Lecture Notes
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Notes
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Video
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| 8/23/22:
Complex number algebraic operations.
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Go here for the 8/23/22 video.
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| 8/25/22:
Complex number geometric properties, polar form, Taylor series, geometric series.
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Go here for the 8/25/22 video.
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| 8/30/22:
Principal argument and multi-valued argument, complex logs and powers.
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Go here for the 8/30/22 video.
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| 9/1/22:
Equation-solving using geometry and algebra in the complex plane, open and closed sets.
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Go here for the 9/1/22 video.
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| 9/6/22:
We discussed topology of the complex plane.
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Go here for the 9/6/22 video.
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9/8/22:
We discussed limits and continuity for complex-valued functions.
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Go here for the sagemath work.
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Go here for the 9/8/22 video.
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| 9/13/22:
We discussed continuous complex-valued functions and the stereographic projection.
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Go here for the 9/13/22 video.
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| 9/15/22:
We discussed continuous complex-valued functions and complex derivatives.
Go here for the class notes (fixed).
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Go here for the 9/15/22 video.
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| 9/20/22:
We discussed complex differentiation, the Cauchy-Riemann equations, and holomorphic functions.
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Go here for the 9/20/22 video.
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9/22/22:
We discussed multivatiable real and complex chain rules, z and z-bar derivatives, and harmonic functions.
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Go here for the sagemath code used in C-R equations.
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Go here for the 9/22/22 video.
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| 9/27/22:
We discussed geometric series and integration over curves in the complex plane.
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Go here for the 9/27/22 video.
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| 9/29/22:
We discussed line integrals, FTCLI, FTCCI, and Cauchy's Theorem.
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Go here for the 9/29/22 video.
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| 10/4/22:
We discussed Cauchy's Theorem and the Deformation Theorem.
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Go here for the 10/4/22 video.
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| 10/10/22:
We discussed review questions for the test.
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Go here for the 10/10/22 video.
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| 10/13/22:
We discussed the Cauchy Integral Theorem with the weak hypothesis, and we proved the Cauchy Integral Formula.
From that we derived the mean value property for both holomorphic and harmonic functions.
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Go here for the 10/13/22 video.
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| 10/18/22:
We discussed the Cauchy integral formula for derivatives, Liouville's theorem,
the Cauchy inequalities, the Fundamental Theorem of Algebra, and the fact that
holomorphic functions are infinitely differentiable.
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Go here for the 10/18/22 video.
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| 10/20/22:
We discussed the maximum modulus principle and the geometric series.
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Go here for the 10/20/22 video.
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| 10/25/22:
We derived the Taylor series formula and consequences.
Go here for the notes (corrected).
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Go here for the 10/25/22 video.
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| 10/27/22:
We derived the identity and uniqueness theorems for holomorphic functions.
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Go here for the 10/27/22 video.
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| 11/1/22:
We discussed examples of using the identity and uniqueness theorem and began the process
of deriving the Laurent series expansion.
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Go here for the 11/1/22 video.
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11/3/22:
We derived the Lauremt series expansion.
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Go here for the 11/3/22 video.
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11/8/22:
We talked more about the Laurent series expansion and types of isolated singularities.
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Go here for the 11/8/22 video.
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11/10/22:
We discussed Laurent series, uniqueness of analytic continuation, term-by-term differentiability of Laurent and Taylor series.
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Go here for the 11/10/22 video.
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11/15/22:
We discussed properties of the three types of isolated singularities of holomorphic functions.
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Go here for the 11/15/22 video.
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11/17/22:
We discussed singularities, the residue theorem, and applications.
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Go here for the 11/17/22 video.
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11/29/22:
We discussed using residues to find values of real integrals. We also introduced the argument principle.
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Go here for the 11/29/22 video.
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12/1/22:
We used residues to find values of another improper real integral. We discussed the argument principle
and Rouche's Theorem.
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Go here for the 12/1/22 video.
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12/6/22:
We discussed Rouche's theorem some more. Also we introduced the gamma function at the Riemann zeta function and their analytic continuations.
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Go here for the 12/6/2 video.
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