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  • Analytic Functions and the Cauchy-Riemann Equations: Where students first encounter the bridge between real and imaginary components.
  • Elementary Functions: Extending exponentials, logs, and trig functions into the complex plane (multi-valued nightmares included).
  • Complex Integration and Cauchy’s Integral Formula: The heart of the subject, requiring meticulous step-by-step algebra.
  • Series and Residues: Laurent series and the Residue Theorem—the tools that solve impossible real integrals.

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