18.03 Lecture Notes - Lecture 24: Cartesian Coordinate System, Thomas More, Taylor Series
Document Summary
Suppose that y(t) is a complex valued function of a real variable t. then y(t) = f(t) +ig(t) for some real valued functions of t. Here f(t) := re y(t) and g(t) := im y(t) Represent complex numbers geometrically as points on the complex plane. Represent complex number algebraically in cartesian form and polar form as a complex exponential using euler"s formula. Perform arithmetic on complex numbers in both cartesian and polar form. Cartesian coordinates (x,y) and polar coordinates (r, ) where x = rcos and y = rsin and r = |z| (r = x + y ) arg z = {all theta such that z = r(cos + isin ) } Example: x = -3i < > (0, -3) and arg z = { -5 /2, - /2, 3 /2, } To make arg a function, restrict the angle: - < . Exponential principle (axiom, postulate): for any constant a, e is (de ned as) the solution of.
