  # Introductory Complex Analysis with Bill Kinney

## Video Lectures

Displaying all 35 video lectures.
 Lecture 1 Play Video Complex Arithmetic, Cardano's Formula(0:00) Introduction. (2:02) Syllabus. (11:00) Module 1: Complex Number Addition and the Complex "Argand" Plane, Activity 1. (19:29) Complex arithmetic examples (the complex number system forms a field under ordinary addition and multiplication). (32:31) Check the division with Mathematica. (34:03) Mention extra videos about Mathematica usage for complex arithmetic (Chapter 1). (36:03) Mobius transformations revealed video. (40:23) Inversion transformation on Mathematica. (42:44) Real cube root function in Mathematica. (43:50) History of complex numbers and a couple examples related to Cardano's formula. (58:11) Ungraded exercise to do before Lecture #2. Lecture 2 Play Video Geometric Interpretations of Complex Arithmetic, Triangle Inequality(0:00) Comments on ungraded exercises to do. (0:30) This lecture will focus on the geometry of complex arithmetic. (1:01) Pre-class and post-class lecture Mathematica notebooks. (1:57) The complex numbers form a field, with field properties of addition and multiplication, such as the commutative and associative properties. (5:03) The natural one-to-one correspondences (associations) between the set of complex numbers, points in the plane, and vectors in the plane. (14:07) Polar coordinates and PolarPlot on Mathematica. (23:19) Exercise to find the polar coordinates of a product in terms of the polar coordinates of the factors. (36:04) Modulus and argument and the general fact relating these things for the product of two complex numbers. (38:27) Mathematica demonstration for this fact. (41:22) Defining the Euclidean (distance) metric using the modulus and its geometric interpretations. (43:50) Parallelogram law for addition and the triangle inequality. (50:12) Complex conjugate and its relation to complex division (in particular, to interpreting 1/z geometrically). (57:48) Complex addition as a mapping that is a translation of the complex plane. Lecture 3 Play Video Polar Form, Principal Value of Arg, Basic Mappings(0:00) Moodle quiz due within two class periods. (0:34) Review rectangular and polar forms for complex numbers. (7:31) Finding the argument of a complex number on Mathematica. (11:02) Principal value of the argument, Arg(z) and Mathematica's extended arctangent function. (14:12) Plot the argument in three dimensions as a function of x and y. (17:34) arg(z) as a multi-valued function and the fact that the argument of a product is the sum or the arguments of the factors. (23:22) Describing circles and disks (both closed and open) in the complex plane. (29:43) Summarize the geometric interpretation of complex multiplication and division (and outline the proof for the moduli of the product, using trigonometric identities). (39:14) Viewing complex arithmetic in terms of mappings of the complex plane (as well as the corresponding real mapping). Examples: 1) Add a complex constant. 2) Multiply by a positive real constant. 3) Multiply by -1. 4) Multiply by a complex conjugate of modulus 1. (55:13) Writing z^2 in terms of real and imaginary parts. Lecture 4 Play Video Mappings, Linear Mappings, Squaring Map, Euler's Identity(0:00) Quiz due by next class period. (0:37) Mathematica project idea (connecting centers of opposing squares constructed on the sides of a quadrilateral generates line segments that are perpendicular). (4:54) The precise connection between complex planar mappings and real planar mappings. (10:21) Linearity for real planar mappings and complex planar mappings. (19:44) The squaring mapping f(z) = z^2, first, for a particular input. (23:34) The real and imaginary parts for f(z) = z^2 and the corresponding real planar mapping. (25:26) Start to explore the mapping properties of f(z) = z^2 by determining the image of the vertical line x = 3. (37:50) Euler's identity derivation via Taylor series centered at zero. (44:52) Derivation using the chain rule and differential equations. (49:56) Defining e^(z) for an arbitrary complex number z. (51:20) Geometric interpretation of the series definition on Mathematica. (56:04) Make sure you know the polar form of complex numbers in terms of the complex exponential, De Moivre's formula, and applications to trigonometric identities and integrals. Lecture 5 Play Video Squaring