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Return to Part VII of the course APMA0340
Introduction to Linear Algebra with Mathematica

Glossary

Bessel functions of Half-integer order

First, we check that

\begin{equation} \label{EqHalf.1} J_{-1/2} (x) = \left( \frac{2}{\pi x} \right)^{1/2} \cos x . \end{equation}
   
Example 1: We check with Mathematica:
DSolve[x^2 *y''[x] + x*y'[x] + (x/4 - 1/16)*y[x] == 0, y[x], x]
{{y[x] -> (E^(I Sqrt[x]) C[1])/x^(1/4) + (I E^(-I Sqrt[x]) C[2])/x^( 1/4)}}
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End of Example 1
Moreover,
\begin{equation} \label{EqHalf.2} J_{n+1/2} (x) = (-1)^n \sqrt{\frac{2}{\pi}}\, x^{n + 1/2} \left( \frac{\text d}{{\text d}x} \right)^n \,\frac{\sin x}{x} , \qquad n=0,1,2,\ldots . \end{equation}
Thus applying a recurrence formula \( \displaystyle \quad J_{\nu +1} (x) = \frac{2\nu}{x}\, J_{\nu} (x) - J_{\nu -1} (x) \quad \) and using the Lommel polynomials yield
\[ J_{n+1/2} (x) = R_{n,\nu} (x)\, J_{1/2} (x) - R_{n-1, \nu +1} \,J_{-1/2} (x) . \]
So we have
\begin{equation} \label{EqHalf.3} J_{n+1/2} (x) = R_{n,\nu}(x) \left( \frac{2}{\pi x} \right)^{1/2} \sin x - R_{n-1, \nu +1} (x) \left( \frac{2}{\pi x} \right)^{1/2} \sin x . \end{equation}
or each fixed ν, the Lommel polynomials are given by
\begin{equation} \label{EqHalf.4} R_{n,\nu} (x) = \sum_{k=0}^{\left\lfloor n/2 \right\rfloor} \frac{(-1)^k (n-k)!\,\Gamma (\nu +n-k)}{k! \,(n-2k)!\, \Gamma (\nu +k)} \left( \frac{2}{x} \right)^{n-2k} , \end{equation}
where ⌊ ν ⌋ is the floor of number ν and
\[ \Gamma (\nu ) = \int_0^{\infty} t^{\nu -1} e^{-t} \,{\text d}t \] is the gamma function.

Eugen Cornelius Joseph von Lommel (1837 – 1899) was a German physicist. He is notable for the Lommel polynomial, the Lommel function, the Lommel–Weber function, and the Lommel differential equation. He is also notable as the doctoral advisor of the Nobel Prize winner Johannes Stark.

These Lommel polynomials have remarkable properties. Since

\[ J_{-1/2} (x) = \left( \frac{2}{\pi x} \right)^{1/2} \cos x , \qquad J_{1/2} (x) = \left( \frac{2}{\pi x} \right)^{1/2} \sin x , \]
and sin²x + cos²x = 1, we have
\begin{equation} \label{EqHalf.5} J^2_{n+1/2} (x) + J^2_{-n-1/2} (x) = 2 \left( -1 \right)^n \frac{1}{\pi x}\,R_{2n, 1/2-n} (x) . \end{equation}
hat is, we have
\[ J^2_{n+1/2} (x) + J^2_{-n-1/2} (x) = \frac{2}{\pi x} \sum_{k=0}^n \frac{(2x)^{2n-2k} (2n-k)! \, (2n-2k)!}{\left[ (n-k)! \right]^2 k!} . \]
A few special cases are
\begin{align*} J^2_{1/2} (x) + J^2_{-1/2} (x) &= \frac{2}{\pi x} , \\ J^2_{3/2} (x) + J^2_{-3/2} (x) &= \frac{2}{\pi x} \left( 1 + \frac{1}{x^2} \right) , \\ J^2_{5/2} (x) + J^2_{-5/2} (x) &= \frac{2}{\pi x} \left( 1 + \frac{3}{x^2} + \frac{9}{x^4} \right) , \\ J^2_{7/2} (x) + J^2_{-7/2} (x) &= \frac{2}{\pi x} \left( 1 + \frac{6}{x^2} + \frac{45}{x^4} + \frac{225}{x^6} \right) . \end{align*}    
Example 2: We check with Mathematica:
DSolve[x^2 *y''[x] + x*y'[x] + 9*x^4 *y[x] == 0, y[x], x]
{{y[x] -> BesselJ[0, (3 x^2)/2] C[1] + 2 BesselY[0, (3 x^2)/2] C[2]}}
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End of Example 2
   
Example 3: We check with Mathematica:
DSolve[y''[x] - x*y[x] == 0, y[x], x]
{{y[x] -> AiryAi[x] C[1] + AiryBi[x] C[2]}}
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End of Example 3

 

 

  1. Carbajal-Dominguez, A., Bernal, J., Martin-Ruiz, A., Martinez-Niconoff, G., Segovia, J., Half-integer order Bessel beams,
  2. Bayman, B. F., “A generalization of the spherical harmonic gradient formula”, J. Math. Phys. 19 (1978), 2558–2562
  8. Watson, G.N., A Treatise on the Theory of Bessel Functions, Cambridge University Press; 2nd edition (August 1, 1995). ISBN-13 ‏ : ‎ 978-0521483919

 

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