# Cauchy-Riemann equations

Here we follow Spiegel (1964) ^{[1]} or Levinson and Redheffer (1970). ^{[2]}

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Any *complex valued function* of the *complex variable* that satisfies the following
system of first-order partial differential equations, known as the Cauchy-Riemann equations.

is called *analytic.* An analytic function is differentiable with respect to .

## Deriving the Cauchy-Riemann equations

Formally we may write the derivative of with respect to using the classical definition of the derivative

Here, of course, we recognize that the theory of limits of complex valued sequences and series can be established.

If we write in terms of its real and imaginary parts, the formal representation of the derivative becomes

where

For the derivative to exist, the limit must exist and be unique. Therefore, it should not matter whether we take the first or then second, or vice versa. We must obtain an equivalent result either way.

If we take the first, then the resulting limiting process is

Recognizing the partial derivative with respect to we have

Alternatively, we may take the and the resulting limit becomes

Thus, we have found two ways to represent .

Because both of these expressions for must be equivalent, we equate the real and imaginary parts of these expressions to obtain

These two equations are known as the Cauchy-Riemann equations. Any complex valued function that satisfies the Cauchy-Riemann equations is called *analytic*.

Analytic functions are not rare. Most, if not all of the functions encountered applied mathematics likely satisfy
satisfy the Cauchy-Riemann equations except possibly at isolated points. These isolated points are *poles,* *branch points,*
and *essential singularities.*

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