The residue of one divided by (one plus z squared) at i is one divided by twice i. The real integral over the line equals two pi i times that residue. What multiple of pi is the result?

Answer: ______________

To evaluate the integral of 1 divided by (1 + x squared) over the whole line by residues, how do you close the contour?

• Close with a unit semicircle kept at fixed radius
• Replace the real line by the unit circle
• Add a large upper semicircle and let its radius tend to infinity
• Close below but drop the orientation sign

To take 1 over 1 plus x squared over the whole line, how do you close the contour?

• Close with a unit semicircle kept at fixed radius
• Replace the real line by the unit circle
• Add a large upper semicircle and let its radius tend to infinity

Jordan lemma discards the large arc for e to the i a z with positive a closed upwards.

Circle one:   True   False

A simple pole sits on the contour and its residue is five plus one. Indenting around it counts half the residue. What is the contribution?

Answer: ______________

You close a positive frequency oscillation downwards. What goes wrong?

• The arc explodes instead of vanishing
• The arc still vanishes exactly as before
• The poles move to the other half plane

When the contour is closed in the upper half-plane, which poles contribute residues?

• Exactly the poles strictly inside the closed contour
• Every pole of the integrand anywhere
• Only the poles on the real axis
• Only the poles in the lower half-plane

Indenting around a simple pole on the contour with a small semicircle counts half the full residue, with sign by detour direction. True or false?

Circle one:   True   False

Your detour rounds a boundary pole clockwise. What weight does it bring?

• Plus one full residue as usual
• Minus half from the clockwise mini loop
• Nothing, boundary poles never count

What is the integral over the whole line of 1 divided by (1 + x to the fourth)?

• pi divided by root 2
• pi
• 2 pi
• 0