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The Helmholtz equation is expressed as $$\nabla^2 \psi + \lambda \psi = 0$$. This equation occurs, for eg., after taking the Fourier transform (with respect to the time coordinate) of the wave equation in free space. While this equation for the case of $\lambda$ real is reasonably well discussed in the literature, the case where the coefficient $\lambda$ is complex-valued is not much discussed in the literature. However, such an equation does come up for the components of the electric and magnetic fields when we try to solve Maxwell's equations within a conductor. So my question is, what is the general solution of the equation $$\nabla^2 \psi + \lambda \psi = 0$$, where $\lambda$ is complex-valued(having both real and imaginary parts), in cylinderical coordinates? Also, can you quote a good reference from where I can get some good information about this?

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The so-called cylindrical waves can be chosen as the basis of the set of solutions (see, e.g., http://www.eecis.udel.edu/~weile/ELEG648Spring06/Resources/Cylindrical.pdf ). I guess the complex-valued $\lambda$ will lead to a complex-valued argument of the Bessel functions in the expressions for the cylindrical waves.

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  • $\begingroup$ Are Bessel functions defined for a complex argument? Will their properties remain the same? Finding the zeroes of these functions are likely to come up while solving an actual problem. Are all the zeroes real? Or are there complex valued zeroes and is this data available in standard tables or handbooks? $\endgroup$ – guru Aug 2 '13 at 23:34
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    $\begingroup$ @guru Yes, Bessel functions are defined for complex argument physics.stackexchange.com/questions/56041/… Mathematica would be able to handle your question about the zeros $\endgroup$ – Michael Brown Aug 3 '13 at 0:30
  • $\begingroup$ @guru: I cannot discuss "an actual problem" unless you tell me what the actual problem is:-) If $\nu>-1$, $J_{\nu}(z)$ only has real roots (books.google.com/… ) $\endgroup$ – akhmeteli Aug 3 '13 at 1:42
  • $\begingroup$ thanks for the info. i cannot reveal details of the actual problem - it is confidential. $\endgroup$ – guru Aug 8 '13 at 9:25
  • $\begingroup$ I understand. No problem, and good luck. $\endgroup$ – akhmeteli Aug 8 '13 at 11:58

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