## Question

###### 4_ Consider the integral$ fce)a:over a closed, positively oriented contour of the function f (2) = 24 + 2. (a) By computing the antiderivative of f (2) , that is the function F(2) such that F' (2) = f (2), show that the integral is zero [3](b) Let the closed contour be the unit circle Izl = 1_ By employing a suitable parametrization of the circle, ~(t) , compute the integral once more and show that it is zero: [5](c) Using the same function f (~) as defined above, find all solutions to the

4_ Consider the integral $ fce)a: over a closed, positively oriented contour of the function f (2) = 24 + 2. (a) By computing the antiderivative of f (2) , that is the function F(2) such that F' (2) = f (2), show that the integral is zero [3] (b) Let the closed contour be the unit circle Izl = 1_ By employing a suitable parametrization of the circle, ~(t) , compute the integral once more and show that it is zero: [5] (c) Using the same function f (~) as defined above, find all solutions to the equation f(2) = 2+26)34+ (-i)44 and write them in polar form_ [6] (d) Use the residue theorem to compute the integral dz 24 + 1 on a positively oriented square with vertices {-2, ~2-2i ,0, -2i} . Sketch the square and the poles of the integrand in the same graph. Simplify your result as much as possible_ [6]