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What are Complex Derivatives?

By Danielle DeLee
Updated May 17, 2024
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Complex derivatives are descriptions of the rates of change of complex functions, which operate in value fields that include imaginary numbers. They tell mathematicians about 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 those fields contains numbers that are part of the field of complex numbers, the function is called a complex function. Complex derivatives come from complex functions, but not every complex function has 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 over one axis for every unit of another axis. For example, a horizontal line on a two-dimensional graph would have a derivative of zero, because for each unit of x, the y value changes by zero. Instantaneous derivatives, which are most often used, give the rate of change at one point on the curve rather than over a range. This derivative is the slope of the straight line that is tangent to the curve at the desired point.

The derivative, however, does not exist everywhere on every function. If a function has a corner in it, for example, the derivative does not exist at the corner. 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. One 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 the conditions called Cauchy-Riemann functions. These require that the complex derivatives are the same regardless of how the function is oriented. If the conditions specified by the functions are fulfilled and the partial derivatives are continuous, then the function is complex differentiable.

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