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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.