MAT341E Theory of Complex Functions
Worksheet 2
1. Sketch the region onto which sector r  1 , 0   

4
31 October 2007
Beycan Kahraman
040020337
is mapped by the transformation
a) w  z 2
z  re i
b) w  z 3
z  re i
c) w  z 4
z  re i
w  r 2 e 2i  e i
w  r 3 e 3i  e i
w  r 4 e 4i  e i
  r2
  r 3   3
  r 4   4
  2
2. Show that when w  z 2 , the image of the closed triangular region formed by the lines
y   x and x  1 is the closed parabolic region bounded on the left by the segment
 2  v  2 of the v axis and on the right by a portion of the parabola v 2  4(u  1) . Verify
the corresponding points on the two boundaries shown in figure ( A, B, C , D and
A' , B ' , C ' , D ' ).
w  u  iv  z 2  ( x  iy ) 2  ( x 2  y 2 )  2ixy

u  x2  y2
v  2 xy
ux  x  0
v  2 xy  2 x 2
0  x 1
AB line: y   x,

0  v  2,
u0
u  x2  x2  0
BC line: x  1, 1  y  1
v  2 xy  2 x 2

0  v  2,
u0
v  2y

v 2  4(u  1)
AC line: y  x,
2
0  x 1
2
u 12  y 2
Point A(0, 0):
Point C(1, 1):
u 0
u 0
v0
v2
Point B(1, -1): u  0
Point D(1, 0): u  1
v  2
v0
3. Show that f ' ( z ) does not exist at any point when
a) f ( z )  z
b) f ( z )  Re z
z  x  iy
u ( x, y )  x
v ( x, y )  0
u x  1,
vy  0
f ( z )  u ( x, y )  iv ( x, y )  x  iy
u ( x, y )  x
v ( x, y )   y
Couchy-Riemann conditions are
ux  vy
v x  u y
Necessary for being differentiable.
u x  1,
v y  1
f ' ( z ) does not exist in
any point.
c) f ( z )  Im z
u ( x, y )  0
v ( x, y )  y
u x  0,
vy  1
f ' ( z ) does not exist in
any point.
f ' ( z ) does not exist.
4. Show that f ' ( z ) and its derivative f ' ' ( z ) exist everywhere and find f ' ' ( z ) when
a) f ( z )  iz  2
b) f ( z )  e  x e iy
u ( x, y )  2  y
v ( x, y )  x
u( x, y)  e  x cos y
v( x, y)  e  x sin y
ux  vy  0
v x  u y  1
u x  v y  e  x cos y v x  u y  e  x sin y
 f ' ( z ) exists everywhere.
 f ' ( z ) exists everywhere.
f ' ( z )  u x  iv x  i  s  it
f ' ( z )  u x  iv x  e  x cos y  ie  x sin y
s ( x, y )  0
t ( x, y )  1
s( x, y)  e  x cos y t ( x, y)  e  x sin y
sx  t y  0
s y  t x  0
s x  t y  e  x cos y
s y  t x  e  x sin y
 f ' ' ( z ) exists everywhere.
 f ' ' ( z ) exists everywhere.
f ' ' ( z )  s x  it x  0
f ' ' ( z )  s x  it x  e  x cos y  ie  x sin y
c) f ( z )  z 3
u( x, y)  x 3  3xy 2
v( x, y)  3x y  y
2
u x  v y  3x  3 y
v x  u y  6 xy
 f ' ( z ) exists everywhere.
f ' ( z )  u x  iv x  3x 2  3 y 2  6ixy
2
2
s( x, y)  3x 2  3 y 2
t ( x, y)  6 xy
s x  t y  6 x s y  t x  6 y
 f ' ' ( z ) exists everywhere.
f ' ' ( z )  s x  it x  6 x  6iy
3
d) f ( z )  cos x. cosh y  i sin x. sinh y
u ( x, y )  cos x. cosh y v( x, y )   sin x. sinh y
u x  v y   sin x. cosh y
v x  u y   cos x. sinh y
 f ' ( z ) exists everywhere.
f ' ( z )  u x  iv x   sin x. cosh y  i cos x. sinh y
s ( x, y )   sin x. cosh y
t ( x, y )   cos x. sinh y
s x  t y   cos x. cosh y
s y  t x   sin x. sinh y
 f ' ' ( z ) exists everywhere.
f ' ' ( z )  s x  it x   cos x. cosh y  i sin x. sinh y