What is (x+iy-1)/(x+iy+1) ?

The solution to (x+iy-1)/(x+iy+1) is ((x-1)(x+1)+y^2)/((x+1)^2+y^2)+(2y)/((x+1)^2+y^2)i

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simplify (x+iy-1)/(x+iy+1)

simplify

\frac{x + i y - 1}{x + i y + 1}

\frac{( x - 1 ) ( x + 1 ) + y ^{2}}{( x + 1 ) ^{2} + y ^{2}} + \frac{2 y}{( x + 1 ) ^{2} + y ^{2}} i

Solution steps

\frac{x + i y - 1}{x + i y + 1}

Rewrite

x + i y - 1

in standard complex form:

(x - 1) + y i

Rewrite

x + i y + 1

in standard complex form:

(x + 1) + y i

= \frac{( x - 1 ) + y i}{( x + 1 ) + y i}

Multiply by the conjugate

\frac{( x + 1 ) - y i}{( x + 1 ) - y i}

= \frac{( ( x - 1 ) + y i ) ( ( x + 1 ) - y i )}{( ( x + 1 ) + y i ) ( ( x + 1 ) - y i )}

Simplify

\frac{( ( x - 1 ) + y i ) ( ( x + 1 ) - y i )}{( ( x + 1 ) + y i ) ( ( x + 1 ) - y i )} : \frac{( x - 1 ) ( x + 1 ) + y ^{2} + 2 y i}{( x + 1 ) ^{2} + y ^{2}}

= \frac{( x - 1 ) ( x + 1 ) + y ^{2} + 2 y i}{( x + 1 ) ^{2} + y ^{2}}

Rewrite in standard complex form:

\frac{( x - 1 ) ( x + 1 ) + y ^{2}}{( x + 1 ) ^{2} + y ^{2}} + \frac{2 y}{( x + 1 ) ^{2} + y ^{2}} i

= \frac{( x - 1 ) ( x + 1 ) + y ^{2}}{( x + 1 ) ^{2} + y ^{2}} + \frac{2 y}{( x + 1 ) ^{2} + y ^{2}} i

3 + \frac{1}{2} - \frac{3}{4}

x^{2} + 11 x + \frac{121}{4} = \frac{125}{4}

ab + a c

3 d > 9

6 x - 18 = 0

What is (x+iy-1)/(x+iy+1) ?
The solution to (x+iy-1)/(x+iy+1) is ((x-1)(x+1)+y^2)/((x+1)^2+y^2)+(2y)/((x+1)^2+y^2)i