Question

Every bounded entire function is constant. This theorem is known as:

• A. Morera's theorem
• B. Liouville's theorem
• C. Cauchy's theorem
• D. Cauchy's integral formula

Hint:

Liouville's Theorem is crucial for proving the Fundamental Theorem of Algebra (every non-constant polynomial has at least one complex root).

Answer 1

The Correct Option is B

Step 1: Concept
This question tests fundamental theorems of complex analysis regarding entire functions.

Step 2: Key Formulas and Approach
Recall the main classical theorems in complex function theory:
1. Liouville's Theorem: If $f: \mathbb{C} \to \mathbb{C}$ is entire and bounded (there exists $M > 0$ such that $|f(z)| \leq M$ for all $z \in \mathbb{C}$), then $f(z)$ is constant.
2. Morera's Theorem: Converse of Cauchy's Theorem; if $f$ is continuous and $\oint_C f(z)dz = 0$ for all closed contours, $f$ is analytic.
3. Cauchy's Theorem: If $f$ is analytic in a simply connected domain, $\oint_C f(z)dz = 0$.
4. Cauchy's Integral Formula: Expresses values of an analytic function inside a domain in terms of boundary integrals.

Step 3: Step-by-step Explanation

• Proof sketch of Liouville's Theorem: By Cauchy's Estimate for derivatives, if $|f(z)| \leq M$ on $\mathbb{C}$, then for any $z_0 \in \mathbb{C}$ and circle $C_R$ of radius $R$ centered at $z_0$: \[ |f'(z_0)| \leq \frac{M}{R} \]
• Taking the limit as $R \to \infty$: \[ |f'(z_0)| \leq \lim_{R \to \infty} \frac{M}{R} = 0 \]
• Hence, $f'(z_0) = 0$ for all $z_0 \in \mathbb{C}$.
• Since $f'(z) = 0$ everywhere on connected domain $\mathbb{C}$, $f(z)$ must be constant.
• This statement is precisely Liouville's Theorem.

Step 4: Final Answer
The assertion "Every bounded entire function is constant" is known as Liouville's theorem. Thus, Option (B) is correct.