Cauchy-Riemann Equations

Let

(1)

where

(2)

so

(3)

The total derivative of with respect to is then

			(4)
			(5)

In terms of and , (5) becomes

			(6)
			(7)

Along the real, or x-axis, , so

(8)

Along the imaginary, or y-axis, , so

(9)

If is complex differentiable, then the value of the derivative must be the same for a given , regardless of its orientation. Therefore, (8) must equal (9), which requires that

(10)

and

(11)

These are known as the Cauchy-Riemann equations.

They lead to the conditions

			(12)
			(13)

The Cauchy-Riemann equations may be concisely written as

			(14)
			(15)
			(16)
			(17)

where is the complex conjugate.

If , then the Cauchy-Riemann equations become

			(18)
			(19)

(Abramowitz and Stegun 1972, p. 17).

If and satisfy the Cauchy-Riemann equations, they also satisfy Laplace's equation in two dimensions, since

(20)

(21)

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