# Introductory Complex Analysis with Bill Kinney

## Video Lectures

Displaying all 35 video lectures.

Lecture 1Play 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 2Play 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 3Play 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 4Play 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 5Play 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 6Play 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 7Play 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 8Play 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 9Play 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 10Play 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 11Play Video |
Areas of Images, Differentiability, Analyticity, Cauchy-Riemann Eqs(0:00) |

Lecture 12Play 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 13Play 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 14Play 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 15Play 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 16Play 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 17Play 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 18Play 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 19Play 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 20Play 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 21Play 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 22Play 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 23Play 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 24Play 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 25Play 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 26Play 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 27Play 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 28Play 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 29Play 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 30Play 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 31Play 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 32Play 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 33Play 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 34Play 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 35Play 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. |