Mapping, Euler's Identity & Trigonometry, 5th Roots Example(0:00) This is a re-do of the morning's lecture. (0:16) Commutative diagram for squaring mapping and the corresponding real planar mapping. (6:13) Visualizing the squaring mapping on Mathematica. Lecture 6 Play Video Exponential Map on Mathematica, Squaring Map, Intro to Topology(0:00) Mathematica project idea (the Riemann sphere and stereographic projection). (1:04) Quiz 2 possible due dates. (1:21) Enabling dynamic updating. (2:49) The real and imaginary parts of the complex exponential function, along with the corresponding real planar mapping. (4:40) Entering the function into Mathematica. (7:23) Animating the geometry of the mapping with Manipulate and ParametricPlot. (12:17) The image of a vertical line (at a fixed value of x) is a circle centered at the origin with radius e^(x). (17:45) The image of a horizontal line (at a fixed value of y) is a ray emanating from the origin (but not including the origin). (25:56) Other kinds of graphs that can be made in complex analysis. (32:44) Dynamic behavior of orbits under iteration of z^2 on Mathematica with NestList. (38:39) Preimages of mappings (of circles through the origin under z^2). (46:49) Modular surfaces of z^2 - 1 and z^2 + 1. (47:59) Overview of images of circles under the reciprocal map 1/z. (49:08) Quick introduction to the topology of the complex plane (focused on domains of typical functions being open and connected sets). Lecture 7 Play Video Exponential & Reciprocal Maps, Domains, Derivative Limit Calculations(0:00) Quiz 2 due date and topics. (0:30) Mathematica demonstration of the stereographic projection onto the Riemann sphere. (4:09) Review the formula for the exponential mapping and animate images of rectangles on Mathematica. (12:28) e^(z) is periodic with period 2*pi*i. (13:51) Making sense of the image from the formula. (15:04) Reciprocal mapping and algebraically verifying images of lines and circles. (28:51) Definition of a "domain" in complex analysis as an open connected subset of the complex plane (a typical "domain of definition" for functions in complex analysis). (30:21) What does it mean for a set to be (path) connected? (32:33) What does it mean for a set to be open (in terms of interior points)? Demonstrate idea with Mathematica. (39:52) Derivative definition and limit calculations for f(z) = z^2 and f(z) = 1/z. Lecture 8 Play Video Topological Definitions, Limits, Continuity, Linear Approximation(0:00) We survived Real Analysis tee shirt. (1:28) Descriptive theory day (topological definitions). (2:23) Facts about linear fractional transformations and a demonstration from the Wolfram Demonstrations project. (5:29) Topological definitions, starting with an open disk (neighborhood) centered at a point. (8:10) Interior point of a set. (9:50) Interior of a set. (10:32) Open sets (also note that the entire complex plane and the empty set are open...though I made a mistake in how I described the empty set as open). (13:03) Polygonal path (or "curve"). Lecture 9 Play Video Facts to Recall, Animations, Continuity Proofs (z^2 and 1/z)(0:00) Why is the empty set open (made a mistake in Lecture 8). (2:17) Typing up loose ends from chapter 1. Lecture 10 Play Video Open Disks are Open, Derivatives, Analyticity, Linear Approximations(0:00) Exam 1 in two class periods. (0:36) Modulus taking is operation preserving with respect to multiplication and division. (1:20) Proof that open disks are open sets. (13:23) Possible proofs on Exam 1. (14:29) Differentiability at a point. (17:16) Calculate the derivative of f(z) = z^3 from the limit definition. (24:21) Be able to derive basic derivative rules (linearity & product rule especially...look up the derivation of the product rule) using the field properties ("ordinary" algebraic properties) of C. (30:57) Even simple complex functions can fail to be differentiable (such as the function that returns the complex conjugate of a given number) which is an example you should study. (32:53) Definition of what it means for a complex function to be analytic at a point. (36:04) Analyticity of a function defined on an open set. (39:19) Polynomials are analytic everywhere on C (they are "entire") and rational functions are analytic everywhere that they are defined. (42:01) This is related to the Cauchy-Riemann equations. (42:34) Linear approximation for real differentiable planar