Let $r = \text{Min}\{\alpha, \beta, \gamma\}$, $R = \text{Max}\{\alpha, \beta, \gamma\}$, $f(z) = \frac{z}{(z-\alpha)(z-\beta)(z-\gamma)}$. $I_1 = \oint_{C_1} f(z)dz$ and $I_2 = \oint_{C_2} f(z)dz$, where $C_1 : |z| < r$ and $C_2 : |z| = R+1$, then $I_1 + I_2 = $}

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?

Let $r = \text{Min}\{\alpha, \beta, \gamma\}$, $R = \text{Max}\{\alpha, \beta, \gamma\}$, $f(z) = \frac{z}{(z-\alpha)(z-\beta)(z-\gamma)}$. $I_1 = \oint_{C_1} f(z)dz$ and $I_2 = \oint_{C_2} f(z)dz$, where $C_1 : |z| < r$ and $C_2 : |z| = R+1$, then $I_1 + I_2 = $}

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The Correct Option is B

Solution and Explanation

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