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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.

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# Complex Analysis: Cauchy Riemann Equations 2

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