Question

Show that \(J_{-n}(z)=(-1)^{n} J_{n}(z) .\) Hint: Let \(v=-n\) in the expansion of \(J_{v}(z)\) and use \(\Gamma(m)=\infty\) for a nonpositive integer \(m\).

Short answer

The given expression \(J_{-n}(z)=(-1)^{n} J_{n}(z)\) is proven true.

Step-by-step solution

Understanding the Bessel Function

Recall the definition of the Bessel function \(J_v(z)\):\[ J_{v}(z)=\left(\frac{z}{2}\right)^{v} \frac{1}{\Gamma(v+1)} \sum_{k=0}^{\infty} \frac{(-1)^{k}}{k ! \Gamma(v+k+1)}\left(\frac{z}{2}\right)^{2 k}\] Let's set \(v=-n\) where \(n\) is a positive integer.

Substituting \(v=-n\) into the Bessel Function

Substitute \(v=-n\) into \(J_v(z)\):\[ J_{-n}(z)=\left(\frac{z}{2}\right)^{-n} \frac{1}{\Gamma(-n+1)} \sum_{k=0}^{\infty} \frac{(-1)^{k}}{k ! \Gamma(-n+k+1)}\left(\frac{z}{2}\right)^{2 k}\] Now reconsider the Gamma function for negative integers by substituting \(m=-n+k\).

Substituting \(\Gamma(m)=\infty\) into the Expansion

Consider the Gamma Function \(\Gamma(m)=\infty\) for \(m=-n+k\):\[\Gamma(-n+k+1) \]= \(\infty\) for \(k<n\). Applying this to the above summation turns all terms with \(k<n\) to zero. Hence the remaining sum becomes: \[\sum_{k=n}^{\infty} \frac{(-1)^{k}}{k ! \Gamma(-n+k+1)}\left(\frac{z}{2}\right)^{2 k}\]

Simplify the remaining expression

Next, simplify the remaining expression:\[\sum_{k=n}^{\infty} \frac{(-1)^{k}}{k ! \Gamma(-n+k+1)}\left(\frac{z}{2}\right)^{2 k} = \sum_{k=n}^{\infty} \frac{(-1)^{k}}{k ! \Gamma(k-n+1)}\left(\frac{z}{2}\right)^{2 k} \]Changing the summation index \(k \to k - n\) results in:\[ =\sum_{k=0}^{\infty} \frac{(-1)^{k+n}}{(k+n) ! \Gamma(k+1)}\left(\frac{z}{2}\right)^{2(k+n)} =\left(\frac{z}{2}\right)^{n}(-1)^{n}\frac{1}{\Gamma(n+1)}\sum_{k=0}^{\infty} \frac{(-1)^{k}}{k ! \Gamma(n+k+1)}\left(\frac{z}{2}\right)^{2 k}\]This equation now equals to \(J_{n}(z)\).

Comparing with the original equation \(J_{n}(z)\).

This equation now equals to \(J_{n}(z)\). So this proves that: \[ J_{-n}(z) = (-1)^n J_{n}(z)\]

Key concepts

Gamma Function

The Gamma function, denoted as \( \Gamma(x) \), is a fundamental aspect of higher mathematics, especially useful in defining complex integrals and products. You might have seen it introduced as an extension of factorial numbers (e.g., \( n! \)). For positive integers, \( \Gamma(n+1) = n! \). It's smooth, continuous, and defined for non-negative integers, positive or negative real numbers, and even complex numbers, excluding non-positive integers.

A fascinating property of the Gamma function is that it interrelates with shifting: \( \Gamma(x+1) = x \cdot \Gamma(x)\). This recursive nature makes it versatile in various calculations, including solving equations involving special functions and Bessel functions.

Importantly for this exercise, the behavior of the Gamma function at non-positive integers is key to understanding simplifications with Bessel functions. In particular, \( \Gamma(m) = \infty \) at non-positive integers through integration results, fundamentally impacting sums and products in mathematical proofs.

Mathematical Proofs

Mathematical proofs are structured logical arguments used to verify that specific mathematical statements are universally true. They're crucial in both theoretical and applied mathematics to ensure consistency and correctness.

In the given exercise, proving the identity \( J_{-n}(z) = (-1)^{n} J_{n}(z) \) requires understanding how to transform and manipulate known series expansions. By applying known properties of the Bessel and Gamma functions, the identity is demonstrated, which further enriches the application of Bessel functions.

Proofs often start with assumptions or identities, like setting \( v = -n \) for this exercise, to guide the logical process. They utilize properties such as symmetry, zero-exclusion in Gamma terms, and index-shifting in infinite sums. Thus, thorough grounding in series expansion and recursive functions is valuable when tackling mathematical proofs in calculus and differential equations.

Special Functions

Special functions are a celebrated class of functions distinguished by their peculiar properties and frequent appearances in practical applications such as physics, engineering, and other scientific disciplines. Examples include the trigonometric functions, the exponential function, and, relevant here, Bessel functions.

Bessel functions, essential in solving certain differential equations, especially within radial and cylindrical systems, are a classic example. They come in two kinds, \( J_v(z) \) and \( Y_v(z) \), based on the situation's boundary conditions and specifics. Their series expansion is what allows linking with the Gamma function for exact solutions or numerical applications.

These functions play a crucial role in expressing waveforms encountered in heat conduction, electromagnetism, and even sound wave analysis. Knowing the attributes and the inter-relation of these functions, as highlighted in proofs and transformations, empowers you to solve complex mathematical and real-world problems effectively.