MATH 2Z03 Study Guide - Final Guide: Ratio Test, Simply Connected Space, Linear Combination

• School: McMaster University
• Department: Mathematics
• Course: MATH 2Z03
• Professor: Jim Reilly

Document Summary

Chapter 1: complex numbers (cid:1878) (cid:1875) =(cid:1878)(cid:1875) (cid:1878) (cid:1875) =(cid:1878) (cid:1875) Open set s: every point is an interior point. Connected set s: there exists a finite path of line segments entirely in s connecting any two points in s. All products, sums, and compositions of continous functions are continuous unless denominator is 0. Removable discontinuities can be fixed by defining a value at z0. A function is analytic at z0 if there"s a disk ce(cid:374)tre at z0 in which f is differentiable. A function analytic in all complex numbers is entire. Polar forms: (cid:1878) =(cid:1870)(cid:1857)(cid:1861) (cid:1878) =(cid:1870)(cid:1857)(cid:1861) (cid:1878)(cid:1866) =(cid:1870)(cid:1866)(cid:1857)(cid:1861)(cid:1866) (cid:1878) 1(cid:1866) = (cid:1878)(cid:1866) =(cid:1870) 1(cid:1866)(cid:1857)(cid:1861)(cid:1866)(cid:1857)(cid:1861)2(cid:1863)(cid:2024)(cid:1866) (cid:1878)1 +(cid:1878)2 (cid:3409) (cid:1878)1 + (cid:1878)2 (cid:1878)1 (cid:1878)2 (cid:3409) (cid:1878)1 (cid:1878)2 (cid:1870) = (cid:1853)2 +(cid:1854)2 tan =(cid:1854)(cid:1853) cos +(cid:1861) sin (cid:1866) = cos(cid:1866) +(cid:1861) sin(cid:1866) Boundary point: any disk around it contains at least a point inside and outside s. Interior point: there exists a disk around it that contains no points outside s.