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.

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 17, 1972.

Arfken, G. "Cauchy-Riemann Conditions." §6.2 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 360-365, 1985.

Knopp, K. "The Cauchy-Riemann Differential Equations." §7 in Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, pp. 28-31, 1996.

Krantz, S. G. "The Cauchy-Riemann Equations." §1.3.2 in Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 13, 1999.

Levinson, N. and Redheffer, R. M. Complex Variables. San Francisco, CA: Holden-Day, 1970.

Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 137, 1997.