Understanding Laurent Series and Bessel Functions Through Complex Analysis

Complex analysis offers elegant ways to represent and manipulate functions, and one such powerful representation is the Laurent series, which can describe functions with singularities. In this blog post, we'll explore how the exponential function

exp⁡[z2(w−1w)]\exp\left[\frac{z}{2}\left(w - \frac{1}{w}\right)\right]
can be expanded as a Laurent series, connecting it to the Bessel functions Jn(z)J_n(z).

Laurent Series and Contour Integrals

To start, let’s recall the Laurent series:

If f(z)f(z) is analytic on an annulus R1<∣z∣<R2R_1 < |z| < R_2, then f(z)f(z) can be expressed as:

f(z)=∑n=−∞∞cnzn,f(z) = \sum_{n=-\infty}^\infty c_n z^n,

where the coefficients are given by:

cn=12πi∫Cf(z)zn+1dz.c_n = \frac{1}{2\pi i} \int_C \frac{f(z)}{z^{n+1}} dz.

Here, CC is a closed contour within the annulus. Our goal is to apply this formula to expand the exponential function as a Laurent series around the origin in the ww-plane.

The Unit Circle in the ww-Plane

We define the unit circle CC as:

w=eiϕ,−π≤ϕ≤π.w = e^{i\phi}, \quad -\pi \leq \phi \leq \pi.

This gives:

1w=e−iϕ,dw=ieiϕdϕ.\frac{1}{w} = e^{-i\phi}, \quad dw = i e^{i\phi} d\phi.

Substituting this into the exponential function:

z2(w−1w)=z2(eiϕ−e−iϕ)=izsin⁡ϕ,\frac{z}{2} \left(w - \frac{1}{w}\right) = \frac{z}{2} \left(e^{i\phi} - e^{-i\phi}\right) = iz \sin\phi,

where we used the identity eiϕ−e−iϕ=2isin⁡ϕe^{i\phi} - e^{-i\phi} = 2i \sin\phi. Thus, the function simplifies to:

f(w)=exp⁡[izsin⁡ϕ].f(w) = \exp\left[iz \sin\phi\right].

Deriving the Laurent Series Coefficients

The coefficients cnc_n are:

cn=12πi∫Cf(w)wn+1dw.c_n = \frac{1}{2\pi i} \int_C \frac{f(w)}{w^{n+1}} dw.

Substituting w=eiϕw = e^{i\phi}, we get:

cn=12πi∫−ππexp⁡[izsin⁡ϕ](eiϕ)n+1⋅ieiϕdϕ.c_n = \frac{1}{2\pi i} \int_{-\pi}^\pi \frac{\exp\left[iz \sin\phi\right]}{(e^{i\phi})^{n+1}} \cdot i e^{i\phi} d\phi.

Simplify the terms:

cn=12π∫−ππexp⁡[izsin⁡ϕ−inϕ]dϕ.c_n = \frac{1}{2\pi} \int_{-\pi}^\pi \exp\left[iz \sin\phi - i n\phi\right] d\phi.

Connecting to Bessel Functions

The integral matches the definition of the Bessel functions:

Jn(z)=12π∫−ππexp⁡[−i(nϕ−zsin⁡ϕ)]dϕ.J_n(z) = \frac{1}{2\pi} \int_{-\pi}^\pi \exp\left[-i(n\phi - z \sin\phi)\right] d\phi.

Thus, the coefficients cn=Jn(z)c_n = J_n(z), leading to the Laurent series:

exp⁡[z2(w−1w)]=∑n=−∞∞Jn(z)wn.\exp\left[\frac{z}{2}\left(w - \frac{1}{w}\right)\right] = \sum_{n=-\infty}^\infty J_n(z) w^n.

Verifying Convergence

The series converges for 0<∣w∣<∞0 < |w| < \infty because:

  1. The function f(w)f(w) is analytic in this region.
  2. The Bessel functions Jn(z)J_n(z) decay rapidly as ∣n∣→∞|n| \to \infty.

Final Result

We have successfully expressed the exponential function in terms of a Laurent series:

exp⁡[z2(w−1w)]=∑n=−∞∞Jn(z)wn,\exp\left[\frac{z}{2}\left(w - \frac{1}{w}\right)\right] = \sum_{n=-\infty}^\infty J_n(z) w^n,

where the coefficients are:

Jn(z)=12π∫−ππexp⁡[−i(nϕ−zsin⁡ϕ)]dϕ.J_n(z) = \frac{1}{2\pi} \int_{-\pi}^\pi \exp\left[-i(n\phi - z \sin\phi)\right] d\phi.

This connection highlights the interplay between contour integrals, series expansions, and special functions like the Bessel functions.

Why This Matters

Laurent series and Bessel functions are fundamental in physics and engineering. For instance:

  • Bessel functions describe wave propagation in cylindrical systems.
  • Laurent series are essential for understanding singularities and residues in complex analysis.

By working through this derivation, we see how abstract mathematics translates into practical tools for solving real-world problems.

This derivation illustrates the elegance of complex analysis. Whether you're a student, teacher, or enthusiast, I hope this walkthrough inspires you to delve deeper into the fascinating world of mathematics!

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