maths.free › Complex Analysis

Complex Analysis

Holomorphic functions, contour integrals, residues, the rigidity of the complex plane.

Zvidzidzo

Introductory The complex plane and Euler's formula z = x + iy as a point, modulus and argument, e^{iθ} = cos θ + i sin θ. e^(i*pi) Core Holomorphic functions and the Cauchy-Riemann equations What complex differentiability demands, and why it is so restrictive. derivative of x^2 - y^2 wrt x Advanced Contour integrals and the residue theorem Cauchy's theorem, the integral formula, and evaluating real integrals with residues. integrate 1/(1+x^2) dx from -oo to oo Advanced Power series and analytic continuation Radius of convergence, the identity theorem, and extending ζ(s) beyond its series. taylor series of 1/(1-x)

Calculus over the complex numbers turns out to be astonishingly rigid: once a function is differentiable on a disc, it is infinitely differentiable, equals its Taylor series, and is determined everywhere by its values on a tiny set. The residue theorem then evaluates real integrals no real method can.

More lessons

The complex plane

Complex analysis Complex number Complex plane Euler's formula Argument (complex analysis)Complex conjugate Riemann sphere Möbius transformation

Holomorphic functions

Holomorphic function Cauchy–Riemann equations Analytic function Harmonic function Conformal map Entire function Meromorphic function Zeros and poles Branch point

Integration

Cauchy's integral theorem Cauchy's integral formula Residue theorem Laurent series Contour integration Argument principle Liouville's theorem (complex analysis)Maximum modulus principle Analytic continuation Riemann mapping theorem Riemann surface

Zvimwe zvezvidzidzo izvi zvakagadzirirwa kubva kune zvakavhurika zvinyorwa: Wikipedia. Ichi chidzidzo chinopa zita remutauro wayo uye chinoratidza chikonzero chekugadzirwa kwake.

Mazita anoshandiswa pano

Tinya chero chiratidzo kuti uone chirevo chakazara, vhidhiyo, uye kuti chii chinoreva chero chiratidzo.

Mabvunzo vanhu vanobvunza

Why is complex differentiability so much stronger than real?

The limit must be the same from every direction in the plane, not just two. That forces the Cauchy-Riemann equations, which in turn force infinitely many derivatives and a convergent Taylor series.

What is a residue?

The coefficient of 1/(z − a) in the Laurent series at a singularity a. The residue theorem says a contour integral equals 2πi times the sum of the residues inside, which evaluates many real integrals in one line.

Une dambudziko rako?
Dzvanya kana kudhonza pane peji re solver: pazuva rimwe nerimwe, pfungwa, mhedzisiro yakachengetwa kusvika iwe unobvunza. _Kugadzirisa →

Other branches

＋ Arithmetic 𝑥 Algebra △ Geometry ∿ Trigonometry ƒ Precalculus σ Statistics & Probability ⊢ Discrete Math & Logic ∫ Calculus ⊞ Linear Algebra ℕ Number Theory ẏ Differential Equations ∇ Multivariable Calculus ε Real Analysis ⋔ Combinatorics & Graph Theory ∈ Set Theory & Logic 𝔾 Abstract Algebra 𝑃 Probability Theory ≈ Numerical Methods ◎ Topology μ Measure Theory ‖·‖ Functional Analysis κ Differential Geometry π₁ Algebraic Topology → Category Theory ∂ Partial Differential Equations ≋ Fluid Dynamics ∞ Frontiers