Complex differentiability

Complex differentiability is the existence of a finite limit for a complex difference quotient as its increment approaches zero from every direction. At a point, it is stronger than ordinary real differentiability of the corresponding two-variable map.

Complex derivative: The limit of a complex difference quotient, when that limit exists. It is the value produced by complex differentiability at a point.
Complex limit: A limit in the complex plane, requiring convergence as the input approaches from every direction. The difference quotient must have a single limit across all paths toward the point.
Real differentiable function: A real-valued function on a real interval whose derivative exists at a point or throughout a domain. Real differentiability tests increments on a line; complex differentiability tests all planar approaches.
Power series: An infinite series whose terms are powers of a variable multiplied by coefficients. Holomorphic functions admit local power-series expansions, a consequence far stronger than real differentiability.
Cauchy–Riemann equations: Partial-derivative identities relating the real and imaginary parts of a complex function. They express a necessary local constraint for complex differentiability when the partial derivatives are continuous.
Difference quotient: A ratio measuring the change in a function’s output relative to a change in its input. Complex differentiability is defined by taking its limit as the increment vanishes.
Wirtinger calculus: A calculus using derivatives with respect to a complex variable and its conjugate as formal coordinates. It represents general real-differentiable maps, including those that fail complex differentiability.
Cauchy integral formula: A formula expressing a holomorphic function’s interior values through its values on a surrounding contour. It yields derivatives of every order once complex differentiability holds throughout a region.
Wirtinger derivatives: Differential operators that combine partial derivatives with respect to the real coordinates of a complex variable. They separate the complex-linear and conjugate-linear parts of a function’s differential.
Real differentiability: Differentiability of a map between real coordinate spaces, described locally by a linear approximation. It permits arbitrary real-linear approximations, while complex differentiability requires complex-linearity.

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