Let \(f_n \colon A \to \mathbb {C}\), \(n \in \mathbb {N}\), be complex-valued functions defined on the same set \(A\). Then the sequence \((f_n)_{n \in \mathbb {N}}\) converges uniformly on \(A\) if and only if for every \(\varepsilon {\gt} 0\) there exists an \(N \in \mathbb {N}\) such that for all \(m,n \ge N\) and all \(z \in A\) we have \(|f_n(z) - f_m(z)| {\lt} \varepsilon \).

(When \((f_n)_{n \in \mathbb {N}}\) satisfies the condition above, it could be called a uniform Cauchy sequence on \(A\).)

Let \(X\) be a metric space (e.g., \(\mathbb {R}\), \(\mathbb {C}\), or a subset of these). If a sequence \((f_n)_{n \in \mathbb {N}}\) of continuous functions \(f_n \colon X \to \mathbb {C}\) converges uniformly to a function \(f \colon X \to \mathbb {C}\), then \(f\) is continuous.

If a sequence \((f_n)_{n \in \mathbb {N}}\) of continuous functions \(f_n \colon [a,b] \to \mathbb {C}\) on a closed interval \([a,b] \subset \mathbb {R}\) converges uniformly to a function \(f \colon [a,b] \to \mathbb {C}\), then we have

Suppose that functions \(f_1, f_2, \ldots \colon U \to \mathbb {C}\) are analytic functions on an open set \(U \subset \mathbb {C}\) and the sequence \((f_n)_{n \in \mathbb {N}}\) converges uniformly on compacts to a function \(f\). Then \(f\) is analytic on \(U\). Moreover, for any \(k \in \mathbb {N}\), the sequence \((f^{(k)}_n)_{n \in \mathbb {N}}\) of \(k\)th derivatives converges uniformly on compacts to \(f^{(k)}\).

If a complex series \(\sum _{n=1}^\infty z_n\) converges, then we have

The geometric series

with ratio \(z \in \mathbb {C}\) converges if and only if \(|z| {\lt} 1\). In that case its sum is

If a complex series converges absolutely, then it converges.

Suppose that \(\sum _{n=1}^\infty z_n\) is a complex series such that the limit

exists. Then:

• If \(r {\lt} 1\), then the series \(\sum _{n=1}^\infty z_n\) converges absolutely.

• If \(r {\gt} 1\), then the series \(\sum _{n=1}^\infty z_n\) does not converge.

Suppose that \(M_1,M_2,\ldots \ge 0\) are nonnegative numbers such that the series \(\sum _{n=1}^\infty M_n\) converges. Suppose also that for each \(n \in \mathbb {N}\), \(f_n \colon X \to \mathbb {C}\) is a function on \(X\) such that \(|f_n(x)| \le M_n\) for all \(x \in X\). Then the series \(\sum _{n=1}^\infty f_n\) converges absolutely and uniformly on \(X\).

Suppose that functions \(f_1, f_2, \ldots \colon U \to \mathbb {C}\) are analytic functions on an open set \(U \subset \mathbb {C}\) such that the series \(\sum _{n=1}^\infty f_n\) converges uniformly on compacts to a function \(f \colon U \to \mathbb {C}\). Then \(f\) is analytic on \(U\). Moreover, for any \(k \in \mathbb {N}\), the series \(\sum _{n=1}^\infty f^{(k)}_n\) of \(k\)th derivatives converges uniformly on compacts to \(f^{(k)}\).

If a power series

converges at \(z = w \in \mathbb {C}\), then it converges absolutely for all \(z \in \mathbb {C}\) such that \(|z-z_0| {\lt} |w-z_0|\).

Suppose that for the coefficients of a power series

the limit

exists. Then the radius of convergence \(R\) of the power series is \(R = \rho \).

Let \(z_0 \in \mathbb {C}\) be a point in the complex plane and let \(a_0,a_1,a_2\ldots \in \mathbb {C}\) be coefficients. Suppose that the power series

has radius of convergence \(R {\gt} 0\). Then it defines an analytic function \(f\) on the disk \(\mathcal{B}(z_0; R)\). The derivative of \(f\) is given by the power series

Moreover, the coefficients \(a_k\) are related to the \(k\)th derivatives of \(f\) at \(z_0\) through the formula

Suppose that two power series \(\sum _{n=0}^\infty a_n \, (z-z_0)^n\) and \(\sum _{n=0}^\infty b_n \, (z-z_0)^n\) converge in a disk \(\mathcal{B}(z_0; r)\) of radius \(r {\gt} 0\) and represent the same function

Then their coefficients must be equal: \(a_n = b_n\) for all \(n\).

Suppose that \(f \colon \mathcal{D}\to \mathbb {C}\) is an analytic function on a connected open set \(\mathcal{D}\subset \mathbb {C}\). If there exists a point \(z_0 \in \mathcal{D}\) such that \(f^{(n)}(z_0) = 0\) for all \(n \in \mathbb {N}\), then \(f\) is a constant function.

A doubly infinite series

of complex numbers converges if and only if both the series \(\sum _{n=0}^\infty z_n\) and \(\sum _{n=1}^\infty z_{-n}\) converge.

Consider a Laurent series

Denote

Then the series \(\sum _{n=-\infty }^\infty a_n (z - z_0)^n\) converges for all \(z\) is the annulus

Moreover, the convergence is uniform on compact subsets of \(\mathcal{A}_{\rho _-,\rho _+}(z_0)\), and the series defines an analytic function \(f(z)\) on the annulus \(\mathcal{A}_{\rho _-,\rho _+}(z_0)\).