Analytic Functions
★★★★☆Undergraduate
📖Definition
An analytic (holomorphic) function is complex differentiable at every point in its domain. It's infinitely differentiable and expressible as Taylor series.
📐Formulas
f(z) = ∑_n=0^∈fty aₙ (z - z₀)ⁿ
Taylor series expansion
aₙ = \fracf^(n)(z₀)n! = (1)/(2π i)∮_C (f(z))/((z-z₀)^n+1)dz
Taylor coefficients
✏️Examples
예제 1
Find Taylor series of e^z at z₀ = 0.
예제 2
What is the series expansion of sin z?
⚡Applications
Physics
Quantum mechanics, statistical mechanics
Engineering
Signal processing, control theory
Mathematics
Number theory, combinatorics
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#해석함수#정칙함수#analytic#holomorphic