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This lesson covers harmonic functions and their relationship to analytic functions.

In this video, we use residues of a complex function to compute the integral of a rational function (with real coefficients) comprised of trigonometric functions.

This video discusses Cauchy's Residue Theorem and provides several examples for the cases of removable singularities, simple poles, higher order poles and essential singularities.

This lesson provides a classification of zeros (i.e., roots) and of isolation singularities for analytic functions. We discuss several theorems related to this topic. Many examples are given.

This lesson entails the definition and examples of Taylor, Maclaurin and Laurent Series for complex functions. It builds on Lesson 18 which covers series in general.

This lesson covers complex sequence, complex series, convergence test such as the ratio test, and several examples.

This lesson covers the Cauchy integral theorem, the Cauchy integral formula, the generalized Cauchy integral formula and several examples. Reference for this lesson are 1. Saff, E.B., Snider, A.D., Fundamentals of complex analysis with applications to engineering and science (3rd edition), Pearson. 2. online notes (not from me) at www.math.bas.bg/~rkovach/lectures/complex7e.pdf

This lesson covers path independence for contour integrals.

This lesson defines and provides some examples of contour integrals for complex functions.

This video covers smooth arcs and contours (the entities over which complex integrals are defined). Subsequent lessons will cover complex integration. The Jordan curve theorem is also summarized in this video.

This lesson is about the complex version of the log function.

Complex versions of exponential, trigonometric and hyperbolic functions are discussed in this video.

This lesson covers complex polynomials and rational functions, including the representation of a polynomial in Taylor series format about a given point, and the representation of a rational function in terms of partial fractions.

This video introduces complex numbers and basic operations on complex numbers (e.g., addition, subtraction, multiplication). Make sure to listen to the introductory video before proceeding.

Lesson 2: Polar Form of a Complex Number, and Euler's Formula

Lesson 3 Basic Identities and the Triangle Inequality

Lesson 4 Roots of Complex Numbers

Lesson 5 Complex Functions

Lesson 6 Limits

Lesson 7 Continuity

Lesson 8 Derivatives

Lesson 9 Cauchy Riemann Equations

Lesson 10 Analytic Functions

Introduction to Complex Analysis

Ricketts Glen - Falls Trail - 3

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Ricketts Glen - Falls Trail - 1

Ricketts Glen State Park 3

Ricketts Glen State Park 2