Math 301 Final Examination – April 24, 2010
THIS EXAM HAS 8 QUESTIONS.
YOU ARE PERMITTED ONE SHEET OF NOTES (DOUBLE SIDED).
NO CELLPHONES, BOOKS OR CALCULATORS.
1. (10pts) Use contour integration to calculate Z∞
0
sin x
xdx, including a brief explanation
of every step.
2. (10 pts)
(a) Construct a conformal map of the unit disk centred at the origin onto itself, that
takes the point i/2 to the origin.
(b) Construct a conformal map of the disk |z−1|<1 onto the whole left half plane.
3. (10 pts) Solve Laplace’s equation for φin the lens-shaped region between the circles
|z−i|= 1 and |z−1|= 1 with the boundary conditions that φ= 0 on the circle
|z−i|= 1 and φ= 1 on the circle |z−1|= 1.
4. (15 pts)
(a) Use residue calculus to verify the sum
∞
X
k=−∞
1
k2+a2=π
acoth(πa).
(b) Use the result of part (a) to calculate
∞
X
k=1
1
k2, explaining all steps in full detail.
5. (15 pts) Let f(z) = (1 −z2)1/2
1 + z2.
(a) Find the residue of f(z) at infinity.
(b) Use an appropriate branch of f(z) with a dogbone contour and residue calculus
to show that
Z1
−1
√1−x2
1 + x2dx = (√2−1)π.
(continued on page 2)
