complexseries

Complexseries refers to infinite sums whose terms are complex numbers. Let a sequence (a_n) with a_n in the complex numbers, and define the partial sums S_N = ∑_{n=0}^N a_n. The series ∑ a_n converges to a limit S in the complex plane if S_N → S as N → ∞. Because C is two-dimensional over R, convergence is equivalent to the convergence of the real parts and the imaginary parts. A series is absolutely convergent if ∑ |a_n| converges; absolute convergence implies convergence and ensures that rearrangements do not affect the sum. If a series is not absolutely convergent, rearrangements can, in general, change the sum, mirroring the real case.

Power series in a complex variable z have the form ∑ c_n z^n with c_n ∈ C. The radius of convergence R is given by 1 / limsup_{n→∞} |c_n|^{1/n}. For |z| < R the series converges absolutely and uniformly on compact subsets, defining a holomorphic function on the disk. Within its domain, the function can be differentiated and integrated term by term: f'(z) = ∑ n c_n z^{n-1}. On and beyond the boundary, convergence behavior can vary and may be delicate.

Laurent series generalize power series to include negative powers: ∑_{n=-∞}^{∞} c_n (z−z0)^n, convergent on an annulus R1 < |z−z0| < R2. They describe functions with isolated singularities, and the principal part determines residues and local behavior.

Applications include Taylor and Laurent expansions, generating functions, and the study of analytic properties in complex analysis. Tests for convergence (ratio, root, and M-test) assist in establishing where a complex series converges and what it represents.