Question

Let \( \Omega \) be a non-empty open connected subset of \( \mathbb{C} \) and \( f: \Omega \to \mathbb{C} \) be a non-constant function. Let the functions \( f^2: \Omega \to \mathbb{C} \) and \( f^3: \Omega \to \mathbb{C} \) be defined by \[ f^2(z) = (f(z))^2 \quad {and} \quad f^3(z) = (f(z))^3, \quad z \in \Omega. \] 
Consider the following two statements: 
S1: If \( f \) is continuous in \( \Omega \) and \( f^2 \) is analytic in \( \Omega \), then \( f \) is analytic in \( \Omega \). 
S2: If \( f^2 \) and \( f^3 \) are analytic in \( \Omega \), then \( f \) is analytic in \( \Omega \). Then, which one of the following is correct?

• A. S1 is TRUE and S2 is FALSE
• B. S2 is TRUE and S1 is FALSE
• C. both S1 and S2 are TRUE
• D. neither S1 nor S2 is TRUE

Hint:

For functions that are powers of analytic functions, if the powers are analytic, the original function must also be analytic. This is a fundamental result in complex analysis.

Answer 1

The Correct Option is C

Step 1: Analyzing Statement S1
Statement S1 asserts that if \( f \) is continuous in \( \Omega \) and \( f^2 \) is analytic in \( \Omega \), then \( f \) is analytic in \( \Omega \). This is TRUE. If \( f^2 \) is analytic in \( \Omega \), then \( f \) must be analytic in \( \Omega \). The reason is that the square root of an analytic function (when the function is non-zero) is also analytic. This can be proven using complex function theory and the fact that the derivative of \( f^2 \) leads to an analytic function for \( f \).

Step 2: Analyzing Statement S2
Statement S2 asserts that if \( f^2 \) and \( f^3 \) are analytic in \( \Omega \), then \( f \) is analytic in \( \Omega \). This is also TRUE. If both \( f^2 \) and \( f^3 \) are analytic, then \( f \) must be analytic. This is a consequence of the fact that the powers of an analytic function are also analytic, and if both \( f^2 \) and \( f^3 \) are analytic, this forces \( f \) to be analytic due to the uniqueness of analytic functions and the fact that the operations preserve analyticity.

Step 3: Conclusion
Since both statements S1 and S2 are TRUE, the correct answer is \( \boxed{C} \).

Final Answer
\[ \boxed{C} \quad \text{both S1 and S2 are TRUE} \]