Question

Consider the following: Let f(z) be a complex valued function defined on a subset \( S \subset \mathbb{C} \) of complex numbers. Then which of the following are correct?
A. The order of a zero of a polynomial equals to the order of its first non-vanishing derivative at that zero of the polynomial
B. Zeros of non-zero analytic function are isolated
C. Zeros of f(z) are obtained by equating the numerator to zero if there is no common factor in the numerator and the denominator of f(z)
D. Limit points of zeros of an analytic function is an isolated essential singularity

• A. A, B and D only
• B. A, B and C only
• C. A, B, C and D
• D. B, C and D only

Hint:

The Identity Theorem is a cornerstone of complex analysis. A key takeaway is that information about an analytic function on a very small set (like a sequence of points converging to a limit point) determines the function's behavior everywhere in its domain. This leads to the principle that zeros must be isolated.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This question tests key theorems and properties concerning the zeros of analytic functions in complex analysis.

Step 2: Detailed Explanation:
A. The order of a zero of a polynomial equals to the order of its first non-vanishing derivative at that zero of the polynomial.
This is correct. If an analytic function \(f(z)\) has a zero of order \(m\) at \(z_0\), its Taylor series expansion around \(z_0\) starts with the term \(a_m(z-z_0)^m\), where \(a_m \neq 0\). This implies that \(f(z_0) = f'(z_0) = ... = f^{(m-1)}(z_0) = 0\) and \(f^{(m)}(z_0) \neq 0\). So, the order of the zero is indeed the order of the first non-vanishing derivative. Statement A is correct.
B. Zeros of non-zero analytic function are isolated.
This is a fundamental property of analytic functions, often called the Identity Theorem or Uniqueness Principle. It states that for any zero \(z_0\) of a non-zero analytic function \(f\), there exists a punctured disk centered at \(z_0\) where \(f\) is non-zero. Statement B is correct.
C. Zeros of f(z) are obtained by equating the numerator to zero if there is no common factor in the numerator and the denominator of f(z).
This statement applies to a rational function \(f(z) = P(z)/Q(z)\). A zero of \(f(z)\) occurs at a point \(z_0\) where \(f(z_0)=0\). This requires \(P(z_0)=0\) and \(Q(z_0) \neq 0\). If there were a common factor \((z-z_0)\), it would be a removable singularity, not a zero. Statement C is correct.
D. Limit points of zeros of an analytic function is an isolated essential singularity.
This statement is incorrect. By the Identity Theorem, if the set of zeros of an analytic function \(f\) has a limit point within the domain of analyticity, then \(f\) must be identically zero on the entire domain. Therefore, a non-zero analytic function cannot have a limit point of zeros in its domain. Statement D is incorrect.

Step 3: Final Answer:
Statements A, B, and C are correct. Therefore, the correct option is (B).