Solutions to Assignment #4
Section 2.4:
1
x
y
1
=
= 2
i 2
z
x + iy
x + y2
x + y2
y
x
and v(x, y) = 2
.
Therefore u(x, y) = 2
x + y2
x + y2
We compute the rst partial derivatives:
(1) (a) f (z) =
(x2 + y 2 ) x(2x)
y 2 x2
= 2
(x2 + y 2 )2
(x + y 2 )2
2xy
uy
Solutions to Assignment #10
Section 4.6:
(1) Problem 7, page 220 Suppose f is entire and there exists a real number r0 > 0
such that |f (z)| |z|2 for all z that satises |z| > r0 . Now x any z0 C. We
will show that f (3) (z0 ) = 0, and therefore conclude t
Solutions to Assignment #8
Section 4.2:
(1) Problem 6, page 171.
(a) The contour can be parametrized by z = 2ei (0 2). In this case,
z = 2ei and dz = 2iei d. So
2
z(t)z ()d
z dz =
0
2
2ei 2iei d
=
0
2
d
= 4i
0
= 8i
(b) The contour is , where was the conto