Laurent Series - Complex Numbers

(a) Locate and classify the singularities (giving the order of any poles) of the function

f(z) = z/(1-e^z)

(b) Let f(z) = zsinh (1/(z+1))

(i) Find the laurent series of f about -1, giving teh general term of the series for odd and even powers of (z+1)

(ii) Write down a punctured open disk D, containing the circle C={z:|z+1|=1} on which f is represented by this series

(iii) State the nature of the singularity of f at -1

(iv) Evaluate

integral zsinh(1/(z+1))dz where C={z:|z+1|=1}

(c) Find the laurent series about 0 for the function

f(z)=7z/((2z+1)(z-3)) on the set {z:|z|.3, giving teh general term of the series