The kernel $k(x,y)=\frac{y}{y^2+x^2}$ is a solution of which equation?

Question

The kernel $$k(x,y)=\frac{y}{y^2+x^2}$$is a solution of

(A) Heat equation

(B) Wave equation

(C) Laplace equation

(D) Lagrange equation

Which are correct ?

I tried through satisfying the equation of heat, wave, Laplace , Lagrange equation but I can't find it..I have no idea how we find it.

Please help...

Answer 1

HINT:

The function $ k(x,y)=\frac{y}{x^2+y^2} $ is in fact a special case of Poisson kernel, which is used to solve Laplace equation. More specifically, $k(x,y)$ is the Poisson kernel defined on the upper half plane.

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Appended:

Technically the term "heat equation" includes steady-state heat equation, which is the Laplace's equation. So if you choose to be picky you can say that the question is not posed correctly.