Exercise 1. a)  Let be a piecewise differentiable path and let be its image under the function (symmetric with respect to the real axis).  Show that if   is a continuous function on , the function is continuous  on and we have

b)  In particular, if is the unit circle, oriented in the counter-clockwise direction, then

Exercise 2.  Let be a continuous path (not necessarily piecewise differentiable). Show that 

where C.

Exercise 3.    Let be a piecewise differentiable path whose image is contained in an open set

Let be a holomorphic function on with values in an open set (in the plane of the complex variable ).  Show that is a piecewise differentiable path and that for each continuous function we have

Is this formula still true in the case where is not differentiable?

Exercise 4.   Let be the (differentiable) path and let be the path   varies in the same interval.  Show that if is continuous in the closed disk , we have