Let
 |
(1) |
where
 |
(2) |
so
 |
(3) |
The total derivative of 
with respect to 
is then
In terms of 
and 
,
(5) becomes
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
The Cauchy-Riemann equations may be concisely written as
where 
is the complex conjugate.
If 
,
then the Cauchy-Riemann equations become
(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.
See also
Analytic Function, Anti-Analytic Function, Cauchy Integral Theorem, Complex Derivative, Conformal Mapping, Entire Function, Monogenic Function, Polygenic Function
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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.
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Cite this as:
Weisstein, Eric W. "Cauchy-Riemann Equations." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Cauchy-RiemannEquations.html