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Use the Fourier transform method to solve this PDE:∂²u/∂x² + ∂²u/∂y² + ∂²u/∂z² = kδ(x,y,z)where k is a constant and δ() is the Dirac delta function. The solution (u(x,y,z)) vanishes sufficiently fast at infinity.a) Find the solution in the Fourier space (kx,ky,kz).b) Write down the integral expression of the solution in real space as the inverse Fourier transform of (kx,ky,kz). Use spherical coordinates to transform the integral and show that it reduces to 1/(k sin(kr)) ∫(a) dk/(2π) ∫(0)^(π) sin(θ) dθ ∫(0)^(2π) dφ.c) (Bonus, extra points) Solve this integral by extending it to the complex plane.
