What is 1/((1+iw)^2) ?

The solution to 1/((1+iw)^2) is (1-w^2)/(w^4+2w^2+1)-(2w)/(w^4+2w^2+1)i

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simplify 1/((1+iw)^2)

simplify

\frac{1}{( 1 + i w ) ^{2}}

\frac{1 - w ^{2}}{w ^{4} + 2 w ^{2} + 1} - \frac{2 w}{w ^{4} + 2 w ^{2} + 1} i

Solution steps

\frac{1}{( 1 + i w ) ^{2}}

Expand

(1 + i w)^{2} : 1 + 2 w i - w^{2}

= \frac{1}{1 + 2 w i - w ^{2}}

Rewrite

1 + 2 w i - w^{2}

in standard complex form:

(1 - w^{2}) + 2 w i

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

Multiply by the conjugate

\frac{( 1 - w ^{2} ) - 2 w i}{( 1 - w ^{2} ) - 2 w i}

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

Simplify

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

= \frac{1 - w ^{2} - 2 w i}{w ^{4} + 2 w ^{2} + 1}

Rewrite in standard complex form:

\frac{1 - w ^{2}}{w ^{4} + 2 w ^{2} + 1} - \frac{2 w}{w ^{4} + 2 w ^{2} + 1} i

= \frac{1 - w ^{2}}{w ^{4} + 2 w ^{2} + 1} - \frac{2 w}{w ^{4} + 2 w ^{2} + 1} i

32 x^{2} - 50

4 x^{5} - 5 x^{4}

\frac{4 π}{3} + π

53 \times 3

\frac{1}{x + 2} - \frac{1}{x}

What is 1/((1+iw)^2) ?
The solution to 1/((1+iw)^2) is (1-w^2)/(w^4+2w^2+1)-(2w)/(w^4+2w^2+1)i