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 t ...continues

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)

Complex integration

Calculate the integral using contour integration. Complete explanation is required integral(o->00)(logx)^2dx/(1+x^2)

Complex integration

Calculate integral using contour integration. Complete explanation is required. integral(o->oo)cos(ax)dx/(x^2+b^2)^2 for a,b positive.

Complex integration

Calculate the integral using contour integration. Complete explanation is required. integral(o->pi/2)d(theta)/(a+sin^2(theta))

Complex integration

Calculate integral using contour integration. Complete explanation are required. integral(o->oo)x^2dx/(x^4+5x^2+6)

Complex integration

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