mathfrak {i}y)\big)\). Assume moreover that that \(u\) and \(v\) are twice continuously differentiable (later it will be shown that this assumption holds automatically by the analyticity of \(f\)). Then \(u\) and \(v\) are harmonic functions, i.e., they satisfy

Let \(B = \mathcal{B}(z_0; r) \subset \mathbb {C}\) be a disk in the complex plane. Suppose that \(u \colon B \to \mathbb {R}\) is harmonic function on \(B\). Then a harmonic conjugate \(v \colon B \to \mathbb {R}\) of \(u\) in the disk \(B\) exists and is unique up to an additive constant.