Complex Analysis
Draft for Information Only Content Complex Analysis source/reference: Complex AnalysisIn general, complex analysis is the study of complex numbers. Complex NumberComplex 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 onedimensional number line to the twodimensional complex plane with the horizontal real axis and the vertical imaginary axis. Function and Complex NumberConsider 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 (14651526) and Tartaglia (14991577), followed by Cardano (15011576), showed that x³=px+q has a solution given by x=∛(√(q²/4p³/27)+q/2)∛(√(q²/4p³/27)q/2). e.g x³=6x+20⇒x=2. However, about 30 years after the discovery of the solution formula, Bombelli (15261572) considered another equation, x³=15x+4 and the solution is x=∛(√(4²/415³/27)+4/2)∛(√(4²/415³/27)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 FunctionA function whose range is in the complex number set is said to be a complex function, or a complexvalued function. Unlike ordinary real function, the domain and range of a complex function are usually represented by two individual complex planes.
TopicsRieman, 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 AlgebraIf 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(zz₀)(zz₁)⋯(zzₙ)
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