Prerequisites & knowledge map
Everything below is a concept Complex Analysis leans on that ISN'T taught inside the course itself — not the whole of another course, just the specific skills.
Calculus III — Multivariable Calculus
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The Fundamental Theorem for line integrals
The real-variable analog -- a line integral of a gradient field depends only on endpoints, the same way a complex integral of an exact derivative does.
helpful
College Algebra
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Calculus I — Differential & Integral Calculus
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Continuity at a point
The real-variable definition this one generalizes.
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Calculus II — Integral Calculus & Infinite Series
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Radius and interval of convergence
Same ratio-test machinery as the real power series case, generalized from an interval to a disk.
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Recursive sequences and fixed points
Classifying a fixed point as attracting or repelling by the size of a derivative is the same idea used there for real recursive sequences, now applied to iterating an analytic map on C.
helpful
Advanced Multivariable Calculus
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Line integrals of vector fields and work
The real-variable line integral this construction generalizes -- a complex contour integral splits into the same real/imaginary parametrized-path sums.
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The Gamma and Beta functions
The real-variable Gamma function from vector-analysis, defined by the same improper integral, before this course continues it off the positive real axis.
helpful