mappings (equations, approximations, and animation of geometric meaning). (53:27) Linear approximation for complex differentiable mappings. Lecture 11 Play Video Areas of Images, Differentiability, Analyticity, Cauchy-Riemann Eqs(0:00) Lecture 12 Play Video Cauchy-Riemann Eqs (Rectangular & Polar), Intro Harmonic Functions(0:00) Announcements. (0:22) Lecture topics. (0:59) Review differentiability and analyticity. (3:19) Derive the Cauchy-Riemann equations symbolically. Lecture 13 Play Video Preimages, Laplace's Equation, Harmonic and Analytic Functions(0:00) Topics for the lecture. (0:32) Level curves of the real and imaginary parts of a complex-valued function are preimages of horizontal and vertical lines under the mapping. (3:26) Symbolically confirm that hyperbolic functions parameterize hyperbolas. (6:33) Animate hyperbolas being mapped under f(z) = z^2. (12:13) Review Cauchy-Riemann equations in polar coordinates and check them for f(z) = z^3 in both rectangular and polar coordinates. (18:21) Laplace's equation (a partial differential equation), notation, overview of applications, harmonic functions, and connections to analytic functions. (27:49) Generate a harmonic function by taking the real part of an analytic function. (30:56) Find the harmonic conjugate by integration and the Cauchy-Riemann equations. (37:34) Example to illustrate a solution to Laplace's equation along the closed unit disk, using polar coordinates to describe the boundary conditions, and then imagine it to be a temperature distribution. (49:41) Infinitesimals, complex derivatives, and the amplitwist concept (due to Tristan Needham). Lecture 14 Play Video Preimages, Mathematica, Maximum Principle (Harmonic), PolynomialsLong division, synthetic division, rational functions and partial fractions decomposition included in the content as well. (0:00) Mathematica project ideas. (4:12) Preimages of rectangles under a mapping are regions between level curves of u and v. (14:36) Maximum principle for harmonic functions and an example over a disk, including optimization analysis along the boundary of the disk. (30:09) Polynomial roots and factorizations. (32:50) Example of a cubic with complex coefficients. Do polynomial long division and synthetic division if you are given one of the roots. Lecture 15 Play Video Review Analytic Functions, Amplitwist Concept, Harmonic Functions(0:00) Celebrate pi day tomorrow (since spring break next week). (0:25) Do the quiz and search for project ideas in coming days. (2:41) Today will be a review day. (3:21) Abstract differentiation rules. (4:57) Local linearity, including a numerical animation made with Manipulate, and a reminder of the distinction between linear approximation for complex differentiable mappings and differentiable real planar mappings. (12:23) Analyticity, lack of analyticity (for functions involving conjugation, taking modulus, real and imaginary parts), polynomials are entire, and rational functions are analytic wherever they are defined. (15:45) Fundamental Theorem of Algebra. (17:55) Graph the modulus of f(z) as a 3-dimensional graph and a contour map to see where the roots are (and where the poles are for a rational function). (24:25) Amplitwist concept (from Tristan Needham's "Visual Complex Analysis") in terms of linear approximation. (32:02) Amplitwist concept in terms of infinitesimals. (35:19) Cauchy-Riemann equations in rectangular and polar coordinates, relationship to gradient vector fields and level curves for u and v. (37:42) Analytic functions are conformal (angle-preserving when the derivative is non-zero. (38:22) Representation of the derivative f' in terms of partial derivatives of u and v and application to the derivative of the complex exponential function being itself. (41:02) The constant function theorem over a (open and connected) domain. (42:12) Finding areas of images of regions under complex analytic mappings (writing the Jacobian determinant in terms of the derivative f'). Application to f(z) = z^2 for small disks near the origin. (52:14) Harmonic functions and Laplace's equation, including an example with discontinuous boundary values. (54:47) Relationship between analytic and harmonic functions. (55:39) Optimizing functions and harmonic functions over compact (closed and bounded) regions (Maximum principle), also look at the form of the determinant of the Hessian matrix. (59:04) Analytic