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(b) Use Cauchy-Riemann equation to show that f is an entire function.
(c) Find the derivative of f (z) in terms of u and v.
2. At what points are the following functions not analytic
(a) z
z−3i . (b) z 2 −2iz
.
z 2 +4 3. Compute f 0 if f = 4z 3 −5z+1
2z−1 . 4. Show that the following functions are not analytic at any
point.
(a) f (z) = y + ix
(b) f (z) = z¯2
(c) f (z) = 2x2 + y + i(y 2 − x)
5. Show that (ez )0 = ez .
6. Express f (z) = e2¯z in u + iv form.
7. Solve ez−1 = −ie2 .
8. Express sin(−2i) in a + ib form.
9. Find all values of z such that cos z = −3i. 1
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Quiz 8
Question 1 1
πx + π + 3 .
(5
) Find the period for the following function: f (x) = 8 cos a) 2π
5 b) π
5 c) 10 d) 5 e) 10 π f) None of the above. Question 2
Find the phase shift for the following function: f (x) = 3 sin a) π units
to the left b) 3 units to the right c) 3 units to the left d) 2 units to the
left e) π units to the right f) None of the above. Question 3 1
πx + π − 2 .
(3