PHZ 3113, Section 3924, Fall 2013, Homework 8
Due at the start of class on Friday, October 25. Half credit will be available for
homework submitted after the deadline but no later than the start of class on Monday,
October 28.
Answer all questions. Please write neatly and include your name on the front page of your
answers. You must also clearly identify all your collaborators on this assignment. To gain
maximum credit you should explain your reasoning and show all working.
1. Let f = u + iv be an analytic function of z = x + iy, with u and v being real functions of x
and y. For the two dimensional vector field F = ui − vj, show that:
a) F is solenoidal (i.e., ∇ · F = 0), and
b) F is irrotational (i.e., ∇ × F = 0).
2. Use complex integration methods to evaluate the following integrals:
∫ π
a)
∫ ∞
dθ
,
2
o (2 + cos θ)
dx
b) P
,
2
−∞ x + 5x + 4
∫ ∞
sin2 x
c)
dx.
2
0 x +1
3. Find all non-zero coefficients an , bn with |n| < 2 in Laurent series for:
F (z) =
z
,
(z + 1)(z − i)
within disks of analyticity centered at
a) z = 0,
b) z = −i.
Each coefficient should be expressed in Cartesian form (i.e., xn + iyn ).
4. Find the determinants and inverses of the following matrices:
(
)
(
)
(
)
0 −1 −1
1 −2 −2
7 −2
a)
,
b)
−1 1 −1 ,
c)
0 −1 −1 ,
−3 1
0
1 −2
1 −1 −2
using the formula A−1 = (det A)−1 C T , where C is the matrix of co-factors.
5. Let F (z) = u + iv = ln(z), where z = x + iy. If ds2 = dx2 + dy 2 , find ds′2 = du2 + dv 2 .
6. Find the residue of the following at z = ∞ :
a)
1
,
z
b)
z
z2 + 1
,
1
c) sin .
z