Thank you for your kind reply! The wikipage is very comprehensive compared with the book. Now I think I have got something, following your advice:
\lim_{m\to 0} \frac{\cos m\pi J_m(x) - J_{-m}(x)}{\sin m\pi}
= \lim_{m\to 0}\frac{
-\pi\sin m\pi J_m(x) + \cos m\pi \frac{\partial}{\partial...
I have to admit I can't solve by hand the Bessel equation
x^2 y'' + x y' + x^2 y = 0
Matlab gives the solution to the equation as
A J_0(x) + B Y_0(x)
Still I don't see how \frac{2}{\pi}\ln(x) emerges from the equations :(
In the same book Y_m is defined as
Y_m(x) = \frac{J_m(x) \cos m\pi - J_{-m}(x)}{\sin m\pi}
while J_m(x) is defined as
J_m(x) = \sum_{r=0}^{\infty} \frac{(-1)^r}{r! \Gamma(r+m+1)}\left(\frac{x}{2}\right)^{2r+m}
Homework Statement
It is stated in "Mathematical methods of Physics" by J. Mathews, 2nd ed, p274, that the Bessel function of the second kind and of order zero, i.e. Y_0(x) can be approximated by \frac{2}{\pi}\ln(x)+constant as x \to 0, but no more details are given in the same text.Homework...