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The exhaustive list of topics in Complex Analysis in which we provide Help with Homework Assignment and Help with Project is as follows:

  • Complex numbers, algebra in the complex plane, conjugation, modulus and inequalities.
  • Polar form, powers and roots of complex numbers, Geometry in the complex plane, the extended complex plane and the Riemann sphere.
  • Topology in $\mathbb{C}$: Interior points, limit points, open sets, closed sets, connected sets, compact sets, Sequences and series of complex numbers and convergence.
  • Complex functions, visualizing complex functions, limits of complex functions, continuity.
  • Differentiation and the Cauchy-Riemann equations, Analytic functions, Harmonic functions and finding harmonic conjugates.
  • Elementary analytic functions and their mapping properties, complex logarithm function, branches of multiple valued functions, complex power or exponent functions, branches of $\sqrt{z}$.
  • Curves, paths and contours, statement of Jordan curve theorem, orientation of closed curves, Contour integrals and its properties, fundamental theorem of calculus.
  • Cauchy’s theorem for a rectangle, Cauchy-Goursat theorem for simply connected domains, Cauchy's integral formula, Cauchy's estimate, Liouville's theorem, fundamental theorem of algebra, higher derivatives of analytic functions, Morera's theorem.
  • Open mapping theorem, maximum modulus theorem.
  • Zeroes of analytic functions, identity theorem, counting zeroes, Singularities and their classification, Taylor's theorem, Casorati-Weierstrass theorem, Residue theorem, Argument principle and Rouche's theorem.
  • Evaluation of some definite integrals.
  • Power series, uniform convergence, Taylor's series, Laurent's theorem, finding Laurent's series.
  • Conformal mappings and its properties.
  • Mobius transformations.
  • Mappings by elementary functions.
  • Riemann mapping theorem, Conformal mapping of polygons.
  • Simply periodic functions, doubly periodic functions.
  • Properties of elliptic functions.
  • Riemann surfaces.