Green's function for Cauchy-Euler equidimensional equation

[anubhab]

Hi,

I am trying to compute the Green's function for a Cauchy-Euler equidimensional equation,
$$\frac{d^2G}{dx^2}+\frac{a}{(x-x_c)^2}G=A_1\delta(x-x')$$
If the impulse is located at a location $$x'\neq x_c$$ then computation of Green's function is not an issue. What happens when $$x'= x_c$$ ?
$$\frac{d^2G}{dx^2}+\frac{a}{(x-x_c)^2}G=A_1\delta(x-x_c)$$
The solution of the homogeneous equation is $$(x-x_c)^{1/2\pm\nu}$$ where, $$\nu=\sqrt{1/4-a}$$
The trouble is if one tries to relate the change in slope of the Green's function with the strength of the impulse one has,
$$\int_{x_c-\epsilon}^{x_c+\epsilon}\frac{d^2G}{dx^2} dx+\int_{x_c-\epsilon}^{x_c+\epsilon}\frac{a}{(x-x_c)^2}G \!dx=A_1$$
From the Frobenius exponents we see that the only for $$a=0$$ is the second integral is a Cauchy principal value integral.
Is it sensible to seek for a Green's function for such cases or one needs to make certain modification?
Thank you,

anubhab

 

[gtoguy97]

Yes, it is sensible to seek for a Green's function for such cases. Usually, the modification required is to make use of the Cauchy principal value integral to calculate the value of the second integral. This is known as the Cauchy Green's function. To calculate this, you need to use the appropriate mathematical techniques to evaluate the integral. Specifically, you might need to use integration by parts or contour integration.