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Complex derivatives are descriptions of the rates of change of complex functions, which operate in fields of values that include imaginary numbers. They tell mathematicians the behavior of functions that are difficult to visualize. The derivative of a complex function f at x0, if it exists, is given by the limit as x approaches x0 of (f(x) – f(x0))/(x – x0).
Functions associate values in one field with values in another field, which is an action called mapping. When one or both of these fields contain numbers that are part of the complex number field, the function is called a complex function. Complex derivatives come from complex functions, but not all complex functions have a complex derivative.
The sets of values that a complex function maps to and from must include complex numbers. These are values that can be represented by a + bi, where a and b are real numbers and i is the square root of negative one, which is an imaginary number. The value of b can be zero, so all real numbers are also complex numbers.
Derivatives are rates of change of functions. Generally, the derivative is a measure of the units of change on one axis for each unit of another axis. For example, a horizontal line on a two-dimensional graph would have a zero derivative, because for each unit of x, the value of y changes by zero. Instantaneous derivatives, which are used more often, provide the rate of change at a point on the curve rather than over an interval. This derivative is the slope of the tangent line to the curve at the desired point.
However, the derivative does not exist everywhere on every function. If a function has an angle, for example, the derivative does not exist at the angle. This is because the derivative is defined by a limit and if the derivative makes a jump from one value to another, then the limit is nonexistent. A function that has derivatives is said to be differentiable. A condition for differentiability in complex functions is that the partial derivatives, or the derivatives for each axis, must exist and be continuous at the point in question.
Complex functions that have complex derivatives must also satisfy conditions called Cauchy-Riemann functions. These require that the complex derivatives be the same regardless of how the function is oriented. If the conditions specified by the functions are met and the partial derivatives are continuous, then the function is complex differentiable.
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