Problem 13 - Complex Analysis

Question

Determine an entire function \(f: \mathbb{C} \rightarrow \mathbb{C}\) with $$ z^{2} f^{\prime \prime}(z)+z f^{\prime}(z)+z^{2} f(z)=0 \quad \text { for all } z \in \mathbb{C} $$ Result: One solution is the BESSEL function of order 0 , $$ f(z):=\mathcal{J}_{0}(z):=1+\sum_{n=1}^{\infty} \frac{(-1)^{n}}{(2 \cdot 4 \cdot 6 \cdots 2 n)^{2}} z^{2 n} $$

Step-by-Step Solution

Answer

The solution is \(f(z) = \mathcal{J}_0(z)\), the Bessel function of order zero.

Step 1: Analyzing the Differential Equation

We have the differential equation \( z^{2} f''(z) + z f'(z) + z^{2} f(z) = 0 \). This is a form of Bessel's equation of order zero. The general form of a Bessel equation is \( x^{2} y'' + x y' + (x^2 - u^2) y = 0 \) for order \( u \). For our equation, \( u = 0 \).

Step 2: Identifying the Bessel Function Solution

The Bessel function of the first kind of order zero, \( \mathcal{J}_0(z) \), satisfies the Bessel differential equation when \( u = 0 \). Thus, our solution to the equation \( z^{2} f''(z) + z f'(z) + z^{2} f(z) = 0 \) is \( f(z) = \mathcal{J}_0(z) \).

Step 3: Verifying the Bessel Series

The series definition for the Bessel function of order zero is \( \mathcal{J}_{0}(z) = 1 + \sum_{n=1}^{\infty} \frac{(-1)^{n}}{(2 \cdot 4 \cdot 6 \cdots 2n)^{2}} z^{2n} \). This series satisfies the initial condition \( \mathcal{J}_0(0) = 1 \) and the Bessel equation. The series terms are derived from the standard form of the Bessel function series expansion.

Key Concepts

Entire Function

An entire function is a function that is complex differentiable at every point in the complex plane. This is a very important concept in complex analysis.
The idea of an entire function extends the idea of a function being analytic everywhere. In simple terms, there are no 'gaps' or 'holes' where the function doesn't work.
Entire functions include well-known functions like the exponential function, sine, and cosine. Bessel functions of the first kind, like the one in our exercise, are also entire functions.
They are defined by a power series expansion that converges for every complex number, ensuring that they are complex differentiable everywhere on the complex plane.

Differential Equations

Differential equations involve unknown functions and their derivatives. In mathematics, they are utilized to describe phenomena that change.
The original exercise presents a form of Bessel's differential equation, an example of a second-order linear differential equation.
These equations are crucial because they model physical systems and processes such as sound waves, heat conduction, and quantum mechanics. In our case, the presented Bessel differential equation is solved by the Bessel function of the first kind, flattening the mathematical description into a soluble form.

• Order of the equation: This tells us the highest derivative present in the equation. Our equation is of the second order.
• Linear differential equation: It means no products of the function and its derivatives occur in the equation.

The role of differential equations is indispensable in framing complex real-world circumstances into solvable mathematical representations.

Series Expansion

Series expansion allows us to express functions as a sum of a sequence of terms.
For Bessel functions, the series expansion is invaluable because it provides a way to represent these functions using simpler polynomial terms.

• The series expansion for the Bessel function of order zero is \[\mathcal{J}_0(z) = 1 + \sum_{n=1}^{\infty} \frac{(-1)^{n}}{(2 \cdot 4 \cdot 6 \cdots 2n)^{2}} z^{2n}\]
• This allows computations with these functions to be more manageable, as each term in the series can be computed individually.

The series expansion satisfies both the differential equation and the initial conditions of the problem. It offers a practical approach to working with functions that may not have simple closed-form expressions.