What is expand ((r+ix)-1)/((r+ix)+1) ?

The solution to expand ((r+ix)-1)/((r+ix)+1) is (-1+r^2+x^2)/((1+r)^2+x^2)+i(2x)/((1+r)^2+x^2)

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expand ((r+ix)-1)/((r+ix)+1)

expand

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

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

Solution steps

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

Remove parentheses:

(a) = a

= \frac{r + i x - 1}{r + i x + 1}

Rationalize

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

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

Rewrite

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

in standard complex form:

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

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

\frac{( 1 + 2 x ) ^{4}}{2}

(1 + 2 e^{i t})^{3}

5 sec (\frac{π}{4} (x + 1))

\frac{18}{23} - \frac{1}{23}

\frac{x ( - x y ^{2} - x ^{2} y + 32 x )}{( y + x ) ^{2}}

What is expand ((r+ix)-1)/((r+ix)+1) ?
The solution to expand ((r+ix)-1)/((r+ix)+1) is (-1+r^2+x^2)/((1+r)^2+x^2)+i(2x)/((1+r)^2+x^2)