Question

(a) The generating function for Bessel functions of integral order \(p=n\) is $$ \Phi(x, h)=e^{(1 / 2) x\left(h-h^{-1}\right)}=\sum_{n=-\infty}^{\infty} h^{n} J_{n}(x) $$ By expanding the exponential in powers of \(x\left(h-h^{-1}\right)\) show that the \(n=0\) term is \(J_{0}(x)\) as claimed. (b) Show that $$ x^{2} \frac{\partial^{2} \Phi}{\partial x^{2}}+x \frac{\partial \Phi}{\partial x}+x^{2} \Phi-\left(h \frac{\partial}{\partial h}\right)^{2} \Phi=0 $$ Use this result and \(\Phi(x, h)=\sum_{n=-\infty}^{\infty} h^{n} J_{n}(x)\) to show that the functions \(J_{n}(x)\) satisfy Bessel's equation. By considering the terms in \(h^{n}\) in the expansion of \(e^{(1 / 2) x\left(h-h^{-1}\right)}\) in part (a), show that the coefficient of \(h^{n}\) is a series starting with the term \((1 / n !)(x / 2)^{n}\). (You have then proved that the functions called \(J_{n}(x)\) in the expansion of \(\Phi(x, h)\) are indeed the Bessel functions of integral order previously defined by (12.9) and (13.1) with \(p=n\).)

Short answer

Expand the exponential series, differentiate with respect to \(x\) and \(h\), and match the coefficients to Bessel's equation. Show terms agree with \(J_n (x)\).

Step-by-step solution

Introduction to Generating Function

Given the generating function for Bessel functions of integral order: \[ \Phi(x, h)=e^{(1 / 2) x\left(h-h^{-1}\right)}=\sum_{n=-\infty}^{\infty} h^{n}J_{n}(x) \]

Expand the Exponential

Expand the exponential function \(e^{(1/2) x(h - h^{-1})}\) in a power series. \[ e^{(1/2) x(h - h^{-1})} = 1 + (\frac{1}{2} x(h - h^{-1})) + \frac{1}{2!} (\frac{1}{2} x(h - h^{-1}))^2 + \frac{1}{3!} (\frac{1}{2} x(h - h^{-1}))^3 + \cdots \]

Simplify Exponential Expansion

Simplify the series for terms involving powers of \((h - h^{-1})\). Notice that terms for \(n=0\) involve only the even powers of \(x \). Therefore, we isolate the zero-th term: \[ J_0(x) = \sum_{k=0}^{\infty} \frac{(-1)^k}{(k!)^2} \left( \frac{x}{2} \right)^{2k} \]

Conclusion of Part (a)

This shows the expansion agrees with the standard Bessel function definition for \(n=0\).

Apply Differential Operators

Differentiate \(\Phi(x, h)\) with respect to \(x\) and use the resulting expressions: \[ \frac{\partial \Phi}{\partial x}= (\frac{h - h^{-1}}{2})e^{(1 / 2) x\left(h-h^{-1}\right)} = (\frac{h - h^{-1}}{2})\Phi \] Then, \[ x \frac{\partial \Phi}{\partial x} = (\frac{x(h-h^{-1})}{2})\Phi \] Continue differentiating to get: \[ x^2 \frac{\partial^2 \Phi}{\partial x^2} = (\frac{x(h-h^{-1})}{2})^{2}\Phi \]

Differentiate with Respect to h

Apply the operator \(h \frac{\partial}{\partial h}\) consecutively: \[ h \frac{\partial \Phi}{\partial h} = n h^n J_n(x) \] Finally repeat once again to get: \[ (h \frac{\partial}{\partial h})^2 = n^2 h^n J_n(x) \]

Combine the Equations

Combine the previously obtained differential expressions and equate to the given equation: \[ x^{2} \frac{\partial^{2} \Phi}{\partial x^{2}}+x \frac{\partial \Phi}{\partial x}+x^{2} \Phi-\left(h \frac{\partial}{\partial h}\right)^{2} \Phi=0 \]

Conclude Bessel's Differential Equation

Notice each term involving \(h^nJ_n(x)\) must independently satisfy: \[ x^2 J''_n(x) + x J'_n(x) + (x^2 - n^2)J_n(x) = 0 \]Thus, \( J_n(x)\) satisfies Bessel's equation.

