4 If f is the complex function defined by fz 1z 1 for z do

4. If f is the complex function defined by f(z) = 1/z + 1 for z does not equal 1, determine real functions u and v such that f = u + iv . Then find f(2 i).

Solution

Solution : The complex no : z= x+y+iy

z+1= x+iy+1 = (x+1) +iy

f(z) = 1/ (z+1) = [ 1 / ( ( x+1) +iy ) ] [ ( x+1-iy) / (x+1- iy) ]

= ( x+1-iy) / [ (x+1)²+y²]

u+iv = [ (x+1) /{(x+1)²+y²}] - iy / { (x+1)²+y²}

equating the real and imaginary parts we get

u = (x+1) / { (x+1)²+y²} , v = - y / { (x+1)²+y²

to find f( 2- i) = 1/ (z+1) = 1 /[ ( 2-i) +1]

= 1/ ( 3-i) = (3+i) / (3²+1) = (3+i) /10 = 3/10 + i /10