What is 1/((x+yi)^2) ?

The solution to 1/((x+yi)^2) is (x^2-y^2)/(x^4+y^4+2x^2y^2)-(2xy)/(x^4+y^4+2x^2y^2)i

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simplify 1/((x+yi)^2)

simplify

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

\frac{x ^{2} - y ^{2}}{x ^{4} + y ^{4} + 2 x ^{2} y ^{2}} - \frac{2 x y}{x ^{4} + y ^{4} + 2 x ^{2} y ^{2}} i

Solution steps

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

Expand

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

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

Rewrite

x^{2} + 2 x y i - y^{2}

in standard complex form:

(x^{2} - y^{2}) + 2 x y i

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

Multiply by the conjugate

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

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

Simplify

\frac{1 \cdot ( ( x ^{2} - y ^{2} ) - 2 x y i )}{( ( x ^{2} - y ^{2} ) + 2 x y i ) ( ( x ^{2} - y ^{2} ) - 2 x y i )} : \frac{x ^{2} - y ^{2} - 2 x y i}{x ^{4} + y ^{4} + 2 x ^{2} y ^{2}}

= \frac{x ^{2} - y ^{2} - 2 x y i}{x ^{4} + y ^{4} + 2 x ^{2} y ^{2}}

Rewrite in standard complex form:

\frac{x ^{2} - y ^{2}}{x ^{4} + y ^{4} + 2 x ^{2} y ^{2}} - \frac{2 x y}{x ^{4} + y ^{4} + 2 x ^{2} y ^{2}} i

= \frac{x ^{2} - y ^{2}}{x ^{4} + y ^{4} + 2 x ^{2} y ^{2}} - \frac{2 x y}{x ^{4} + y ^{4} + 2 x ^{2} y ^{2}} i

\frac{( x - 1 ) ^{3}}{3}

2 (x + Δ x)

(x + \frac{1}{2}) (x - \frac{3}{4})

\frac{\frac{d}{d x} ( ( x ^{2} - 1 ) ^{2} )}{( 5 x + 8 ) ^{2}}

\frac{x - 2}{( x - 1 ) ^{2}}

What is 1/((x+yi)^2) ?
The solution to 1/((x+yi)^2) is (x^2-y^2)/(x^4+y^4+2x^2y^2)-(2xy)/(x^4+y^4+2x^2y^2)i