Introduction to Complex Analysis
Introduction to Complex Analysis. Instructor: Prof. William Kinney, Math and Computer Science, Bethel University. Complex analysis involves extending the real number system by introducing an "imaginary unit." It is an extremely beautiful subject with lots of unexpected treasures and that, perhaps surprisingly, also has real-life applications. You are encouraged to explore some real-life applications in your project. Main topics dealt with in this course include: complex arithmetic and coordinate geometry; complex functions (especially as "mappings"); derivatives of complex functions and their interpretations; integrals of complex functions and their interpretations; infinite series of complex numbers and complex variables; residue theory and applications to evaluation of definite integrals; conformal mapping and Mobius transformations.
Lecture 29 - Uniform Convergence, Taylor Series Facts
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Lecture 01 - Complex Arithmetic, Cardano's Formula
Lecture 02 - Geometric Interpretations of Complex Arithmetic, Triangle Inequality
Lecture 03 - Polar Form, Principal Value of Argument, Basic Mappings
Lecture 04 - Mathematical Mappings, Linear Mappings, Squaring Map, Euler's Identity
Lecture 05 - Squaring Mapping, Euler's Identity and Trigonometry, 5th Roots Example
Lecture 06 - Exponential Map on Mathematica, Squaring Map, Intro to Topology
Lecture 07 - Exponential and Reciprocal Maps, Domains, Derivative Limit Calculations
Lecture 08 - Topological Definitions, Limits, Continuity, Linear Approximation
Lecture 09 - Facts to Recall, Animations, Continuity Proofs
Lecture 10 - Open Disks are Open, Derivatives, Analyticity, Linear Approximations
Lecture 11 - Areas of Images, Differentiability, Analyticity, Cauchy-Riemann Equations
Lecture 12 - Cauchy-Riemann Equations, Intro to Harmonic Functions
Lecture 13 - Preimages, Laplace's Equation, Harmonic and Analytic Functions
Lecture 14 - Preimages, Mathematica, Maximum Principle (Harmonic), Polynomials
Lecture 15 - Review of Analytic Functions, Amplitwist Concept, Harmonic Functions
Lecture 16 - Taylor Polynomials, Complex Exponential, Trigonometric and Hyperbolic Functions
Lecture 17 - Complex Logarithm, Functions as Sets, Multivalued Functions
Lecture 18 - Branches of Arg, Harmonic Functions over Washers, Wedges and Walls
Lecture 19 - Complex Powers, Inverse Trigonometric Functions, Branch Cuts
Lecture 20 - Invariance of Laplace's Equation, Real and Imaginary Parts of Complex Integrals
Lecture 21 - Conformality, Riemann Mapping Theorem, Vector Fields, Integration
Lecture 22 - Complex Integrals, Cauchy-Goursat Theorem
Lecture 23 - Real Line Integrals and Applications, Complex Integration
Lecture 24 - Integration, Cauchy-Goursat Theorem, Cauchy Integral Formula
Lecture 25 - Cauchy Integral Formula, Applications, Liouville's Theorem
Lecture 26 - Sequences and Series of Functions, Maximum Modulus on Mathematica
Lecture 27 - Review: Cauchy's Theorem, Cauchy Integral Formulas, and Corollaries
Lecture 28 - Taylor Series Examples (Tricks), Graphs of Partial Sums, Ratio Test
Lecture 29 - Uniform Convergence, Taylor Series Facts
Lecture 30 - Laurent Series Calculations, Visualize Convergence on Mathematica
Lecture 31 - Laurent Series, Poles of Complex Functions, Essential Singularities
Lecture 32 - More Laurent Series, Review: Integrals and Cauchy Integral Formula
Lecture 33 - Integrating 1/(1 + Z^2), Mathematica Programming, Residue Theorem Intro
Lecture 34 - Series, Zeros, Isolated Singularities, Residues, Residue Theorem
Lecture 35 - Residue Theorem Examples, Principal Values of Improper Integrals
Lecture 36 - Review for Complex Analysis Final Exam