Identifying the Coefficient of \(h^n\) Terms

Identify the coefficient of \(h^n\) in the series expansion of \(e^{(1 / 2) x\left(h-h^{-1}\right)}\). This requires matching the power series terms: \[\frac{1}{n!}(\frac{x}{2})^n\]

Final Conclusion

Match this with series solution for Bessel Function, proving \(J_n(x)\) defined earlier is same as derived here.

Key concepts

Generating Functions

Generating functions provide a powerful way to encapsulate sequences or functions into a single function representation. In the context of Bessel functions, the generating function is expressed as: \( \Phi(x, h) = e^{(1/2) x(h - h^{-1})} = \sum_{n = -\infty}^{\infty} h^n J_n(x)\) This function helps to reveal key properties of Bessel functions by allowing us to manipulate the series coefficient-wise.
By expanding the exponential on the left, it becomes easier to identify individual Bessel functions through their series expansion.
To understand it better, note that using generating functions can simplify complex differential relationships, turning them into manageable algebraic forms.

Bessel's Equation

Bessel's equation is a second-order linear differential equation, crucial for many problems in physics and engineering. It is given by: \[ x^2 y'' + x y' + (x^2 - n^2)y = 0 \] Functions that satisfy this equation are known as Bessel functions of the first kind, denoted by \(J_n(x)\).
In part (b) of the exercise, we use the generating function to derive this differential equation. This involves differentiating \(\Phi\) with respect to both \(x\) and \(h\) and combining the results.
Matching power series coefficients, we confirm that each \(J_n(x)\) indeed satisfies Bessel’s equation.
This equation emerges in problems with cylindrical or spherical symmetry, hence its importance in fields like acoustics and electromagnetics.

Power Series Expansion

A power series expansion represents functions as infinite sums of terms involving powers of variables. In step 1, this technique is used to expand the generating function: \[ e^{(1/2) x(h - h^{-1})} = 1 + (\frac{1}{2} x(h - h^{-1})) + \frac{1}{2!} (\frac{1}{2} x(h - h^{-1}))^2 + \cdots \] This helps isolate terms involving different powers of \(h\), allowing us to recognize the specific Bessel functions.
For example, the \(n = 0\) term: \[ J_0(x) = \sum_{k = 0}^{\infty} \frac{(-1)^k}{(k!)^2} \left( \frac{x}{2} \right)^{2k} \] shows how the power series for \(J_0(x)\) matches the standard definition of a Bessel function of the first kind.
Such expansions make it easier to relate the given functions to known series representations.

Differential Operators

Differential operators are tools that apply differentiation with respect to a variable. In this context, these operators help verify that the Bessel function satisfies specific differential equations. For instance:\( \frac{\partial \Phi}{\partial x} = \left(\frac{h - h^{-1}}{2}\right)e^{(1 / 2) x\left(h-h^{-1}\right)} = \left(\frac{h - h^{-1}}{2}\right)\Phi\) Another crucial operator is \(x\frac{\partial}{\partial x}\), which transforms the expression into manageable algebraic forms used to show: \[ x^2 \frac{\partial^2 \Phi}{\partial x^2} + x \frac{\partial \Phi}{\partial x} + x^2 \Phi - \left(h \frac{\partial}{\partial h}\right)^2 \Phi = 0 \] These operators systematically simplify complex expressions, vital for proving properties like those of Bessel functions.
The order of operations and results from applied differential operators ensure the coherence of the derived equations with Bessel functions.