Cauchy-Riemann equations

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

References

  1. Spiegel, Murray R. "Theory and problems of complex variables, with an introduction to Conformal Mapping and its applications." Schaum's outline series (1964).
  2. Levinson, Norman, and Raymond M. Redheffer. "Complex variables." (1970), Holden-Day, New York.