18.04 Complex analysis with applications

1 Complex analysis with applicationsSpring 2020 lecture notesInstructor: R. R. RosalesThese notes are an adaption and extension of the original notes for by Andre Nachbinand Jeremy Orloff and later on by J orn Dunkel. Credit for course design and contentshould go to them; responsibility for typos and errors lies with will be updating and modifying the the date to make sure that you have the most current 20, 2020 Contents1 Brief course Topics needed from prerequisite math classes .. Level of mathematical rigor .. Speed of the class ..62 Complex algebra and the Complex Motivation .. Fundamental theorem of algebra .. Terminology and basic arithmetic .. The Complex plane .. geometry of Complex numbers .. triangle inequality .. Polar coordinates .. Euler s Formula .. exponential behaves like a true exponential .. exponentials and polar form .. or Complex replacement .. roots .. geometry of N-th roots.

2 Inverse Euler formula .. de Moivre s formula .. Representing Complex multiplication as matrix multiplication ..173 Complex The exponential function .. Complex functions as mappings .. The argument function .. functions .. of arg(z) .. principal branch of arg(z) .. Concise summary of branches and branch cuts .. The function log(z) .. showing w=log(z) as a mapping .. powers ..314 Analytic The derivative: preliminaries .. Open disks, open deleted disks, open regions .. Limits and continuous functions .. of limits .. functions .. of continuous functions .. The point at infinity .. involving infinity .. projection from the Riemann sphere .. Derivatives .. rules .. Cauchy-Riemann equations .. derivatives as limits .. Cauchy-Riemann equations .. the Cauchy-Riemann equations .. Complex derivative as a 2 by 2 matrix .. Geometric interpretation and linear elasticity theory.

3 Cauchy-Riemann all the way down .. Gallery of functions .. of functions, derivatives and properties .. few proofs .. Branch cuts and function composition .. Appendix: Limits .. Limits of sequences .. The limit of f(z) as z approaches z0 .. Connection between limits of sequences and limits of functions ..5525 Line integrals and Cauchy s Complex line integrals .. Fundamental theorem for Complex line integrals .. Path independence .. Examples .. Cauchy s theorem .. Extensions of Cauchy s theorem ..636 Cauchy s integral Cauchy s integral for functions .. Cauchy s integral formula for derivatives .. approach to some basic examples .. examples .. triangle inequality for integrals .. Proof of Cauchy s integral formula .. useful theorem .. of Cauchy s integral formula .. Proof of Cauchy s integral formula for derivatives .. Consequences of Cauchy s integral formula.

4 Of derivatives .. s inequality .. s theorem .. modulus principle ..797 Introduction to harmonic Harmonic functions .. Del notation .. functions have harmonic pieces .. conjugates .. A second proof that u and v are harmonic .. Maximum principle and mean value property .. Orthogonality of curves ..878 Two dimensional hydrodynamics and Complex Velocity fields .. Stationary flows .. Physical assumptions, mathematical consequences .. assumptions .. Complex potentials .. functions give us incompressible, irrotational flows .. , irrotational flows always have Complex potential func-tions .. Stream functions .. points .. More examples ..999 Taylor and Laurent Geometric series .. to Cauchy s integral formula .. Convergence of power series .. test and root test .. Taylor series .. of a zero .. series examples .. of Taylor s theorem.

5 Singularities .. Laurent series .. of Laurent series .. Digression to differential equations .. Poles .. of poles .. 11910 Residue Poles and zeros .. Words: Holomorphic and meromorphic .. Behavior of functions near zeros and poles .. Picard s theorem and essential singularities .. Quotients of functions .. Residues .. Residues at simple poles .. Residues at generic poles .. cot(z) .. Cauchy Residue Theorem .. Residue at infinity .. 13611 Definite integrals using the residue Integrals of functions that decay .. Integrals over the full real line and a half real line .. Trigonometric integrals .. Integrands with branch cuts .. Cauchy principal value .. Integrals over portions of circles .. Primer on integral convergence .. Fourier transform .. Solving ODEs using the Fourier transform.

