Functions of a complex variable / MAT 3223.001/ Spring 2005
1.
123−1−2−3
1
2
3
−1
−2
−3
x
y
z1+z2
z2/z1
z1= 1 + i=√2eiπ/4,
z2= 2ei2π/3=−1 + i√3
z1+z2=i(1 + √3) ≈i2.732
z2/z1=−1 + i√3
1−i=−1 + i√3
1−i·1 + i
1 + i
=(−1 + i√3)(1 + i)
2=−1 + √3
2+i√3−1
2
≈ −1.366 + i0.366
2. f(z) = 3eiπ/3z+ 3i
3. If f(x+iy) = x2+y2+i2xy, then R[f] = u=x2+y2and I[f] = v= 2xy, so
uxuy
vxvy=2x2y
2y2x
The first Cauchy-Riemann equation ux=vyis always satisfied, but the second uy=−vx
only holds when y= 0. Thus, the set where fis complex differentiable is the xaxis.
4.
1−1
1
x
y
Γx
y=cos t
sin t+ 1,−π≤t≤0, so dx
dy=−sin t
cos tdt,
ZΓ
y dx −xdy =Z0
−π
(sin t+1)(−sin t)dt−cos tcos t dt =Z0
−π
[−(sin t)2−sin t−(cos t)2]dt
=Z0
−π
(−1−sin t)dt = [−t+ cos t]0
−π= 2 −π
5.
1 2
1
x
y
ΩZZΩ
xy2dx dy =Z2
0"Z−x
2+1
0
xy2dy#dx =Z2
0xy3
3−x
2+1
0
dx
=Z2
0
x(−x
2+ 1)3
3dx =Z2
0
x
3−x3
8+3x2
4−3x
2+ 1dx =Z2
0−x4
24 +x3
4−x2
2+x
3dx
=−x5
120 +x4
16 −x3
6+x2
62
0
=−4
15 + 1 −4
3+2
3=1
15
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