Complex Analysis by Muir Jerry R.;

Author:Muir, Jerry R.;
Language: eng
Format: epub, pdf
Publisher: John Wiley & Sons, Incorporated
Published: 2014-11-14T00:00:00+00:00

Proof.

If , we have established that the Cauchy–Riemann equations hold for u and v. Since is continuous, it follows from (3.6.1 ) and (3.6.2 ) that u and v have continuous partial derivatives and hence are differentiable.

For the converse, let . Using (3.6.4 ), the Cauchy–Riemann equations, and writing , we have

Then for ,

Applying (3.6.5 ) (using moduli), we have

Therefore exists for all , and f is analytic by Goursat's theorem.

3.6.3 Example.

Consider the function given by , where . Note that and have continuous partial derivatives and hence are differentiable. From the above work, it is clear that f has a complex derivative for each at which the Cauchy–Riemann equations hold. Now (3.6.3 ) becomes

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