What is 1/((wi+0.5)^2) ?

The solution to 1/((wi+0.5)^2) is (-w^2+0.25)/(w^4+0.5w^2+0.0625)-w/(w^4+0.5w^2+0.0625)i

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simplify 1/((wi+0.5)^2)

simplify

\frac{1}{( w i + 0.5 ) ^{2}}

\frac{- w ^{2} + 0.25}{w ^{4} + 0.5 w ^{2} + 0.0625} - \frac{w}{w ^{4} + 0.5 w ^{2} + 0.0625} i

Solution steps

\frac{1}{( w i + 0.5 ) ^{2}}

Expand

(w i + 0.5)^{2} : - w^{2} + w i + 0.25

= \frac{1}{- w ^{2} + w i + 0.25}

Rewrite

- w^{2} + w i + 0.25

in standard complex form:

(- w^{2} + 0.25) + w i

= \frac{1}{( - w ^{2} + 0.25 ) + w i}

Multiply by the conjugate

\frac{( - w ^{2} + 0.25 ) - w i}{( - w ^{2} + 0.25 ) - w i}

= \frac{1 \cdot ( ( - w ^{2} + 0.25 ) - w i )}{( ( - w ^{2} + 0.25 ) + w i ) ( ( - w ^{2} + 0.25 ) - w i )}

Simplify

\frac{1 \cdot ( ( - w ^{2} + 0.25 ) - w i )}{( ( - w ^{2} + 0.25 ) + w i ) ( ( - w ^{2} + 0.25 ) - w i )} : \frac{- w ^{2} + 0.25 - w i}{w ^{4} + 0.5 w ^{2} + 0.0625}

= \frac{- w ^{2} + 0.25 - w i}{w ^{4} + 0.5 w ^{2} + 0.0625}

Rewrite in standard complex form:

\frac{- w ^{2} + 0.25}{w ^{4} + 0.5 w ^{2} + 0.0625} - \frac{w}{w ^{4} + 0.5 w ^{2} + 0.0625} i

= \frac{- w ^{2} + 0.25}{w ^{4} + 0.5 w ^{2} + 0.0625} - \frac{w}{w ^{4} + 0.5 w ^{2} + 0.0625} i

(\frac{x}{2}) + 3

e^{x + 2} (2 x + 2)

\frac{( x + 5 )}{( x + 2 ) ( x + 3 ) ( x + 4 )}

β e (4 - θ^{2})^{2} - 2 t^{2} e

2 x 2 + 7 x - 15

What is 1/((wi+0.5)^2) ?
The solution to 1/((wi+0.5)^2) is (-w^2+0.25)/(w^4+0.5w^2+0.0625)-w/(w^4+0.5w^2+0.0625)i