Complex Functions: Norm Equivalence

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Covers complex functions, linear and anti-linear functions, harmonic polynomials, rational functions, and exponential functions.

Covers normed spaces, dual spaces, Banach spaces, Hilbert spaces, weak and strong convergence, reflexive spaces, and the Hahn-Banach theorem.

Explores differentiating under the integral sign and continuity of functions in integrals.

Discusses Laurent series, residue theorem, and their applications in complex analysis.

Explores holomorphic functions, Cauchy-Riemann conditions, and principal argument values in complex analysis.

Covers the integration of differential forms on smooth manifolds, including the concepts of closed and exact forms.

Covers the Cauchy theorem, complex functions, and contour integrals.

Covers the calculation of integrals over unclosed curves, focusing on essential singularities and residue calculation.

Explores Taylor series in complex analysis, emphasizing the behavior around singular points.