Problem 14 - Complex Analysis

Question

Let the BESSEL function of order \(m, m \in \mathbb{N}_{0}\), be defined by the formula $$ \mathcal{J}_{m}(z)=\sum_{n=0}^{\infty} \frac{(-1)^{n}(z / 2)^{2 n+m}}{n !(m+n) !} $$ Show: Each \(\mathcal{J}_{m}\) is an entire function.

Step-by-Step Solution

Answer

Each \( \mathcal{J}_m(z) \) is entire because its series converges for all complex \( z \).

Step 1: Understanding the Function

The Bessel function of order \( m \), denoted as \( \mathcal{J}_m(z) \), is given by the series \( \mathcal{J}_{m}(z)=\sum_{n=0}^{\infty} \frac{(-1)^{n}(z / 2)^{2 n+m}}{n !(m+n) !} \). This is an infinite series involving powers of \( z \).

Step 2: Definition of Entire Function

A function is entire if it is complex differentiable at every point in the complex plane, which means it is analytic everywhere. This typically involves showing that the radius of convergence of the power series for the function is infinite.

Step 3: Checking Convergence

To show that \( \mathcal{J}_{m}(z) \) is entire, we need to examine the convergence of the series. We use the ratio test: for the series \( \sum a_n \), if \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L < 1 \), the series converges absolutely. Here, \( a_n = \frac{(-1)^n (z/2)^{2n+m}}{n! (m+n)!} \).

Step 4: Applying the Ratio Test

Calculate:\[\left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{(-1)^{n+1} (z/2)^{2n+2+m}}{(n+1)! (m+n+1)!} \cdot \frac{n! (m+n)!}{(-1)^n (z/2)^{2n+m}} \right| = \frac{(z/2)^2}{(n+1)(m+n+1)}\]Since \( \left| \frac{z^2}{4(n+1)(m+n+1)} \right| \to 0 \) as \( n \to \infty \) for any \( z \), the series converges for all \( z \).

Step 5: Conclusion of Entirety

The series for \( \mathcal{J}_m(z) \) converges for all complex \( z \) because the ratio test shows it convergences absolutely for all \( z \). Thus, \( \mathcal{J}_m(z) \) is an entire function since it is analytic everywhere in the complex plane.

Key Concepts

Entire Function

An entire function is a special type of complex function. It is defined as a function that is complex differentiable at every single point in the entire complex plane.
This is a significant property because it means the function is also analytic everywhere in the complex plane. In simpler terms, there are no "holes" or interruptions in its domain of definition.
The Bessel function of order \( m \), \( \mathcal{J}_m(z) \), meets this criterion.

• The reason is that, when represented as a power series, it converges everywhere in the complex plane.
• This infinite convergence ensures the function remains smooth and differentiable anywhere you might plot it in the complex domain.

Such functions are important in many fields including physics and engineering, often used to solve differential equations in circular or cylindrical domains.

Complex Differentiable

Complex differentiability is a crucial part of defining entire functions. A function is complex differentiable at a point if it satisfies certain conditions with respect to the complex number plane.
This involves differentiating the function in a way that is similar to real functions, but considering both real and imaginary components.

• If a function is complex differentiable at every point on the complex plane, it is not just complex differentiable, but also an entire function.
• For the Bessel function \( \mathcal{J}_m(z) \), due to its infinite series representation, it follows that the function is smooth and has derivatives of all orders for all complex numbers.

This smoothness and infinite differentiability are what make the Bessel function an entire function.

Absolute Convergence

Absolute convergence is a key concept to understand when working with infinite series like the Bessel function.
When a series converges absolutely, you can rearrange its terms without changing the sum.

• In the context of \( \mathcal{J}_m(z) \), absolute convergence is shown using the ratio test.
• This ensures the power series converges for every possible complex number.

The importance here is that absolute convergence of \( \mathcal{J}_m(z) \) leads to its property of being an entire function. Because of this property, the function is defined not just at select points, but across the entire complex plane, a hallmark of analytical prowess in mathematical functions.

Ratio Test

The ratio test is a tool used to determine the convergence of an infinite series.
It's pivotal in proving the entirety of a function like \( \mathcal{J}_m(z) \). The test involves examining the limit:\[\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \]

• If this limit is less than 1, the series is absolutely convergent, making it converge for all inputs (in this case, all values of \( z \)).
• For the Bessel function, the ratio test shows that is it convergent for any complex number \( z \) because the limit approaches 0 as \( n \) goes to infinity.

This guarantees that the series representation of the function remains defined and differentiable across the entire complex plane, reinforcing that \( \mathcal{J}_m(z) \) is an entire function.