Question

Derive the series expansion of the Bessel function of the first kind from that of the confluent hypergeometric series and the expansion of the exponential. Check your answer by obtaining the same result by substituting the power series directly in the Bessel DE.

Short answer

The Bessel function of the 1st kind can be expressed as power series by using the definitions of the hypergeometric and exponential series. The verification involves substituting the obtained series into Bessel DE and checking if it satisfies this equation.

Step-by-step solution

Definition of the Confluent Hypergeometric and the Exponential Series

By definition, the exponential series expansion is given by \(e^z = \sum_{n=0}^{\infty} \frac{z^n}{n!}\) and the confluent hypergeometric series is given by \(_1F_1(a; c; z) = \sum_{n = 0}^{\infty} \frac{(a)_n}{(c)_n n!} z^n\), where (a)_n denotes the Pochhammer symbol, the rising factorial.

Obtaining the Series Expansion of Bessel Function from that of the Confluent Hypergeometric Series and Exponential

The Bessel function of the first kind, \(J_\alpha (x)\), is defined in terms of the confluent hypergeometric series as \(J_\alpha (x) = \frac{1}{\pi} \Gamma(\alpha + 1)\ _1F_1(-\alpha; 2\alpha + 2; -x^2/4)\). Knowing this identity, one can substitute the series definitions from Step 1 into this equation. By doing this, the Bessel function is expanded in terms of known series.

Verification by Substituting in Bessel DE

After obtaining the power series for the Bessel function in terms of the confluent hypergeometric and exponential series, it's now time to verify it by substituting it into Bessel's DE. If this expansion satisfies Bessel's DE, then it is indeed a solution, thus verifying the derivation.

Key concepts

Confluent Hypergeometric Series

The confluent hypergeometric series, denoted as \(_1F_1(a; c; z)\), is a function that generalizes the hypergeometric series by merging two of its parameters into one, hence the term "confluent." It arises in many areas of mathematical physics and has applications in solving differential equations like the Bessel differential equation. The series form is given by:

• \(_1F_1(a; c; z) = \sum_{n = 0}^{\infty} \frac{(a)_n}{(c)_n n!} z^n\)

Here, \(a\) and \(c\) are parameters, \(z\) is the variable, and \((a)_n\) denotes the rising factorial, also known as the Pochhammer symbol. This series converges for all finite \(z\) and is a powerful tool for finding solutions that are otherwise hard to express. Particularly, in the context of Bessel functions, it assists in expanding functions in a more approachable form for computations.

Power Series Expansion

A power series expansion is a way of expressing a function as an infinite sum of terms, calculated from the values of its derivatives at a point. It’s a core concept in calculus and analysis and provides vital tools for approximating functions. The general form of a power series can be written as:

• \((x) = \sum_{n=0}^{\infty} a_n (x - c)^n\)

where \(a_n\) are coefficients and \(c\) is the center of expansion. By substituting in the Bessel differential equation, a power series provides a systematic method to derive solutions. This method is vital for understanding the behavior and properties of special functions in mathematical physics, such as the exponential function and the Bessel function of the first kind.

Bessel's Differential Equation

Bessel's differential equation is a second-order linear ordinary differential equation crucial in various scientific fields like physics and engineering. It is often written in the form:

• \(x^2 y'' + x y' + (x^2 - \alpha^2) y = 0\)

where \( y = J_\alpha (x) \) are Bessel functions of the first kind. These functions describe the wave propagation in cylindrical or spherical systems and appear in problems such as heat conduction, vibrations, and acoustics. Verifying a solution to this equation, like expanding a Bessel function in terms of a power series, is crucial for ensuring the solution's validity. Solving Bessel's differential equation helps us explore how these functions behave under various conditions.

Rising Factorial

The rising factorial, also known as the Pochhammer symbol \( (a)_n \), is an important notation used in the study of special functions. It is defined as a product of \(n\) consecutive integers starting from \(a\):

• \( (a)_n = a (a+1) (a+2) \cdots (a+n-1) \).

This representation arises naturally in combinatorics, calculus, and fractional calculus. It simplifies the notation and manipulation within series expansions, particularly in hypergeometric series and thus in the computation of Bessel functions. The rising factorial enables the compact representation of terms in sequences and series, making analytical work more manageable.