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 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 |