Cauchy-Riemann Equations
المؤلف:
Abramowitz, M. and Stegun, I. A.
المصدر:
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover
الجزء والصفحة:
...
18-10-2018
2094
Cauchy-Riemann Equations
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.
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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