suppose that f and g are continuous functions on a closed bounded region R and let D be the interior of R if f and g are analytic throughout D with f(z)=g(z) for all z in the boundary of R
prove that :
f(z)=g(z) for all z in R

2. Let C the line that limits the region D. If f(*) is analytic inside and on C and a a point internal to C is...

$f(a)= \frac{1}{2\pi i} \oint_{C} \frac{f(z)}{z-a} dz$ (1)

If f(*) and g(*) are both analytic and on C is f(z)=g(z), then for the (1) the same holds in any point of D...

Kind regards

$\chi$ $\sigma$