Complex derivative

The complex derivative is the limit of a difference quotient as a complex increment approaches zero. Its existence requires the same limit from every direction in the complex plane.

Difference quotient: A ratio of a function's change to a change in its input, used to define derivatives. The complex derivative is the limit of this ratio as its complex increment vanishes.
Cauchy's integral formula: A formula expressing a holomorphic function's values inside a contour through its values on the contour. It follows from holomorphy and produces powerful consequences for complex derivatives.
Complex number: A number expressible as a real part plus an imaginary part. The increment can approach zero through every direction in the complex plane.
Complex conjugation: The map sending each complex number to its conjugate by reversing the sign of its imaginary part. It is real-differentiable everywhere but complex-differentiable nowhere.
Cauchy–Riemann equations: Equations relating the first partial derivatives of a complex function's real and imaginary parts. They characterize complex differentiability when the relevant partial derivatives are continuous.
Taylor series: A power series whose coefficients encode a function's derivatives at a point. Every holomorphic function has a local Taylor expansion, unlike a merely smooth real function.
Limit of a function: A value that a function approaches as its input approaches a specified point. The derivative is defined by a limit whose input is the complex increment.
Absolute value: The nonnegative magnitude of a real or complex number. The complex modulus function fails to be complex-differentiable at every point.
Holomorphic function: A complex function that is complex differentiable at every point of an open domain. A complex derivative at every point of an open set makes the function holomorphic there.
Identity theorem: A theorem stating that holomorphic functions agreeing on a set with an accumulation point agree throughout a connected domain. Complex differentiability helps make values on a small set determine an entire function.

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