Expand  in a Laurent series valid for

i)     ii) 

Find the radius of convergence of the following series.

i)   ii)        

Evaluate  where e is the eight like figure shown in Fig. 1. 

Find the maximum modulus  on the closed circular region defined by

Evaluate de where e is the circle

Find all the singularities of the function

Find all solutions to the equation sin z=5.

5.a) Find the constant c such that  can be extended to be analytic at z =1 , when n∈ N is fixed.

then show that there exists a real R>0 such that  for 

Find the image of the circle under the mapping  What happens when r = 1 ?

Evaluate the following integrals:

i)

ii)

Find the points where the function  is not analytic. 

Consider  and the closed circular region  Find points in R where |  | has its maximum and minimum values.

  is entire such that in C then show that  has the form  where , b are constants with Re=0.

Determine whether each of the following statement is true or false. Justify your answer with a short proof or a counter example.

i)  where a and b are integers, the 

ii)  are analytic functions in a domain, then f is necessarily a constant

iii) A real-valued function u(x, y) is harmonic in D iff w(x,y) is harmonic in D.

iv) 

v) The inequality  holds for 

vii) If a power series  converges for and if is such that  then  converges for r | z |  1.

viii) If  is entire and  for all, then there exists an entire function g such that 

ix) A mobius transformation which maps the upper half plane onto itself and fixing 0, and no other points, must be of the form for some  and 

x) If  is entire and Re  is bounded as  then  is constant.

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