## Cauchy-Riemann Equations

Let

 (1)

where
 (2)

so
 (3)

The total derivative of with respect to may then be computed as follows.
 (4) (5)

so
 (6) (7)

and
 (8)

In terms of and , (8) becomes
 (9)

Along the real, or x-Axis, , so
 (10)

Along the imaginary, or -axis, , so
 (11)

If is Complex Differentiable, then the value of the derivative must be the same for a given , regardless of its orientation. Therefore, (10) must equal (11), which requires that
 (12)

and
 (13)

These are known as the Cauchy-Riemann equations. They lead to the condition
 (14)

The Cauchy-Riemann equations may be concisely written as
 (15)

In Polar Coordinates,
 (16)

so the Cauchy-Riemann equations become
 (17) (18)

If and satisfy the Cauchy-Riemann equations, they also satisfy Laplace's Equation in 2-D, since
 (19)

 (20)

By picking an arbitrary , solutions can be found which automatically satisfy the Cauchy-Riemann equations and Laplace's Equation. This fact is used to find so-called Conformal Solutions to physical problems involving scalar potentials such as fluid flow and electrostatics.

Arfken, G. Cauchy-Riemann Conditions.'' §6.2 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 3560-365, 1985.