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?

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?

cdquestions Admin · Accepted Answer

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 $$ \boxed{C} \quad \text{both S1 and S2 are TRUE} $$

Answer

S1 is TRUE and S2 is FALSE

Answer

S2 is TRUE and S1 is FALSE

Answer

both S1 and S2 are TRUE

Answer

neither S1 nor S2 is TRUE

Show Hint

The Correct Option is C

Solution and Explanation

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x	0	1	2	3	4
y(x)	1	3	6	9	12