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Cauchy-Riemann Equations


Let

 f(x,y)=u(x,y)+iv(x,y),
(1)

where

 z=x+iy,
(2)

so

 dz=dx+idy.
(3)

The total derivative of f with respect to z is then

(df)/(dz)=(partialf)/(partialx)(partialx)/(partialz)+(partialf)/(partialy)(partialy)/(partialz)
(4)
=1/2((partialf)/(partialx)-i(partialf)/(partialy)).
(5)

In terms of u and v, (5) becomes

(df)/(dz)=1/2[((partialu)/(partialx)+i(partialv)/(partialx))-i((partialu)/(partialy)+i(partialv)/(partialy))]
(6)
=1/2[((partialu)/(partialx)+i(partialv)/(partialx))+(-i(partialu)/(partialy)+(partialv)/(partialy))].
(7)

Along the real, or x-axis, partialf/partialy=0, so

 (df)/(dz)=1/2((partialu)/(partialx)+i(partialv)/(partialx)).
(8)

Along the imaginary, or y-axis, partialf/partialx=0, so

 (df)/(dz)=1/2(-i(partialu)/(partialy)+(partialv)/(partialy)).
(9)

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

 (partialu)/(partialx)=(partialv)/(partialy)
(10)

and

 (partialv)/(partialx)=-(partialu)/(partialy).
(11)

These are known as the Cauchy-Riemann equations.

They lead to the conditions

(partial^2u)/(partialx^2)=-(partial^2u)/(partialy^2)
(12)
(partial^2v)/(partialx^2)=-(partial^2v)/(partialy^2).
(13)

The Cauchy-Riemann equations may be concisely written as

(df)/(dz^_)=1/2[(partialf)/(partialx)+i(partialf)/(partialy)]
(14)
=1/2[((partialu)/(partialx)+i(partialv)/(partialx))+i((partialu)/(partialy)+i(partialv)/(partialy))]
(15)
=1/2[((partialu)/(partialx)-(partialv)/(partialy))+i((partialu)/(partialy)+(partialv)/(partialx))]
(16)
=0,
(17)

where z^_ is the complex conjugate.

If z=re^(itheta), then the Cauchy-Riemann equations become

(partialu)/(partialr)=1/r(partialv)/(partialtheta)
(18)
1/r(partialu)/(partialtheta)=-(partialv)/(partialr)
(19)

(Abramowitz and Stegun 1972, p. 17).

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

 (partial^2u)/(partialx^2)+(partial^2u)/(partialy^2)=partial/(partialx)((partialv)/(partialy))+partial/(partialy)(-(partialv)/(partialx))=0
(20)
 (partial^2v)/(partialx^2)+(partial^2v)/(partialy^2)=partial/(partialx)(-(partialu)/(partialy))+partial/(partialy)((partialu)/(partialx))=0.
(21)

By picking an arbitrary f(z), 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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Cauchy-Riemann Equations

Cite this as:

Weisstein, Eric W. "Cauchy-Riemann Equations." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Cauchy-RiemannEquations.html

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