Question

Show that if u(x,y) and v(x,y) each have equal mixed second partials, and if u and v satisfy the Cauchy -Riemann equations, then u, v, and u+v satisfy Laplace's equation.

Answer 1

the Cauchy-Riemann eq:
du/dx=dv/dy and du/dy=-dv/dx

Answer 2

Laplace 's eq:
d^2z/dx^2 + d^2z/dy^2=0

Answer 3

@zepdrix help?

Answer 4

help?

Answer 5

@tiffanymak1996 what is mean't by that u and v have equal mixed second partials? Is it just implying that u(x,y) and v(x,y) have the same second partial derivative with respect to x or y?

Answer 6

so ill show you an example for u :
using cauchy eq:
du/dx = dv/dy
take derivative with respect to x :
A. d^2u/dx^2 = d^2v/(dxdy)
using cauchy second eq:
du/dy = -dv/dx
take derivative with respect to y:
d^2u/dy^2 = -d^2v/(dydx)
now v have equal mixed partials so :
d^2v/(dydx) = d^2v/(dxdy)
so that
B. d^2u/dy^2 = -d^2v/(dxdy)
using A +B we get:
d^2u/dx^2 + d^2u/dy^2 = 0

Answer 7

@Coolsector Thanks.