and harmonic functions turn out to be infinitely differentiable. (1:00:12) Partial derivatives of harmonic functions are harmonic. Lecture 16 Play Video Taylor Polynomials, Complex Exponential, Trig & Hyperbolic Functions(0:00) Goals and a demonstration of what happens near where the derivative of an analytic function is zero. (5:42) Visualizing the amplitwist based on the modulus of the derivative and the argument of the derivative. (9:35) Find a (finite) Taylor expansion of a polynomial function about z = 2 in two ways: 1) substitution, 2) Taylor's formula. (20:07) Comments about functions versus multi-valued functions. (21:20) The complex exponential function: formula, periodicity, mapping properties. (28:05) The complex cosine and sine functions: derivatives, identities, modulus of cosine along the imaginary axis. (39:34) The complex hyperbolic functions: identities, relationships to trigonometric functions, geometric/numerical demonstration of sin(iz) = i*sinh(z). (46:15) Introduction to the complex logarithm Lecture 17 Play Video Complex Logarithm, Functions as Sets, Multivalued FunctionsThe focusing gets better about 30 seconds in. (0:00) Lecture topics and Exam 2 date. (1:47) Summary of the definition of the complex logarithm as a multivalued function and corresponding single valued branches. (11:34) Properties of logarithms still hold, with a caveat that an appropriate branch must be chosen. (18:37) Functions as sets (subsets of Cartesian products...special relations). In effect, we are saying the graph of the function "is" the function (also includes a discussion of Naive Set Theory versus Axiomatic Set Theory). Lecture 18 Play Video Branches of Arg, Harmonic Functions over Washers, Wedges and Walls(0:00) Taking (single-valued) branches of multivalued functions. (0:35) Two-parameter plot of the "full graph" of the multivalued argument function arg(z) with ParametricPlot3D. (4:10) What does it mean to take a branch? There's a lot of flexibility. The branch cuts don't even have to be straight lines.(17:08) Examples of harmonic functions (satisfying Laplace's equation) over various domains. (24:11) These examples can help us solve Laplace's equation on washers, wedges, and walls, which can ultimately help us solve Laplace's equation on more general regions. (36:40) Project ideas and readings. (40:53) Introduction to power functions and inverse trigonometric functions. Lecture 19 Play Video Complex Powers, Inverse Trigonometric Functions, Branch Cuts(0:00) Announcements. (0:31) Overview of the subjects and applications. (1:25) Definition of complex powers. (6:00) Example calculations with multivalued powers. (16:25) Inverse trigonometric functions. Lecture 20 Play Video Invariance of Laplace's Eq, Real & Im Parts of Complex Integrals(0:00) Exam 2 next week (advising/assessment day this week). (0:48) Plan for the lecture. (1:42) Invariance of Laplace's equation (basic set up). (7:24) The proof that psi (the newly-defined function) is harmonic. Lecture 21 Play Video Conformality, Riemann Mapping Theorem, Vector Fields, IntegrationPictures of the class were being taken the first few minutes. Also sorry about the clicking noises from the camera tripod. (0:00) Upcoming schedule and project information. (1:11) Amplitwist and Chain-Rule related reasoning for why analytic mappings are conformal when the derivative is nonzero. (7:10) Riemann mapping theorem statement. (10:43) Prelude to complex integration: complex-valued functions as vector fields and their graphs. (17:03) The vector field "rotates" around circles, and this can be used to define the index of a vector field at a singularity (which can be taken to be a zero or a pole). (25:36) It will be more natural to view complex functions as vector fields for the purposes of integration and the applications will switch over to work done by a force, flux of a (2-dimensional) fluid flow across a curve (1-dimensional) membrane. (26:56) The squaring function as a vector field. (28:41) Integrating a complex function over a (parabolic) contour via a parameterization. (37:43) Doing the same integral with the complex version of the Fundamental Theorem of Calculus. (42:23) Approximating the answer via a Riemann sum and a start at visualizing it. Lecture 22 Play Video Complex Integrals, Cauchy-Goursat Theorem, Quick Exam 2 Review(0:00) Exam 2 next class period. (1:26) Various ways