A complex functionfz - u- iv is defined as fz - z-iz-1 where z- x-iy then the imaginary part of fz is

F(x+iy) = x+iy ix+iy+1 = [x+i(y 1)][x+1 iy](x+1)^2+y^2 = x^2+y^2+x y+i(y x 1)x^2+y^2+2x+1 Imag (f(x+iy)) = y x 1x^2+y^2+2x+1 ...

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Question

A complex function
$f (z) = u + i v$ is defined as $f (z) = \frac{z - i}{z + 1}$ where $z = x + i y$ then the imaginary part of f(z) is

\frac{x^{2} + y^{2} + x - y}{x^{2} + y^{2} + 2 x + 1}

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\frac{y - x - 1}{x^{2} + y^{2} + 2 x + 1}

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\frac{- 1}{\sqrt{x^{2} + y^{2}} + 1}

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\frac{\sqrt{x^{2} + y^{2}}}{\sqrt{x^{2} + y^{2}} + 1}

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Solution

The correct option is B $\frac{y - x - 1}{x^{2} + y^{2} + 2 x + 1}$
$f (x + i y) = \frac{x + i y - i}{x + i y + 1}$
$= \frac{[x + i (y - 1)] [x + 1 - i y]}{(x + 1)^{2} + y^{2}} = \frac{x^{2} + y^{2} + x - y + i (y - x - 1)}{x^{2} + y^{2} + 2 x + 1}$
$I m a g (f (x + i y)) = \frac{y - x - 1}{x^{2} + y^{2} + 2 x + 1}$

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A complex function f(z)=u+iv is defined as f(z)=z−iz+1 where z=x+iy then the imaginary part of f(z) is

The correct option is B y−x−1x2+y2+2x+1f(x+iy)=x+iy−ix+iy+1 =[x+i(y−1)][x+1−iy](x+1)2+y2=x2+y2+x−y+i(y−x−1)x2+y2+2x+1 Imag(f(x+iy))=y−x−1x2+y2+2x+1

A complex function
$f (z) = u + i v$ is defined as $f (z) = \frac{z - i}{z + 1}$ where $z = x + i y$ then the imaginary part of f(z) is