Potential Outline

NOTE: EVERYTHING IS VERY MUCH SUBJECT TO CHANGE - even titles, structure, etc.! Change however you think is best :)

A Brief Introduction to Bessel Functions

A short abstract.

Introduction

2-3 pages of introduction.

Chapter 1: Derivation of the Bessel Differential Equation

to be written by William Fayers.

• Physical and Mathematical Origins:
  • Discuss physical models (e.g. wave equations in cylindrical coordinates, heat conduction) that lead naturally to the Bessel differential equation.
  • Derive the standard form of the Bessel equation: $x^2{y}^{\prime \prime }+x{y}^{\prime }+(x^2-{\nu }^2)y=0$.
• Foundational Techniques:
  • Introduce singular differential equations.
  • Provide a brief overview of the Frobenius method as a tool for obtaining series solutions.
• Suggested Detail: Aim for approximately 3500–4000 words, with integrated examples (possibly some short code snippets) to illustrate the derivation process.

Chapter 2: Bessel Functions as Solutions

Short overview paragraph, incl. comparison

Chapter 2.1: Bessel Functions of the First Kind

to be written by Tom Ward.

• Deep Dive into $J_{\nu }(x)$:
  • Present the Frobenius series solution for $J_{\nu }(x)$, discussing convergence issues and the treatment of integer versus non-integer orders.
  • Analyse the behaviour of $J_{\nu }(x)$ at $x=0$ (regular, finite) and as $x\to \infty$ (oscillatory decay).
• Key Properties Specific to $J_{\nu }(x)$:
  • Detail relevant recurrence relations and the Wronskian for $J_{\nu }(x)$.
  • Include some of the integral representations that are particularly useful for $J_{\nu }(x)$.
• Programming Integration:
  • Offer computational examples (using, say, Python or MATLAB) to illustrate the series computation and visualise the function’s behaviour.
• Suggested Detail: Approximately 4000–5000 words.

Chapter 2.2: Bessel Functions of the Second Kind

to be written by Alex Rushworth.

• Exploration of $Y_{\nu }(x)$:
  • Explain the derivation of $Y_{\nu }(x)$ via a limiting process, particularly for non-integer orders.
  • Discuss the singular behaviour at $x=0$ and the corresponding physical interpretations.
• Complementary Properties:
  • Cover recurrence relations and Wronskian aspects that complement those of $J_{\nu }(x)$.
  • Present integral representations pertinent to $Y_{\nu }(x)$.
• Programming Integration:
  • Provide examples that contrast $Y_{\nu }(x)$ with $J_{\nu }(x)$, emphasising numerical challenges (such as handling singularities).
• Suggested Detail: Approximately 4000–5000 words.

Chapter 3: Modified and Spherical Bessel Functions

to be written by Hayden Loyseau.

• Modified Bessel Functions $I_{\nu }(x)$ and $K_{\nu }(x)$:
  • Derive these functions from the modified Bessel equation: $x^2{y}^{\prime \prime }+x{y}^{\prime }-(x^2+{\nu }^2)y=0$.
  • Compare their asymptotic behaviours—exponential growth versus decaying trends.
• Spherical Bessel Functions $j_n(x)$ and $y_n(x)$:
  • Explain their origin from the Helmholtz equation in spherical coordinates.
  • Illustrate the connection with sine and cosine functions and discuss elementary representations.
• Programming Integration:
  • Embed computational examples that demonstrate how to compute and visualise these functions, highlighting the contrasts with the standard Bessel functions.
• Suggested Detail: Roughly 4000–4500 words.

Chapter 4: Mathematical Properties and Tools

to be written by Torin Anderson.

• Orthogonality and Series Expansions:
  • Discuss the weighted orthogonality of functions like $J_{\nu }(kx)$ over intervals defined by their zeros.
  • Explore Fourier–Bessel series expansions.
• Zeros and Asymptotic Analysis:
  • Analyse the zeros of Bessel functions, including their asymptotic spacing and methods for numerical estimation.
  • Present series expansions for small $x$ (Taylor-like series) and large $x$ (Hankel’s asymptotic forms).
• Suggested Detail: Approximately 4000–5000 words, with possible programming demonstrations on how to compute zeros and plot asymptotic behaviour.

Chapter 5: Boundary Conditions and Eigenvalue Problems

to be written by Katie Arnold.

• Impact of Boundary Conditions:
  • Examine how boundary conditions select between the finite solutions (such as $J_{\nu }$) and the singular solutions (such as $Y_{\nu }$) in various domains.
• Eigenvalue Problems:
  • Apply Bessel functions to model eigenvalue problems in contexts like vibrating circular membranes (drumheads) and waveguide cut-off frequencies.
• Programming Integration:
  • Integrate computational approaches to numerically solve eigenvalue problems and simulate physical systems.
• Suggested Detail: Approximately 3500–4500 words.

Chapter 6: Applications in Science and Engineering

to be written by Daniel Morris.

• Real-World Applications:
  • Detail applications in physics and engineering: cylindrical waveguides (electromagnetism), quantum particles in cylindrical boxes, and heat transfer in radial geometries.
• Cross-Disciplinary Perspectives:
  • Discuss the role of Bessel functions in signal processing (e.g. Bessel filters) and in optics (diffraction patterns).
• Programming Integration:
  • Provide case studies and computational examples that demonstrate how Bessel functions are applied in practice.
• Suggested Detail: Roughly 4000–5000 words.

Summary

1-3 pages of summary.

Bibliography & Appendices

Following IEEE style and including minutes.