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When we want to look at different singular points for e.g Bessel's eq. $$u´´(x) + \frac{u'(x)}{x} + (1- \frac{n^2}{x^2})u(x)$$.

We usually evaluate the equation letting x= 1/z. But I don't algebraically see how such a substitution ends up with $$w´´(z) +( \frac{2}{z}- \frac{1}{z^2})z*w'(z) + \frac{1}{z^4}(1- n^2 z^2)w(z)$$.

Letting x= 1/z, and derive both sides gives ##1/z^2 z'## but I simply don't know how to go from u(x) to w(z) which is very central and should be very basic and just one microstep in long calculations lol.