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: Concept Overview:
This question assesses knowledge of theorems and properties related to the zeros of analytic functions in complex analysis.

Step 2: Detailed Analysis:
A. The order of a zero of a polynomial equals the order of its first non-vanishing derivative at that zero.
This is accurate. If an analytic function \(f(z)\) has a zero of order \(m\) at \(z_0\), the Taylor series around \(z_0\) starts with \(a_m(z-z_0)^m\), where \(a_m eq 0\). This implies \(f(z_0) = f'(z_0) = ... = f^{(m-1)}(z_0) = 0\) and \(f^{(m)}(z_0) eq 0\). Thus, the zero's order is the first non-vanishing derivative's order. Statement A is correct.
B. Zeros of a non-zero analytic function are isolated.
This is a fundamental property, often related to the Identity Theorem or Uniqueness Principle. For any zero \(z_0\) of a non-zero analytic function \(f\), there's a punctured disk around \(z_0\) where \(f\) isn't zero. Statement B is correct.
C. Zeros of f(z) are found by setting the numerator to zero if the numerator and denominator of f(z) share no common factors.
This applies to rational functions \(f(z) = P(z)/Q(z)\). A zero of \(f(z)\) occurs where \(f(z_0)=0\), which requires \(P(z_0)=0\) and \(Q(z_0) eq 0\). A common factor \((z-z_0)\) would create 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 zero set of an analytic function \(f\) has a limit point within the domain of analyticity, then \(f\) is identically zero. Therefore, a non-zero analytic function cannot have a limit point of zeros in its domain. Statement D is incorrect.

Step 3: Conclusion:
Statements A, B, and C are correct. Therefore, the answer is (B).