Complex Analysis: Cauchy Riemann Equations 2

[Darth Frodo]

Hi All,

I was reading through Kreyzeig's Advanced Engineering Mathematics and came across two theorems in Complex Analysis.

Theorem 1:
Let f(z) = u(x,y) + iv(x,y) be defined and continuous in some neighborhood of a point z = x+iy and differentiable at z itself.
Then, at that point, the first-order partial derivatives of u and v exist and satisfy the Cauchy–Riemann equations.

Theorem 2:
If two real-valued continuous functions and of two real variables x and y have continuous first partial derivatives that satisfy the Cauchy–Riemann equations in some domain D.
Then the complex function is analytic in D.

It seems that the hypothesis of Theorem 1 is similar to the conclusion of Theorem 2. Can these two theorems be modified into one iff statement?

Thanks.

 

[Hawkeye18]

Theorem 1 states that if a function is complex differentiable at one point, then it has partial derivatives at this point, and these partial derivatives satisfy Cauchy--Riemann equations. BTW, continuity of $f$ in a neighborhood of $z$ is not needed in Theorem 1, $f$ only needs to be defined there and to be complex differentiable at the point $z$. The proof is very easy, just a trivial computation.

Theorem 2 says that if partial derivatives exist and continuous in a domain $D$, then the function is analytic in this domain.

Of course, one can say that a function is analytic in a domain if and only if the partial derivatives exist, are continuous in the domain and satisfy the Cauchy--Riemann equations there; this is a true theorem. But many authors like to emphasize that complex differentiability at one point implies the Cauchy--Riemann equations at this point.

 

[WWGD]

I think the standard example of differentiable but not analytic is $f(z)=|z| ; |z|:= (x^2+ y^2)^{1/2} ; z=x+iy$