GATE 2024 Mathematics Solutions | Q11: Entire Functions and Conditions

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🔥 GATE 2024 Mathematics | Entire Functions and Analytic Polynomials Problem | Complete Solution 🔥

In this video, we solve an advanced complex analysis problem based on entire functions and their properties. The problem requires analyzing a condition ∣f(z)∣=1∣f(z)∣=1 for all z∈Cz∈C such that Im(z)=0Im(z)=0. This condition (denoted as (P)(P)) is applied to determine which of the following statements about ff is correct:

Options:
(A) There is a non-constant analytic polynomial ff satisfying (P)(P).
(B) Every entire function ff satisfying (P)(P) is a constant function.
(C) Every entire function ff satisfying (P)(P) has no zeroes in CC.
(D) There is an entire function ff satisfying (P)(P) with infinitely many zeroes in CC.

👉 Problem Objective:
Determine which statement about ff satisfying the given condition (P)(P) is correct.
🔍 What You’ll Learn in This Video:
1️⃣ The role of the modulus condition ∣f(z)∣=1∣f(z)∣=1 in complex analysis.
2️⃣ Application of Liouville's Theorem, Maximum Modulus Principle, and properties of entire functions.
3️⃣ How to analyze and classify functions based on their analytic behavior.
4️⃣ Step-by-step reasoning to identify the correct option with logical clarity.

This problem is perfect for strengthening your understanding of complex analysis and entire functions, essential topics for GATE 2024 Mathematics and other competitive exams like CSIR NET, TIFR, NBHM, and JAM.
💡 Why Watch This Video?

Detailed Explanation: Each step is explained with clear reasoning and no shortcuts.
GATE-Focused Content: Tailored for students preparing for GATE Mathematics 2024.
Conceptual Depth: Dive deep into the properties of entire functions and their real-world implications.

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