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Complex Analysis

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# Content

` Complex Analysis  Complex Number Function and Complex Number  Complex Function  Topics   The Fundamental Theorem of Algebra`

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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)`

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