= + ∈ℂ, for some , ∈ℝ 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. Remember a real part is any number OR letter that … + z2 2! 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. 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! Basic rule: if you need to make something real, multiply by its complex conjugate. Complex numbers are built on the concept of being able to define the square root of negative one. For instance, for any complex numbers α,β,γ, we have • Commutative laws: α+β= β+αand αβ= βα. Rationalizing: We can apply this rule to \rationalize" a complex number such as z = 1=(a+ bi). Noether (1882{1935) gave general concept of com- The real numbers … 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 + ::: = 1 + z 1 + z2 2! Example: 3i If a ≠0 and b ≠ 0, the complex number is a nonreal complex number. In this T & L Plan, some students 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. Several elds were studied in mathematics for some time including the eld of real numbers the eld of rational number, and the eld of complex numbers, but there was no general de nition for a eld until the late 1800s. 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 Rings also were studied in the 1800s. Example: 7 + 2i A complex number written in the form a + bi or a + ib is written in standard form. + = ez Then jeixj2 = eixeix = eixe ix = e0 = 1 for real x. Basic Arithmetic: … • Associative laws: (α+β)+γ= γ+(β+γ) and (αβ)γ= α(βγ). Complex Numbers and the Complex Exponential 1. (Note: and both can be 0.) Questions can be pitched at different levels and can move from basic questioning to ones which are of a higher order nature. The representation is known as the Argand diagram or complex plane. Basic Concepts of Complex Numbers If a = 0 and b ≠ 0, the complex number is a pure imaginary number. Complex numbers can be represented as points in the plane, using the cor-respondence x + iy ↔ (x, y). (See chapter2for elds.) The real complex numbers lie on the x–axis, which is then called the real axis, while the imaginary numbers lie on the Complex numbers are often denoted by z. 2. Basic rules of arithmetic. Complex Number – any number that can be written in the form + , where and are real numbers. The union of the set of all imaginary numbers and the set of all real numbers is the set of complex numbers. Complex numbers obey many of the same familiar rules that you already learned for real numbers. 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