Math complexes

Complex Analysis Lecture Notes Dan Romik About this document These notes were created for use as primary reading material for the graduate course Math 205A: Complex Analysis at UC Davis The current 2020 revision (dated June 15 2021) updates my earlier version of the notes from 2018

DEFINITION 5 1 1 A complex number is a matrix of the form x −y y x where x and y are real numbers x 0 Complex numbers of the form are scalar matrices and are called 0 x real complex numbers and are denoted by the symbol {x} The real complex numbers {x} and {y} are respectively called the real x

complex numbers by adding their real and imaginary parts:-(a+bi)+(c+di)= (a+c)+(b+d)i (a+bi)−(c+di)= (a−c)+(b−d)i We can multiply complex numbers by expanding the brackets in the usual fashion and using i2 = −1 (a+bi)(c+di)=ac+bci+adi+bdi2 =(ac−bd)+(ad+bc)i To divide complex numbers we note firstly that (c+di)(c−di)=c2 +d2 is

7 1 Operations on complex numbers real part Re(x + yi) := x imaginary part Im(x + yi) := y (Note: It is y not yi so Im(x + yi) is real) complex conjugate x + yi := x yi (negate the imaginary component) One can add subtract multiply and divide complex numbers (except for division by 0)

In Math 112 at Reed College students learn to write proofs while at the same time learning about binary operations orders elds ordered elds complete elds complex numbers sequences and series We also review limits continuity di erentiation and integration My aim for these notes is to constitute a self-contained book that covers the

The complex numbers z for which jzj= 5 holds constitute the circle with radius 5 and center 0 Verify that the complex numbers z for which jz 1j= 5 holds constitute the circle with radius 5 and center 1 Draw the following circles in the complex plane and for each circle give its center and its radius 1 14 a jzj= 4 b jz 1j= 3 c jz 2j= 2 d