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I was trying to find the fundamental solutions for the Helmholtz equation in $\mathbb{R}^d$ when I found this answer. Here, and in some other places, it is stated that Hankel functions represent travelling (ingoing and outgoing) waves.

Of course, it probably has to do with their asymptotic behaviour at infinity, and with the Sommerfeld Radiation Condition: $$\lim_{|x| \to \infty} |x|^{\frac{n-1}{2}} \left( \frac{\partial}{\partial |x|} - ik \right) u(x) = 0.$$

However, I can not really grasp the details. Any help would be very much appreciated.

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  • $\begingroup$ What exactly details do you not grasp? Have you tried animating the plot of $\Re\{u(x)\exp(i\omega t)\}$ with changing $t$? $\endgroup$
    – Ruslan
    Mar 16 at 14:52
  • $\begingroup$ I was trying to understand mathematically why Hankel functions are said to represent travelling waves. I tried to show that they fulfill the radiation condition and I couldn't. I spotted my mistake, though, it is solved now. Thank you for trying to help. $\endgroup$ Mar 17 at 11:11

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