Question

square joining the points \( z=1, z=2, z=i+1 \) and \( z=l+2 \). [2] (a) Determine the region in the complex plane represented by each of the following: (i) \( |z|<1 \). [3 Marks] (ii) \( 1<|z+2 i| \leq 2 \). [4 Marks] (iii) \( \frac{\pi}{3}<\arg z \leq \frac{\pi}{2} \). [3 Marks] (b) Express each of the following complex functions in the form \( u(x, y)+i v(x, y) \), where \( u \) and \( v \) are real: (i) \( \sin z \) [2.5 Marks] (ii) \( \frac{1}{1+z^{2}} \) [2.5 Marks] (iii) \( e^{32} \) [2.5 Marks] [3] (a) Suppose that \( f \) is complex function in a region \( \Omega \) and it has no zero in \( \Omega \). Compute the Laplacian of \( \log |f| \).. [5.5 Marks]

          square joining the points \( z=1, z=2, z=i+1 \) and \( z=l+2 \).
[2] (a) Determine the region in the complex plane represented by each of the following:
(i) \( |z|<1 \).
[3 Marks]
(ii) \( 1<|z+2 i| \leq 2 \).
[4 Marks]
(iii) \( \frac{\pi}{3}<\arg z \leq \frac{\pi}{2} \).
[3 Marks]
(b) Express each of the following complex functions in the form \( u(x, y)+i v(x, y) \), where \( u \) and \( v \) are real:
(i) \( \sin z \)
[2.5 Marks]
(ii) \( \frac{1}{1+z^{2}} \)
[2.5 Marks]
(iii) \( e^{32} \)
[2.5 Marks]
[3] (a) Suppose that \( f \) is complex function in a region \( \Omega \) and it has no zero in \( \Omega \).
Compute the Laplacian of \( \log |f| \)..
[5.5 Marks]

square joining the points z=1, z=2, z=i+1 and z=l+2.
[2] (a) Determine the region in the complex plane represented by each of the following:
(i) |z|<1.
[3 Marks]
(ii) 1<|z+2 i| ≤ 2.
[4 Marks]
(iii) (π)/(3)< z ≤(π)/(2).
[3 Marks]
(b) Express each of the following complex functions in the form u(x, y)+i v(x, y), where u and v are real:
(i) sin z
[2.5 Marks]
(ii) (1)/(1+z^2)
[2.5 Marks]
(iii) e^32
[2.5 Marks]
[3] (a) Suppose that f is complex function in a region Ω and it has no zero in Ω.
Compute the Laplacian of log |f|..
[5.5 Marks]

Added by Gregg M.

Question

Please give Ace some feedback