Complex Analysis

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• The complex plane and regions ~16 min
• Distance, circles, and disks ~16 min
• Lines, half-planes, and loci ~16 min
• Inequalities and regions ~18 min
• Connectedness and domains ~16 min

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• Functions of a complex variable ~16 min
• Limits of complex functions ~18 min
• Limits via real and imaginary parts ~18 min
• Continuity ~16 min
• Limits and the point at infinity ~16 min

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• The complex derivative ~16 min
• Analytic, holomorphic, and entire functions ~14 min
• The Cauchy-Riemann equations — necessary conditions ~18 min
• The Cauchy-Riemann equations — sufficient conditions ~16 min
• Cauchy-Riemann in polar coordinates ~15 min
• Zero derivative implies constant ~14 min

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• Harmonic functions and Laplace's equation ~16 min
• Harmonic conjugates ~18 min
• The maximum principle and mean value property ~16 min
• The identity theorem ~16 min
• The Schwarz reflection principle ~15 min

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• The complex exponential function ~20 min
• The logarithm and its branches ~22 min
• Complex powers ~20 min
• Trigonometric and hyperbolic functions ~22 min
• Inverse trigonometric and hyperbolic functions ~20 min

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• Contours and parametrization ~20 min
• Contour integrals ~25 min
• Properties of contour integrals and the ML inequality ~25 min
• Antiderivatives and path independence ~25 min
• The fundamental theorem for contour integrals ~20 min

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• The Cauchy-Goursat theorem ~24 min
• Simply and multiply connected domains ~20 min
• Contour deformation ~20 min
• The Cauchy integral formula ~24 min
• The derivative formula ~22 min
• Liouville's theorem and the Fundamental Theorem of Algebra ~22 min
• The maximum modulus principle ~22 min

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• Convergence of complex sequences and series ~22 min
• Taylor's theorem for analytic functions ~22 min
• Uniform convergence and term-by-term calculus ~20 min
• Uniqueness of power series representations ~18 min
• Multiplying and dividing power series ~20 min

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• Laurent's theorem ~24 min
• Isolated singularities -- removable, pole, and essential ~24 min
• The pole order and residue shortcut formula ~24 min
• Zeros of analytic functions ~22 min
• Behavior near isolated singularities ~22 min

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• The residue ~20 min
• Cauchy's residue theorem ~22 min
• Residue at infinity ~20 min
• Computing residues, several methods ~22 min

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• Real improper integrals via semicircular contours ~24 min
• Trigonometric integrals via the unit circle ~22 min
• Indented contours and principal values ~24 min
• Jordan's lemma ~24 min
• The argument principle ~24 min
• Rouché's theorem ~26 min
• Inverse Laplace transforms via residues ~26 min

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• Linear transformations and inversion ~18 min
• Möbius transformations ~20 min
• Conformal mapping and angle preservation ~18 min
• The elementary mapping catalogue ~18 min
• The Riemann sphere and the extended complex plane ~18 min
• Schwarz's lemma ~18 min
• The Schwarz-Pick theorem ~20 min
• Harmonic conjugates revisited ~20 min

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• Steady-state temperature and electrostatic potential ~24 min
• Two-dimensional fluid flow ~24 min
• The Schwarz-Christoffel transformation ~26 min
• The Poisson integral formula and the Dirichlet problem ~28 min
• The Neumann problem ~20 min

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• Natural boundaries ~20 min
• The identity theorem, revisited ~18 min
• Continuation along a curve ~20 min
• Branch points and why continuation can fail ~20 min

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• The Gamma function ~22 min
• The functional equation and the reflection formula ~24 min
• An introduction to complex dynamics ~20 min
• Julia sets and the Mandelbrot set, qualitatively ~22 min