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  • Task info Open for submissions from Wednesday, 27 July 2016, 9:00 PM (2766 days ago)
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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?