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
Laurent series generalize power series to include negative powers: ∑_{n=-∞}^{∞} c_n (z−z0)^n, convergent on an annulus
Applications include Taylor and Laurent expansions, generating functions, and the study of analytic properties in complex