Course Information

Home
Video Lectures
About The Professor

Course

Code:	MATH 3160
Subject:	Mathematics
Topic:	Complex Analysis
Views:	98,518

Educator

Name:	University of Utah (U of U)
Type:	University

Visit Official Website

Instructor

Name

William H. Nesse

Introduction to Applied Complex Variables

Video Lectures

Displaying all 50 video lectures.

Lecture 1 Play Video	Math 3160 introduction We describe the exegesis for complex numbers by detailing the broad goal of having a complete algebraic system, starting with natural numbers and broadening to integers, rationals, reals, to complex, to see how each expansion leads to greater completion of the algebra.
Lecture 2 Play Video	Basic Complex Algebra We define the basic algebraic properties of complex numbers, including sums, products, identities, inverses, and division.
Lecture 3 Play Video	Moduli, conjugates, triangle inequality, and polar coordinates We define important terminology and operations related to geometric properties of complex numbers.
Lecture 4 Play Video	Products and quotients in exponential form We detail how polar or exponential form of complex numbers can aid in basic calculations in the complex plane.
Lecture 5 Play Video	Roots of complex numbers We illustrate roots of z
Lecture 6 Play Video	Functions of complex variables and mappings We detail the basic structure of complex functions and go over examples of how functions map sets in the plane to image sets
Lecture 7 Play Video	Regions in the complex plane We define useful terminology related to sets and properties of sets in the complex plane
Lecture 8 Play Video	Mappings by the exponential function We discuss the basics of the exponential function in the complex plane and how it maps sets.
Lecture 9 Play Video	Limits of complex functions we establish the definition of limits and go through several examples of how to establish limits in the complex plane
Lecture 10 Play Video	Limits at infinity We explore the meaning and techniques to compute limits at infinity of a complex variable.
Lecture 11 Play Video	The derivative of a complex function We define and compute examples of derivatives of complex functions and discuss aspects of derivatives in the complex plane
Lecture 12 Play Video	Differentiation formulas for complex functions We state the standard differentiation rules for functions of a complex variable that will be familiar to any calculus student
Lecture 13 Play Video	Cauchy-Riemann equations We derive the condition for differentiability of complex functions to be the Cauchy-Riemann equations.
Lecture 14 Play Video	Analytic functions We define the meaning of analytic functions of a complex variable and use the Cauchy-Riemann equations to determine analyticity for examples.
Lecture 15 Play Video	Harmonic functions and analytic functions We establish the relationship between real-valued harmonic functions and analytic functions of a complex variables and show how to create analytic functions out of harmonic conjugates
Lecture 16 Play Video	The complex exponential and logarithm functions We go over the basics of exponentials and logs in the complex plane, and discuss the branches of the log function.
Lecture 17 Play Video	Complex log identites We go over standard identities of logarithm in the complex variable.
Lecture 18 Play Video	The information in analytic functions Without going into details, we describe that analytic functions can be "discovered" by knowing only values on a small subset of points within its domain. That is, the amount of information that creates f(z) is much smaller than the set of all correspondences that define the mapping.
Lecture 19 Play Video	Applications to signal processing We establish how complex variable tools can help characterize RLC circuit affects on inputs by characterizing amplifications in terms of complex moduli, and delays as phase shifts of the complex argument
Lecture 20 Play Video	Applications of analytic functions to fluid flow We show how analytic functions represent steady state fluid flow solutions, where u is a fluid potential, and the level curves of v are the stream lines.
Lecture 21 Play Video	Complex exponents We describe how to compute complex numbers raised to a complex power and establish the power rule for derivatives of functions involving complex powers
Lecture 22 Play Video	Complex trigonometric functions We define and state basic properties of complex trigonometric and hyperbolic functions.
Lecture 23 Play Video	Inverse trigonometric functions of a complex variable We derive inverse complex sine, and state standard identities of inverse trigonometric and hyperbolic functions, including derivatives
Lecture 24 Play Video	Derivatives and integrals of complex functions w(t) We examine complex functions of a real variable w(t) and show how to take derivatives and integrals.
Lecture 25 Play Video	Contours and arc length in the complex plane We define the basic terminology for contours in the plane as well as define the integral that computes the arc length of contours
Lecture 26 Play Video	Contour integrals of complex functions We derive the contour integral of complex functions and give examples.
Lecture 27 Play Video	Closed circle integral of 1/z and branch cuts We examine integrals of z^n over closed circular contours and highlight the important issues involving integrations of functions requiring functions with branch cuts
