Question

$\underset{x\to 0}{lim}{J}_1(x)/x=\frac{1}{2}$

Short answer

The limit is $\frac{1}{2}$.

Step-by-step solution

Understand the Problem

The task is to find the limit of the function $\frac{J_1(x)}{x}$ as $x$ approaches 0, where it is given that this limit equals $\frac{1}{2}$. $J_1(x)$ is a Bessel function of the first kind which typically requires special techniques to analyze.

Identify Known Information

According to the problem, $\frac{J_1(x)}{x}\to \frac{1}{2}$ as $x$ approaches 0. This suggests that we need to understand the behavior of $J_1(x)$ near zero.

Recall the Series Expansion

Bessel functions of the first kind $J_1(x)$ have a series expansion around $x=0$. Specifically, $J_1(x)$ can be approximated by $\frac{x}{2}-\frac{x^3}{16}+O(x^5)$ for values of $x$ close to 0.

Simplify the Function

Using the series expansion, replace $J_1(x)$ in the limit:$$\frac{J_1(x)}{x}=\frac{\frac{x}{2}-\frac{x^3}{16}+O(x^5)}{x}$$

Simplify the Expression

Simplify the expression inside the limit:$$\frac{J_1(x)}{x}=\frac{x}{2x}-\frac{x^3}{16x}+O(\frac{x^5}{x})=\frac{1}{2}-\frac{x^2}{16}+O(x^4)$$

Evaluate the Limit

As $x$ approaches 0, the higher-order terms (those involving $x^2$ and $x^4$) go to 0, so:$$\frac{J_1(x)}{x}\to \frac{1}{2}$$Thus, the limit is indeed $\frac{1}{2}$.

Key concepts

Bessel functions

Bessel functions are a family of solutions to Bessel's differential equation. They are named after Friedrich Bessel, who studied them in the early 19th century. Bessel functions often appear in problems involving wave propagation, static potentials, and heat conduction in cylindrical geometries.

The most common types are Bessel functions of the first kind, denoted as $J_n(x)$, and of the second kind, denoted as $Y_n(x)$. In particular, $J_1(x)$ is the Bessel function of the first kind of order one. These functions are significantly used in physics and engineering.

The specific form of $J_1(x)$ around zero can be approximated using a series expansion, making the analysis more manageable for small values of $x$. This approximation is essential for deriving limits, such as the one given in the exercise.

limit calculation

Calculating limits is a fundamental concept in calculus that helps us understand the behavior of functions as the input approaches a certain value. For this problem, we need to find the limit of $\frac{J_1(x)}{x}$ as $x$ approaches zero.

Limits are crucial for determining the values that functions take and for understanding their long-term behaviors. In our case, the limit calculation shows how $J_1(x)$ behaves when $x$ is near 0.

The calculated limit is based on the series expansion of the Bessel function and simplifies the function to a more manageable form. This helps in seeing that the higher-order terms in the series become negligible as $x$ approaches zero, leaving only the leading terms, thus confirming the given limit $\frac{1}{2}$.

series expansion

A series expansion is a way to write a function as an infinite sum of terms calculated from the values of its derivatives at a single point. For the Bessel functions, specifically $J_1(x)$, the series expansion around $x=0$ is given by:

$$J_1(x)=\frac{x}{2}-\frac{x^3}{16}+O(x^5)$$

This series provides an approximation that becomes increasingly accurate as $x$ gets closer to 0. The first few terms offer a simple yet potent tool for analyzing the limit behavior of the function. The expansion shows that for small $x$, $J_1(x)$ is approximately linear. Higher-order terms (like $x^3$) contribute less as $x$ shrinks.

Using the series expansion simplifies the operation of taking a limit, as it transforms the function into a form where leading terms dominate, and the limit can be directly observed from these terms.

Mathematical methods in physics

Mathematical methods in physics involve using advanced mathematical techniques and tools to solve physical problems. Bessel functions, limits, and series expansions are part of these methods and are essential in solving complex differential equations often encountered in physics.

These methods bridge the gap between theoretical physics and practical calculations. For instance, Bessel functions help solve problems related to vibration modes in circular membranes, wave propagation in cylindrical cavities, and heat conduction.

Understanding these methods equips students with the tools to tackle a wide range of physical problems. The exercise involving the limit calculation of a Bessel function exemplifies how these methods underpin much of the theoretical physics used in practical applications.