I have an equation that relates two variables:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]k(\mathbf{x},\mathbf{x}') =exp(-(\mathbf{x}-\mathbf{x}')^2)[/tex]

If I want to determine the value of this equation where x' is kept constant and x is actually the set of every real number then I can express the function as the integral where the integrand relates x' to the integration variable u between the interval of minus infinity to infinity:

[tex]f(\mathbf{x}') = \int_{-\infty}^{\infty} exp(-(\mathbf{u}-\mathbf{x}')^2) d\mathbf{u}[/tex]

and the solution to this will be some sort of error function.

Now, a slight variation on this. I need to include an additional term that's like a weighting term which decays with distance from x. So I'm trying to find a solution for the following equation:

[tex]g(\mathbf{x},\mathbf{x}') = \int_{-\infty}^{\infty} \frac{exp(-(\mathbf{u}-\mathbf{x}')^2)}{|\mathbf{x}-\mathbf{u}|} d\mathbf{u}[/tex]

The problem I'm having is that when [tex]\mathbf{u} = \mathbf{x}, [/tex]

then the integrand goes to infinity. I think I can get around it by possibly converting to spherical coordinates (all of vectors here are 3-D vectors) but I also need to evaluate the function, h, when x' is also integrated from minus infinity to infinity and a second weighting term is introduced:

[tex]h(\mathbf{x},\mathbf{x}') = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \frac{exp(-(\mathbf{u}-\mathbf{v})^2)}{|\mathbf{x}-\mathbf{u}||\mathbf{x}'-\mathbf{v}|} d\mathbf{u}d\mathbf{v},[/tex]

and here the spherical coordinate approach doesn't seem to help.

How do I deal with this singularity? Someone suggested complex analysis but I'm not very familiar with that area.

Any suggestions would be greatly appreciated. I can post how I evaluate g(.,.) using spherical coordinates if people think it'd help.

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# Singularity in Integrand

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