Re: Complex analysis question.-
On 30-06-2005 14:15, koundinya.k@xxxxxxxxx wrote:
"Prove that if u(x,y) and v(x,y) have
continuous first order parital derivatives which satisfy the CR
equations in a region D, then f(z)=u(z) + iv(z) is analytic with a
continuous derivative f'z and conversly".
How do I go about proving this ?
Proof of => : Since the partial derivatives are continuous functions,
f is differentiable when seen as a function from an open subset of R^2
into R^2. Since, furthermore, the Cauchy-Riemann equations are
satisfied, f is differentiable as a complex function. The function f'
is continuous since its components (du/dx and dv/dx) are continuous.
Proof of <= : Whenever a function f = u + iv is analytic, u and v
satisfy the Cauchy-Riemann equations. Saying that f' is continuous is
the same thing as saying that du/dx and dv/dx are continuous. Since
du/dy = -dv/dx and dv/dy = du/dx, it follows that the partial
derivatives are continuous.
Best regards,
Jose Carlos Santos
.
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