Lecture 28 Play Video	Moduli of complex integrals and integral bounds We state the means of bounding the modulus of an complex integral using the triangle inequality and give an example of how the bounds are used in the practice of contour integral computation.
Lecture 29 Play Video	Complex antiderivatives and the fundamental theorem We illustrate the fundamental theorem of calculus for functions of a complex variable and give examples
Lecture 30 Play Video	Proof of the antiderivative theorem for contour integrals We prove the main theorem for contour integrals using a multi-part proof.
Lecture 31 Play Video	Cauchy-Goursat theorem We state and partially prove the theorem using Green's theorem, to show that analyticity of f implies independence of path, and antiderivatives exists for f.
Lecture 32 Play Video	Simply and multiply connected domains We define simply and multiply connected domains as a property of sets and show how this property relates and extends the Cauchy-Goursat theorem.
Lecture 33 Play Video	Cauchy integral formula We state the formula and show how it relates to the mean value property of harmonic functions, as well as prove it. We then give an example of how we can use it to compute specific difficult integrals very simply.
Lecture 34 Play Video	Cauchy Integral Results We use and extend the Cauchy integral formula to establish a host of results concerning analytic functions.
Lecture 35 Play Video	The fundamental theorem of algebra revisited We specify why the FTA is relevant and important to studying signal processing and transfer functions using the Cauchy integral formula. NOTE, there is one misstatement in this video. For a general polynomial with complex coefficients, the roots do not necessarily come in conjugate pairs---the roots can be placed anywhere in the plane. Conjugate pair roots only arise when the coefficients are real.
Lecture 36 Play Video	Harmonic oscilators in the complex plane (optional) We illustrate how to model a mass-spring system or an RC-circuit can be modeled simply as a differential equation of a complex variable. This result will be used to understand important aspects Schrodinger's equation, which we detail in later videos.
Lecture 37 Play Video	How Schrodinger's equation works (optional) We take a purely mathematical approach to understanding the basic workings of Schrodinger's equation and show how it relates to harmonic oscillators. For concreteness and clarity, we avoid physical interpretations and its relation to quantum mechanics. Also, apologies for mis-spelling Schrodinger as "Shrodinger" in the video title....
Lecture 38 Play Video	Sequences and series involving complex variables We define the basic definitions of sequences and series of complex variables and give some examples.
Lecture 39 Play Video	Taylor series for functions of a complex variable We state and prove Taylor's theorem using direct calculation, which is a direct result of Cauchy's integral formula. We then illustrate several example series.
Lecture 40 Play Video	Laurent series We derive the Laurent series for functions that are non analytic at a point z_0 by utilizing the same technique as Taylor series, where we use the Cauchy integral formula. Because there is a point of non-analyticity, the contours form a multiply connected domain that yields additional terms beyond that of a standard Taylor series.
Lecture 41 Play Video	Examples of Laurent series computations We go through several examples of how to compute Laurent series. This video is highlights how a number of our integral theorems come into play in order to study important functions like transfer functions.
Lecture 42 Play Video	Aspects of complex power series convergence We go over the major statements about convergence of complex power series but do not prove them.
Lecture 43 Play Video	Singularities and residues of complex functions We define and give example calculations of residues of complex functions
Lecture 44 Play Video	The residue theorem We state and prove the residue theorem and give examples.
Lecture 45 Play Video	Residues at infinity NOTE: I made a sign error by misdescribing the orientation of the contour integral on the projected complex plane on the sphere. The correct identity is below: Res_{z=infinity}(f) = -Res(f(1/z)/z^2). Of course, the integral about the singularities the positive sign: integral_C f(z) dz = Res(f(1/z)/z^2). When the contour integral encloses all the singularities of the function, one compute a single residue at infinity rather than use the standard residue theorem involving the sum of all the individual residues.
Lecture 46 Play Video	Taxonomy of singularities of complex functions We define the three types of isolated singularities
Lecture 47 Play Video	Aspects of zeros and poles of analytic functions We describe that analytic functions that are non-constant cannot have level curves in the plane.
Lecture 48 Play Video	Zeros and poles of rational functions We describe a way to compute residues of rational functions with simple poles.
Lecture 49 Play Video	Applications of residues to improper real integration We use the residue theorem and contour integral bounds to compute integrals of real variables that would otherwise be impossible.
Lecture 50 Play Video	Fourier type integrals using residues We use the residue theorem and some analysis tools to compute Fourier transforms of certain common rational functions.