PHYS 325
HW#2
F08
1. Locate the numbers z1  z 2 and z1  z 2 vectorially when
2
a) z1  2i, z 2   i b) z1  (2i), z 2  (2,0) c) z1  2  3i, z 2  2  3i
3
2. Evaluate a) 1  i 3 b) 3  4i 
1
2
c) ln 1  i  d) tan 1 1  i 
3. Derive the following trigonometric equations by using complex algebra
a) cos 3  cos 3   3 cos  sin 2 
b) sin 3  3 cos 2  sin   sin 3 
4. Show that a) i sin z  sinh iz b) sin iz  i sinh z c) cos z  cosh iz
d) cos iz  cosh z e) z n  e n ln z
5. How do circles centered on the origin in the z-plane transform for
1
w z   z 
(for z  0 ).
z
6. What part of z-plane corresponds to the interior of a unit circle in the w-plane if
z 1
w
.
z 1
7. Find the region in z-plane whose image under the transformation w  z 2 is the
rectangular domain in w-plane bounded by the lines u  1, u  2, v  1, v  2.
8. For f ( z )  z 2  1 z  2 
a) Locate the branches points.
b) Indicate two different methods of drawing branch cuts such that f(z) is single
valued in the cut plane.