Complex differentiability secretly forces infinite smoothness
Complex analysis studies functions of complex variables, especially holomorphic maps that are complex-differentiable. Unlike real differentiability, holomorphicity implies infinite differentiability and local power series. Roots run through Euler, Gauss, Riemann, and Cauchy from the eighteenth century onward.
The subject reaches far beyond pure mathematics. Number theorists, algebraic geometers and combinatorialists use it, and so do physicists working on fluid flow, thermodynamics, quantum mechanics and twistor theory, along with aerospace, nuclear and electrical engineers. Conformal mappings have many physical uses, string theory studies conformal invariants, and fractal images made by iterating holomorphic functions gave the field a modern popularity boost. Weierstrass joins Euler, Gauss, Riemann and Cauchy among its founders.
Splitting a function into real and imaginary parts turns it into two real functions of two variables. Continuity carries over between the two views, but differentiability does not, because a complex derivative must give the same limit whichever direction the input approaches from. That demand is so strong that a single derivative guarantees derivatives of every order, whereas a real function can have infinitely many derivatives and still be analytic nowhere. The real and imaginary parts of a holomorphic function obey the Cauchy–Riemann conditions, though without extra continuity assumptions those equations alone do not guarantee holomorphy.
Polynomials, the exponential and trigonometric functions are holomorphic on the whole plane, earning the name entire. Rational functions fail only where their denominator vanishes, making them meromorphic. Two holomorphic functions agreeing near one point agree across any connected shared domain, the basis of analytic continuation, which defines the complex logarithm among others. Liouville's theorem says a bounded entire function is constant, giving a short proof that every polynomial has a complex root, and Picard showed a non-constant entire function misses at most one complex value.
Contour integrals depend only on how a path winds around singularities, so a path can be deformed into a handier one. Residues at poles, points where a function blows up, then evaluate hard real integrals.
Source: Complex analysis
