Cauchy Riemann Conditions Question

[Ed Quanta]

Ok, I am told in a complex analysis book that the gradient squared of u is equal to the gradient squared of v which is equal to 0.

We know the derivate of w exists, and w(z)=u(x,y) + iv(x,y)

Thus the Cauchy Riemann conditions must hold. (When I use d assume that it refers to a partial derivative)

So du/dx=dv/dy and du/dy=-dv/dx by Cauchy Riemann

We know the gradient squared of u is equal to d^2u/dx^2 + d^2u/dy^2 is equal to d^2v/dy^2 - d^2v/dx^2

We know the gradient squared of v is equal to d^2v/dx^2 + d^2v/dy^2 which is equal to d^2u/dx^2 - d^2u/dy^2

Am I correct in assuming this? And if I am, I don't see how the gradient squared of u is equal to the gradient squared of v. Any suggestions?

 

[HallsofIvy]

Cauchy-Riemann conditions:
(a) du/dx= dv/dy and (b) du/dy= - dv/dx

Differentiate both sides of (a) with respect to x:
d²/dx²= d²v/dydx

Differentiate both sides of (b) with respect to y:
d²u/dy²= -d²v/dydx

Adding: d²/dx²+ d²u/dy²= d²v/dydx -d²v/dydx= 0.

Differentiate (a) with respect to y and (b) with respect to x and subtract to get the formula for v.

The "Laplacian" (what you are calling "gradient squared") of u and v are not just equal, they are both 0. The real and imaginary parts of an analytic function are always "harmonic functions", i.e. they satisfy
$\Delta u=0$ and $\Delta v= 0$.

 

[M98Ranger]

Yes, you are correct in assuming that the Cauchy Riemann conditions must hold for a complex function to be differentiable. The Cauchy Riemann conditions state that the partial derivatives of the real and imaginary parts of a complex function must satisfy certain equations, which you have correctly stated as du/dx=dv/dy and du/dy=-dv/dx.

However, it is important to note that the gradient squared of u and the gradient squared of v are not necessarily equal. The Cauchy Riemann conditions only guarantee that the partial derivatives of u and v are related in a specific way, but they do not necessarily determine the values of the second order partial derivatives of u and v.

In fact, the equations you have written for the gradient squared of u and v are not necessarily true. They only hold if the second order partial derivatives of u and v are equal, but this is not always the case. So while the Cauchy Riemann conditions are necessary for a function to be differentiable, they are not sufficient to determine the values of the second order partial derivatives.

I hope this helps clarify your understanding of the Cauchy Riemann conditions and their relationship to the gradient squared of u and v.