Complex analytic functions {holomorphic function} {holomorphic map} {differential function} {complex differentiable function} {regular function} can be differentiable for all derivatives.
equations
Complex-function real-part derivative with variable real part can equal complex-function imaginary-part derivative with variable imaginary part. Complex-function real-part derivative with variable imaginary part can equal negative of complex-function imaginary-part derivative with variable real part {Cauchy-Riemann equations, holomorphic function}. If Cauchy-Riemann equations hold, integrating on any path between two real complex-plane points obtains same result.
holomorphy
Paths can be deformable into each other or not {holomorphy}. Deformability allows canceling by going over same portion in opposite directions {homology, holomorphy}. Non-deformability does not allow canceling {homotopy, holomorphy}. If path has singularity, different paths are not homologous. At infinitesimal limit, holomorphic functions can be conformal, non-reflective, and orientation-preserving, so the only transformations are additions and multiplications. Reciprocal functions are holomorphic. Transformations (a*z + b)/(c*z + d) {bilinear transformation} {Möbius transformation, holomorphy} are holomorphic. Laplace equation in two dimensions has holomorphic function solutions.
Mathematical Sciences>Algebra>Function>Kinds>Complex
3-Algebra-Function-Kinds-Complex
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Date Modified: 2022.0224