Teaching Haryana (HPSC) Assistant Professor Mock Test 2025 Mathematical Science Complex Analysis Theorems of Complex analysis
Let ( f ) be a holomorphic function defined on a domain\( D = \{z \in \mathbb{C} : |z - 1| < 2\}\) and ( g ) a holomorphic function defined on a domain \(E = \{z \in \mathbb{C} : |z + 1| < 2\}\) and there exists an overlapping region \(O \subset D \cap E \) where f(z) = g(z) for all ( z ) in an infinite sequence {zn} within O converging to a point ( z0 ) in O. What can be concluded about ( f ) and ( g ) on their respective domains?
1
( f ) and ( g ) are identical on D ∪ E since they are equal on O
2
( f ) and ( g ) are identical throughout O but cannot be assumed to be identical in non-overlapping regions of D and E.
3
( f ) and ( g ) cannot be identical in either D or E unless O equals D ∩ E.
4
( f ) and ( g ) are not identical throughout O
5
Question Not Attempted