- #1
Darth Frodo
- 212
- 1
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.
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.