Hello, I am having trouble finding the proper justification for being able to pass the derivative through the integral in the following:(adsbygoogle = window.adsbygoogle || []).push({});

## u(x,y) = \frac{\partial}{\partial y} \int_0^\infty\int_{-\infty}^\infty f(x') K_0( \sqrt{ (x - x')^2 + (y-y')^2 } \, dx' dy' ##

##K_0## is the Modified Bessel function of the second kind with properties:

1. ## K_0(z) \approx - \ln z \text{ as } z \to 0 .##

2. ## \mathop {\lim }\limits_{|z| \to \infty} K_\nu (z) = 0 .##

3. ##K_\nu(z)## is real and positive for ##\nu > -1## and ## z \in \mathbb{R} >0 .##

4. ##K_{-\nu}(z) = K_\nu(z) .##

5. ## \frac{\partial K_\nu (z) }{\partial z } = -\frac 12 \left( K_{\nu -1 }(z) + K_{\nu+1}(z)\right).##

I have seen it done, but my advisor asked me to justify the step.

The problem occurs at the singularity ## x= x', y = y'.##

I also know that ##K_0## has an equivalent representation in the Fourier transform domain of:

##c \int_{-\infty}^\infty \frac { e ^{ - \sqrt{ \xi^2+\alpha^2 } |y- y'|}}{\sqrt{ \xi^2+\alpha^2 }} e^{i (x-x') \xi } d\xi##

for some scaling constant c.

In a few articles, they discuss the properties of pseudo-differential operators...which may be a hint for justifying the derivative.

I am really stumped on this one...any insight would be helpful. I feel like there is something intuitive that I am simply missing.

Thanks.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Differentiating Integral with Green's function

**Physics Forums | Science Articles, Homework Help, Discussion**