6 156412 Conformal Geometric definition of conformal mappings .. Tangent vectors as Complex numbers .. Analytic functions are conformal .. Digression to harmonic functions .. Riemann mapping theorem .. Fractional linear transformations .. Examples .. Lines and circles .. Correspondence with matrices .. Mapping three z s to three w s .. Reflection and symmetry .. Reflection and symmetry across a line .. Reflection and symmetry across a circle .. Reflection across the unit circle .. Solving the Dirichlet problem for harmonic functions .. Harmonic functions on the upper half-plane .. Harmonic functions on the unit disk .. Flows around cylinders .. Milne-Thomson circle theorem .. Examples .. of conformal maps and exercises .. 17751 Brief course descriptionComplex analysis is a beautiful, tightly integrated subject.

7 It revolves around complexanalytic functions. These are functions that have a Complex derivative. Unlike calculususing real variables, the mere existence of a Complex derivative has strong implications forthe properties of the analysis is a basic tool in many mathematical theories. By itself and throughsome of these theories it also has a great many practical are a small number of far-reaching theorems that we will explore in the firstpart of the class. Along the way, we will touch on some mathematical and engineeringapplications of these theorems. The last third of the class will be devoted to a deeper lookat main theorems are Cauchy s Theorem, Cauchy s integral formula, and the existenceof Taylor and Laurent series. Among the applications will be harmonic functions, twodimensional fluid flow, easy methods for computing (seemingly) hard integrals, Laplacetransforms, and Fourier transforms with applications to engineering and Topics needed from prerequisite math classesWe will review these topics as we need them: Limits Power series Vector fields Line integrals Green s Level of mathematical rigorWe will make careful arguments to justify our results.

8 Though, in many places we will skiptechnical details if they get in the way of understanding the main point. However, we willnote what was left Speed of the class(Borrowed from R. Rosales OCW 1999)Do not be fooled by the fact things start slow. This is the kind of course where thingskeep on building up continuously, with new things appearing rather often. Nothing is reallyvery hard, but the total integration can be staggering and it will sneak up on you if youdo not watch it. Or, to express it in mathematically sounding lingo, this course is locallyeasy but globally hard . This means that if you keep up-to-date with the homework andlectures, and read the class notes regularly, you should not have any Complex algebra and the Complex planeWe will start with a review of the basic algebra and geometry of Complex numbers. Mostlikely you have encountered this previously in or MotivationThe equationx2= 1 has no real solutions, yet we know that this equation arises naturallyand we want to use its roots.

9 So we make up a new symbol for the roots and call it acomplex symbols iwill stand for the solutions to the equationx2= 1. We willcall these new numbers Complex numbers. We will also write 1 = iwhereiis also called animaginary is a historical term. These are perfectlyvalid numbers that don t happen to lie on the real number re going to look at thealgebra, geometry and, most important for us, the exponentiation of Complex starting a systematic exposition of Complex numbers, we ll work a simple the equationz2+z+ 1 = :We can apply the quadratic formula to getz= 1 1 42= 1 32= 1 3 12= 1 :Do you know how to solve quadratic equations by completing the square? This is howthe quadratic formula is derived and is well worth knowing! Fundamental theorem of algebraOne of the reasons for using Complex numbers is because allowing Complex roots meansevery polynomial has exactly the expected number of roots.

10 This is thefundamentaltheorem of algebra:A polynomial of degreenhas exactlyncomplex roots (repeated roots are counted withmultiplicity).In a few weeks, we will be able to prove this theorem as a remarkably simple consequenceof one of our main typically usejinstead ofi. We ll follow mathematical custom in motivation for using Complex numbers isnotthe same as the historical motivation. Historically,mathematicians were willing to sayx2= 1 had no solutions. The issue that pushed them to accept complexnumbers had to do with the formula for the roots of cubics. Cubics always have at least one real root, andwhen square roots of negative numbers appeared in this formula, even for the real roots, mathematicianswere forced to take a closer look at these (seemingly) exotic Terminology and basic arithmeticDefinitions Complex numbersare defined as the set of all numbersz=x+yi,wherexandyare real numbers. We denote the set of all Complex numbers byC.