ACM 95/100aAdam Neumann

Sample problems for recs or reviewNovember 17, 2014

Problem 1

For the complex function

w(z) =1

z 3 +1

z 41. Find the Laurent expansion about z = 0 valid for jzj < 32. Find the Laurent expansion about z = 0 valid for 3 < jzj < 43. Find the Laurent expansion about z = 1 valid for 2 < jz 1j < 3

Problem 2

Find the rst three terms of the Laurent series for

w = log

z + 1

z 1

valid for jzj > 1.

Problem 3

For the function

w(z) =tan z

z =2Find a few terms of the Laurent series expansion around z = 0 that is a valid representation of w(z) atz = . Indicate the region of convergence for this series.

Problem 4

Calculate the integral Z 10

x3

x5 + 1dx

Problem 5

Calculate the integral Z 10

1

x2 + 5x+ 6dx

by evaluating the integral ZC

log x

x2 + 5x+ 6dx

for an appropriate choice of C.