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Hello, I am having trouble finding the proper justification for being able to pass the derivative through the integral in the following:
## 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.
## 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.