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Complex Analysis

In general, complex analysis is the study of complex numbers.

Complex Number

Complex numbers are numbers of the form a+b𝑖, where a and b are real numbers and 𝑖 is the imaginary unit that is equal to the square root of -1. Algebraically, a complex number can be considered as the algebraic extension of an ordinary real part of a number by an imaginary part 𝑖. Geometrically, complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane with the horizontal real axis and the vertical imaginary axis.

Function and Complex Number

Consider a quadratic equation, x²=mx+b that representing the intersection of graphs y=x² and y=mx+b. The solutions are eqaul to x=m/2±√(m²/4+b). However, when m²/4+b<0, there is no real solutions since the graphs y=x² and y=mx+b do not inersect in this case. Sometime, it is the simplest case used to argue that the concept of complex number with 𝑖= √-1 is introduced to solve the no real solution problems.

Consider a cubic equation, x³=px+q that representing the intersection of graphs y=x³ and y=px+q. Unlike quadratic equations, there must always be a solution for the cubic equation. Del Ferro (1465-1526) and Tartaglia (1499-1577), followed by Cardano (1501-1576), showed that x³=px+q has a solution given by x=∛(√(q²/4-p³/27)+q/2)-∛(√(q²/4-p³/27)-q/2). e.g x³=-6x+20⇒x=2.

However, about 30 years after the discovery of the solution formula, Bombelli (1526-1572) considered another equation, x³=15x+4 and the solution is x=∛(√(4²/4-15³/27)+4/2)-∛(√(4²/4-15³/27)-4/2)=∛(2+√-121)+∛(2-√-121). Bombelli further discovered that ∛(2+√-121)=2+√-1 and ∛(2-√-121)=2-√-1 since (2+√-1)³=2+√-121 and (2-√-1)³=2-√-121. And therefore x=4. Bombelli's discovery demonstrated that perfectly real problems also require complex arithmetic for formulate the solution.

Complex Function

A function whose range is in the complex number set is said to be a complex function, or a complex-valued function. Unlike ordinary real function, the domain and range of a complex function are usually represented by two individual complex planes.

 

Topics

Rieman, Weuerstrass, Cauchy, 𝑖, lim, e open set

compex dynamics: Mandelbrot set, Julia setss

complex function, continuity, complex differentiation

conformal mappings, Mobius transformations, Riemann mapping theorem

complex integration, Cauchy theory and consequences.

power series representation of analytic functions, Riemann hypothesis

The Fundamental Theorem of Algebra

If a₀, a₁, ⋯, aₙ are complex numbers with aₙ≠0, then the polynomial

p(z)=aₙzⁿ+aₙ₋₁zⁿ⁻¹+⋯+a₂z²+a₁z+a₀

has n roots z₀, z₁, ⋯, zₙ in ℂ and can be factored as

p(z)=aₙ(z-z₀)(z-z₁)⋯(z-zₙ)