to integrate z^2 over a parabolic path, done on Mathematica, first in terms of complex quantities. (5:02) Relate to real line integrals from Multivariable Calculus and calculate with Mathematica. (14:10) Confirm the answer with the Fundamental Theorem of Calculus. (15:45) Independence of path, including calculation over alternative oriented contours. (22:45) Integrals around closed loops. (24:57) Cauchy's Integral Theorem (Cauchy-Goursat Theorem), including a description of a simply connected domain. (30:02) Outline of proof of weaker version with Green's Theorem and the Cauchy-Riemann equations (give statement of Green's Theorem as well). (40:48) Analytic functions are infinitely differentiable, which doesn't happen for real functions when you have differentiability over a neighborhood of a point. (43:09) Review for Exam 2 (including a review of the Chain Rule in polar coordinates). Lecture 23 Play Video Real Line Integrals and Applications, Complex Integration(0:00) Summer school course on Financial Math. (1:21) Plan for the lecture. (2:42) Scalar line integrals (application: mass of a thin wire). (10:23) Example to find the mass of a thin wire (first approximate it, then find the exact mass). (16:06) Vector line integrals (application: work done by a force). Lecture 24 Play Video Integration, Cauchy-Goursat Theorem, Cauchy Integral Formula(0:00) Project idea (flows for vector fields). (5:49) Review the basics of complex integration over an oriented contour. Lecture 25 Play Video Cauchy Integral Formula, Applications, Liouville's TheoremThe lecture also includes a proof of a theorem that can be used to prove the generalized Cauchy integral formula (once you assume the truth of the Cauchy integral formula). (0:00) Mathematical beauty and Philippians 4:8. (0:54) Animation of the flow of a linear vector field (you could relate these ideas to vector fields generated by analytic functions). (3:23) Cauchy integral formula. (5:58) Generalized Cauchy integral formula. (7:06) Application to evaluation of integrals and check the answers with parameterizations on Mathematica. (19:20) Theorem statement about differentiating a special kind of integral (which can be used to prove the generalized Cauchy integral formula from the ordinary Cauchy integral formula) and an example on Mathematica. (30:47) Verbally describe Liouville's Theorem and its proof. (33:56) Liouville's Theorem can be used to prove the Fundamental Theorem of Algebra (and describe basic idea of proof). (36:12) Verbal description of (two forms of) the Maximum Modulus Principle. (37:44) Go back to some details of the proof of theorem about differentiating a special kind of integral. Lecture 26 Play Video Sequences and Series of Functions, Maximum Modulus on Mathematica(0:00) Plan for the lecture. (0:44) Initial example for series: expand f(z) = 5/(3+7z) as a Taylor series about z = 8 by using some algebra tricks and the formula for the sum of a geometric series. (12:32) Sequences of real-valued functions of a real variable and pointwise convergence (examples on Mathematica). (24:55) Definition of convergence of a (complex) series as the limit of a sequence of partial sums (when the limit exists). (30:40) Geometric series example (find the function and find the disk of convergence, (almost) confirming the disk of convergence with the Ratio Test). (38:20) Give an illustration of the maximum modulus principle on Mathematica, optimizing the modulus over a closed disk by analyzing the behavior along the boundary of the disk. Lecture 27 Play Video Review Cauchy's Theorem, Cauchy Integral Formulas, and Corollaries(0:00) Journal catch-up time and Mathematica project time. (1:08) This lecture will mostly be a review of what we've done since the last exam to try to get these ideas organized and solidified in our minds. (2:00) Cauchy's Theorem. (7:11) Verbal description of continuous deformation theorem. (8:58) Cauchy integral formula. Lecture 28 Play Video Taylor Series Computations, Graphs of Partial Sums, Ratio Test(0:00) Quiz ready. (0:22) Purely algebraic proof of the Fundamental Theorem of Algebra (involves Field Theory, Galois Theory, Group Theory, and the Sylow Theorems). (2:32) Plan for the day. (3:11) Taylor series for an analytic function centered at a point z0 (at which it is analytic). (6:58) Find the Taylor series for the sine function centered at z0 = 0. (12:07) Show graphs