ASSIGNMENT 1 2012 PAST

1. Find all values of z for which the equation

holds.

2. Show that is analytic everywhere except at the points , and on the ray .

3. Construct a branch of that is analytic at the point and takes on the value there.

4. For what values of z is it true that

? Why?

5. Find the principal value of

6. For , show that the principal branch of

is given by the equation

,

where .

7. Establish that for all z.

8. Show that for all z, .

9. Find all values for the function

10. By equating the real and imaginary parts, find all roots of the equation