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

The solution to 1/((x+iy)^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+iy)^2)

simplify

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

Expand

(x + i y)^{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

e^{2 t} u (- t + 2)

tan^{2} (θ) (cot^{2} (θ) - cos^{2} (θ))

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

\frac{2}{( x - 1 ) ^{3}}

(x - 23) (x + 23)

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