Math 311
Sample Final Examination
1. Let Cbe the boundary of the square with vertices at the points z= 0, z = 1, z = 1 + i,
z=iand with counterclockwise orientation. Evaluate
IC
z2dz.
2. Evaluate I|z|=1
Log(z+ 2)
z2dz.
(the circle |z|= 1 is oriented counterclockwise)
3. Evaluate I|z|=2
tan (z/2)
(z−1)2dz.
(the circle |z|= 2 is oriented counterclockwise)
4. Evaluate
Z3+i
i
(z−1)3dz.
5. (a) Given functions u(x, y ) and v(x, y) state sufficient conditions (on the partial derivatives) for
f(z) = u(x, y) + iv(x, y )
to be analytic at a point z0.
(b) State the Cauchy Integral Formula.
6. Obtain the first four (4) non-zero terms of the Laurent series expansion of the function
f(z) = 1
ez−1,
valid in the domain 0 <|z|<2π.
7. Obtain the expansion of the function
f(z) = (z−1)
z2
into its Laurent series, valid in the domain 0 <|z|<∞.
8. Using residues, show that
Z∞
−∞
x2
(x2+ 1)2dx =π
2.
