Cauchy-Riemann Equations

Author

Eric W. Weisstein
September 23, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/ComplexAnalysis/Cauchy-RiemannEquations.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Cauchy-RiemannEquations.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Partial Derivative Identities

z==(x^2)

Expand[(x+I y)^2]

u[x_,y_]:=x^2-y^2
v[x_,y_]:=2x y

{D[#,{y,2}],D[#,{x,2}]}&/@{u[x,y],v[x,y]}

z==cos(x+I y)

ComplexExpand[Cos[x+I y]]

u[x_,y_]:=Cos[x] Cosh[y]
v[x_,y_]:=-Sin[x] Sinh[y]

{D[#,{y,2}],D[#,{x,2}]}&/@{u[x,y],v[x,y]}