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One solution to the bessel equation

$$\left[x^2\partial_x^2 + x \partial_x + x^2 - \alpha^2\right] y(x) = 0$$

are the Hankel functions $H^{(1)}_\alpha(x)$ and $H^{(2)}_\alpha(x)$. I am interested in their asymptotic limit for complex numbers $z$ with $|z|\rightarrow \infty$. According to Wikipedia the leading terms are then

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What I do not understand is the range of the arguments of the complex number $z$. Why is it necessary to define these limits for a range $-2\pi < \arg(z) < 2\pi$? Naively I would assume that $\exp(i\pi) = \exp(-i\pi)$, why is the distinction still important?

physicsGuy
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    I think it has something to do with analytic continuation. From what I can gather, the Hankel functions seem to be a bit like the complex logarithm $\log(z) = \ln|z|+i \arg z$, which can be thought of as a multivalued function of $z$ (with different values depending on what value you assign to $\arg z$), or as a singlevalued function on a suitable Riemann surface, if you prefer that. – Hans Lundmark Jul 08 '15 at 15:08
  • That should be the reason, still I was wondering for the strange range $[-\pi,2\pi]$ instead of $[0,2\pi]$. I guess there we would need an expert on Bessel/Hankel functions. – physicsGuy Jul 09 '15 at 16:27

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