Perturbative Methods for Modelling (M2)

Exercice 3.68, p. 143, Reference Book

Example  3, p. 111, Reference Book

Show that the Bessel function  $\fs2J_\nu = \left({x\over 2}\right)^\nu \, \sum_{n=0}^\infty{ (-x^2/4)^n  \over n!\, \Gamma(\nu + n + 1)}$ is solution of the Bessel equation $\fs2 x^2\, y'' + x \, y' + (x^2-\nu^2) \, y=0$ .

Show that $\fs2 J_\nu(x) = e^{-i\, \nu\, \pi/2} \, I_\nu\left(x \, e^{i\, \pi/2}\right)$ .

Is it correct to write  $\fs2 y \sim c_1 \, x^{-1/2} \,\cos\left( x- {1\over 2} \nu\,\pi - {1\over 4} \,\pi\right)$ and  $\fs2 y \sim c_2 \, x^{-1/2} \,\sin\left( x- {1\over 2} \nu\,\pi - {1\over 4} \,\pi\right)$ for $\fs2 x\to + \infty$? Why?

We denote by $\fs2 Y_\nu(x)$  the solution of the Bessel equation  whose graph ``closely
resembles'' to the one of  $\fs2(2/\pi)^{1/2} \,x^{-1/2} \,\sin\left( x- {1\over 2} \nu\,\pi - {1\over 4} \,\pi\right)$. Why is
it unique?