(Note: and both can be 0.) 1 Basic Theorems of Complex Analysis 1.1 The Complex Plane A complex number is a number of the form x + iy, where x and y are real numbers, and i2 = −1. Basic rule: if you need to make something real, multiply by its complex conjugate. Complex Numbers and the Complex Exponential 1. If z= a+biis a complex number, we say Re(z) = ais the real part of the complex number and we say Im(z) = bis the imaginary part of the complex number. Basic Concepts of Complex Numbers If a = 0 and b ≠ 0, the complex number is a pure imaginary number. Basic Arithmetic: … The representation is known as the Argand diagram or complex plane. Let 2=−බ ∴=√−බ Just like how ℝ denotes the real number system, (the set of all real numbers) we use ℂ to denote the set of complex numbers. Example: 3i If a ≠0 and b ≠ 0, the complex number is a nonreal complex number. Addition / Subtraction - Combine like terms (i.e. Complex numbers are built on the concept of being able to define the square root of negative one. = + ∈ℂ, for some , ∈ℝ Complex Number – any number that can be written in the form + , where and are real numbers. + = ez Then jeixj2 = eixeix = eixe ix = e0 = 1 for real x. In this T & L Plan, some students Rationalizing: We can apply this rule to \rationalize" a complex number such as z = 1=(a+ bi). Complex numbers can be represented as points in the plane, using the cor-respondence x + iy ↔ (x, y). Complex numbers are often denoted by z. The real complex numbers lie on the x–axis, which is then called the real axis, while the imaginary numbers lie on the Basic rules of arithmetic. • Associative laws: (α+β)+γ= γ+(β+γ) and (αβ)γ= α(βγ). Paul Garrett: Basic complex analysis (September 5, 2013) Proof: Since complex conjugation is a continuous map from C to itself, respecting addition and multiplication, ez = 1 + z 1! 2. + z2 2! If two complex numbers are equal then the real parts on the left of the ‘=’ will be equal to the real parts on the right of the ‘=’ and the imaginary parts will be equal to the imaginary parts. Complex numbers The equation x2 + 1 = 0 has no solutions, because for any real number xthe square x 2is nonnegative, and so x + 1 can never be less than 1.In spite of this it turns out to be very useful to assume that there is a number ifor which one has 6.1 Video 21: Polar exponential form of a complex number 41 6.2 Revision Video 22: Intro to complex numbers + basic operations 43 6.3 Revision Video 23: Complex numbers and calculations 44 6.4 Video 24: Powers of complex numbers via polar forms 45 7 Powers of complex numbers 46 7.1 Video 25: Powers of complex numbers 46 For instance, for any complex numbers α,β,γ, we have • Commutative laws: α+β= β+αand αβ= βα. Complex numbers obey many of the same familiar rules that you already learned for real numbers. 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