Complex Analysis
The branch of mathematical analysis studying functions of complex variables f(z) = u(x,y) + i v(x,y).
Governing formula
∮_C f(z) dz = 2 π i ∑ Res(f, z_k) (Residue Theorem)
SI unit
Complex Function Theory
In depth
Holomorphic complex functions satisfy Cauchy-Riemann equations. Cauchy's Residue Theorem evaluates difficult real definite integrals in physics by integrating along complex contour paths enclosing poles.
Examples in the real world
Evaluating quantum scattering integrals and 2D fluid potential flow around airfoils using conformal mapping.