Activity 1

a)  Let f  be a continuous function on a contour C. Use Morera’s theorem to prove that the function F defined by

is analytic off C.

b)  Let f  be a continuous function on [a,b]. Use Morera’s theorem to prove that the function F defined by

is entire.

Activity 2

Complete the following mathematical argument by filling in the blanks :

Recall that for real numbers and we have

.

Thus

holds when ________________ are real.

But for any given real number, both sides of the identity are ______________ functions of.

By the _______________________________ the identity holds for all __________.

Now given any complex value to .  Since the identity holds for real , and both sides are _____________  functions of , the identity holds for all _____________

Consequently,

  holds when and are ____________.