Laplace operator, gradient vectors

Let f(z)=u+iv be an analytic function, phi(u,v) any function with second order partial derivatives and g(u,v) any function with first order partial derivatives. a) Let L_x,y be the Laplace operator in x,y coordinates and L_u,v be the Laplace operator in u,v coordinates. Show that L_x,y(phi o f)=L_u,v |f'(z)|^2 b)Let G_u,v be the gradient vector in u,v, and G_x,y be the gradient vector in x,y coordinates. Show that |G_x,y(g o f)|^2=|G_u,v(g)|^2 |f'(z)|^2 where the | | is the euclidean norm in C.

Residue problem

Residue problem. See attached file for full problem description.

Residue/integrating using contour integrals

Residue/integrating using contour integrals. See attached file for full problem description.

Integration using contour integrals

Calculate the integral using contour integration. Complete explanation is required integral(o->oo) dx/(x^3+1)

Complex integration

Calculate the integral using contour integration. Complete explanation is required integral(o->oo)cosxdx/(x^2+1)