Question

If a function is analytic in a domain, then it is necessarily:

• A. Continuous only
• B. Differentiable only once
• C. Infinitely differentiable
• D. Bounded

Hint:

In complex analysis, remember: {Analytic $\Rightarrow$ infinitely differentiable $\Rightarrow$ power series expansion}.

Answer 1

The Correct Option is C

To understand why an analytic function in a domain is necessarily infinitely differentiable, let's break down the concept of analytic functions and their properties.

1. Definition of Analytic Function: An analytic function is a function that is locally given by a convergent power series. This means in some neighborhood of every point in its domain, the function can be expressed as a sum of its Taylor series.
2. Infinitely Differentiable: A key property of analytic functions is that they are not only differentiable but differentiable an infinite number of times. This arises from the fact that the Taylor series representation of the function converges to the function itself. Each term in the Taylor series represents a derivative of the function at a point, and since the series is convergent, the number of derivatives is infinite.
3. Comparing Options:
   
   • Continuous only: While analytic functions are continuous, they have stronger properties such as differentiability, which makes this option insufficient.
   • Differentiable only once: Analytic functions are differentiable infinitely, not just once.
   • Infinitely differentiable: This is correct because of the convergence of the Taylor series representation, making analytic functions differentiable any number of times.
   • Bounded: Analytic functions are not necessarily bounded; they can grow without bound in certain domains, such as the exponential function.
4. Conclusion: Given the properties of analytic functions, especially their nature of being locally expressible as a power series, they are infinitely differentiable. Thus, the correct answer is that a function analytic in a domain is necessarily Infinitely differentiable.