What is expand i/((x+iy)^2) ?

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

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expand i/((x+iy)^2)

expand

\frac{i}{( x + i 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}}

Solution steps

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

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

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

Rationalize

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

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

Rewrite

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

in standard complex form:

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

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

4 x (x + 3) (x - 3)

(x + \frac{1}{4} y)^{5}

(a - b - c - d - e)^{2}

(t + 3)^{2} d t

sec^{2} (x) + tan (x) (sin (x) - tan (x))

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