of partial sums on the real line converging to the graph of the sin(x). (14:40) Write down the Taylor series for the cosine function centered at z0 = 0 and graph the partial sums on the real line. (16:36) Write down the Taylor series for the exponential function e^z centered at z0 = 0 and graph the partial sums on the real line. (18:47) Taylor series for 1/(1 - z) centered at z0 = 0 (geometric series), for z within 1 unit of 0. (21:18) Use the Ratio Test to prove the convergence of the Taylor series for e^z for all z (though this doesn't prove it converges to e^z) and graph the partial sums on the real line. (29:41) Taylor series for 1/(1+z^2). (32:21) Graph of partial sums on the real line. Why is the interval of convergence just the open interval from -1 to 1? The function has a pole at z = i and z = -i. (38:28) Integrate the Taylor series for 1/(1+z^2) term-by-term to get the Taylor series for the principal value of the arctangent function. (42:29) Approximations of pi with the Taylor series for arctangent (including Machin's formula). (46:56) Differentiate the series for 1/(1+z^2) term-by-term to get the Taylor series for -2z/((1+z^2)^2). Lecture 29 Play Video Uniform Convergence, Taylor Series Facts(0:30) Quiz 5 due by midnight. (0:58) Exam 3 in one week. (2:44) Plan for the lecture. (3:07) Definition of pointwise convergence of a sequence of complex functions over some subset of the complex plane. Lecture 30 Play Video Laurent Series Calculations, Visualize Convergence on Mathematica(0:00) Lecture plan and the coming weeks. (1:33) Taylor series for f(z) = z/(z^2 + z -12) centered at z = 0 (which will also be the Laurent series centered at the same point) via partial fraction expansion and addition of Taylor series. (16:50) Check answer with "Series" on Mathematica. (24:30) Find the Laurent series centered at z = 0 in the annulus between the circles of radius 3 and 4. (31:43) Use Plot3D and Manipulate on Mathematica to visualize the modulus of f(z) and the modulus of partial sums of the Laurent series. (49:48) The order of a pole. We will relate these topics to integrals. Lecture 31 Play Video Laurent Series, Poles, Essential Singularities(0:00) Exam 3 date change. (0:32) Review expansion and visualization of f(z) = z/(z^2 + z - 12) as a Laurent series in a disk centered at z = 0 and an annulus centered at z = 0. (11:20) Find the Laurent series centered at z = -4 (converging on a punctured disk of radius 7 centered at -4). (27:13) The pole at z = -4 has order 1. (27:59) New example with a pole of order 2 at z = -4. Lecture 32 Play Video More Laurent Series, Review Integrals & Cauchy Integral Formula(0:00) Find one more Laurent series for f(z) = z/(z^2 + z - 12) (for z strictly outside of the circle of radius 7 centered at z = -4), comments about the order of a pole are made. Lecture 33 Play Video Integrating 1/(1+z^2), Mathematica programming, ResiduesSorry that the camera has trouble focusing the first 2 minutes. (0:00) Plan for the last 2 weeks. (2:11) Consider whether God's plan for your life could include exploring complex analysis further. (3:26) Antiderivative of 1/(1+x^2) and graphs. (6:10) Area under the graph of 1/(1+x^2) evaluated as an improper integral which is defined as the sum of two improper integrals (and these integrals do converge). (10:39) Integrate 1/(1+z^2) over various closed contours using Mathematica to visualize the corresponding vector fields. Lecture 34 Play Video Series, Zeros, Isolated Singularities, Residues, Residue Theorem(0:00) Exam information. (1:28) "Holomorphic function" means "analytic function"...we'll look at "meromorphic functions" as well. (2:26) A function that is analytic at a point has a Taylor series that converges to it in the largest open disk centered at the point over which the function is analytic. (5:32) Given a power series with a positive radius of convergence, it defines an analytic function whose Taylor series is the given power series. (7:30) Some people define analyticity in terms of power series. This also can be done to define real analytic functions, though that is not equivalent to being infinitely differentiable. (9:04) Laurent series representation in an annulus. Lecture 35 Play Video Residue Theorem Examples, Principal Values of Improper Integrals(0:00) Schedule before final exam. (1:25) The principal value of the integral of 1/(1+x^2) using the Residue Theorem.