Evaluate Integral for Reduced Green's Function: Semi-Infinite Plates

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The integral in question diverges, as confirmed by Mathematica, due to the behavior of the integrand at large z', specifically resembling 1/|z'|. This divergence is problematic for obtaining the reduced Green's function for two semi-infinite plates meeting at a right angle on the z-axis. A potential solution involves a substitution where z-z' is expressed in terms of hyperbolic functions. This approach aims to manage the divergence and facilitate the evaluation of the integral. The discussion emphasizes the need for careful handling of the integral to achieve meaningful results.
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Can someone help me evaluate this integral?

<br /> \int\limits_{-\infty}^{\infty} dz&#039; \frac{1}{\sqrt{(x-x&#039;)^2 + (y-y&#039;)^2 + (z-z&#039;)^2}}<br />

Mathematica is telling me that this guy diverges. But it CAN'T! This is supposed to give me the reduced Green's function for two semi-infinite plates that meet at a right angle on the z-axis.
 
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For large z' the integrand behaves like 1/|z'|, leading to divergence.
 
It does indeed diverge.

To solve you could use
z-z&#039;=\sqrt{(x-x&#039;)^2+(y-y&#039;)^2}\sinh t
 

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