MTH632 Quiz Solution

Exponential from: · 4 cos · sin

i

z · e · z · i · i · z

Express the function f(x) = �ഥ Im(z) + z Re (z) + z �ഥ in form f(z) = u(x,y)

+iv(x,y), where z = x+iy

Solution:

z = x + iy so �ത = x - iy

Re(z) = x Im(z) = y

f(x) = (x – iy)y + (x + iy)x + (x + iy) (x – iy) f(x) = xy – iy2 + x2 + ixy + x2 – i2 y2 f(x) = xy – iy2 + x2 + ixy + x2 + y2 f(x) = 2x2 + y2 + xy – iy2 + ixy u(x ,y) = 2x2 + y2 + xy iv(x ,y) = y2 + xy

Find the 4th roots of -1

Solution: Z0 = -1

Z0 = -1 + i So x = -1 and y = 0