What is (iw+a)/((iw+a)^2+b^2) ?

The solution to (iw+a)/((iw+a)^2+b^2) is

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simplify (iw+a)/((iw+a)^2+b^2)

simplify

\frac{i w + a}{( i w + a ) ^{2} + b ^{2}}

\frac{a ^{3} + a w ^{2} + a b ^{2}}{w ^{4} + a ^{4} + b ^{4} - w ^{2} a ^{2} - w ^{2} b ^{2} + 3 a ^{2} w ^{2} + a ^{2} b ^{2} - b ^{2} w ^{2} + b ^{2} a ^{2}} + \frac{- w ^{3} + w b ^{2} - w a ^{2}}{w ^{4} + a ^{4} + b ^{4} - w ^{2} a ^{2} - w ^{2} b ^{2} + 3 a ^{2} w ^{2} + a ^{2} b ^{2} - b ^{2} w ^{2} + b ^{2} a ^{2}} i

Solution steps

\frac{i w + a}{( i w + a ) ^{2} + b ^{2}}

Expand

(i w + a)^{2} + b^{2} : - w^{2} + 2 a w i + a^{2} + b^{2}

= \frac{i w + a}{- w ^{2} + 2 a w i + a ^{2} + b ^{2}}

Rewrite

i w + a

in standard complex form:

a + w i

Rewrite

- w^{2} + 2 a w i + a^{2} + b^{2}

in standard complex form:

(- w^{2} + a^{2} + b^{2}) + 2 a w i

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

Multiply by the conjugate

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

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

Simplify

\frac{( a + w i ) ( ( - w ^{2} + a ^{2} + b ^{2} ) - 2 a w i )}{( ( - w ^{2} + a ^{2} + b ^{2} ) + 2 a w i ) ( ( - w ^{2} + a ^{2} + b ^{2} ) - 2 a w i )} : \frac{a ^{3} + a w ^{2} + a b ^{2} + ( - w ^{3} + w b ^{2} - w a ^{2} ) i}{w ^{4} + a ^{4} + b ^{4} - w ^{2} a ^{2} - w ^{2} b ^{2} + 3 a ^{2} w ^{2} + a ^{2} b ^{2} - b ^{2} w ^{2} + b ^{2} a ^{2}}

= \frac{a ^{3} + a w ^{2} + a b ^{2} + ( - w ^{3} + w b ^{2} - w a ^{2} ) i}{w ^{4} + a ^{4} + b ^{4} - w ^{2} a ^{2} - w ^{2} b ^{2} + 3 a ^{2} w ^{2} + a ^{2} b ^{2} - b ^{2} w ^{2} + b ^{2} a ^{2}}

Rewrite in standard complex form:

\frac{a ^{3} + a w ^{2} + a b ^{2}}{w ^{4} + a ^{4} + b ^{4} - w ^{2} a ^{2} - w ^{2} b ^{2} + 3 a ^{2} w ^{2} + a ^{2} b ^{2} - b ^{2} w ^{2} + b ^{2} a ^{2}} + \frac{- w ^{3} + w b ^{2} - w a ^{2}}{w ^{4} + a ^{4} + b ^{4} - w ^{2} a ^{2} - w ^{2} b ^{2} + 3 a ^{2} w ^{2} + a ^{2} b ^{2} - b ^{2} w ^{2} + b ^{2} a ^{2}} i

= \frac{a ^{3} + a w ^{2} + a b ^{2}}{w ^{4} + a ^{4} + b ^{4} - w ^{2} a ^{2} - w ^{2} b ^{2} + 3 a ^{2} w ^{2} + a ^{2} b ^{2} - b ^{2} w ^{2} + b ^{2} a ^{2}} + \frac{- w ^{3} + w b ^{2} - w a ^{2}}{w ^{4} + a ^{4} + b ^{4} - w ^{2} a ^{2} - w ^{2} b ^{2} + 3 a ^{2} w ^{2} + a ^{2} b ^{2} - b ^{2} w ^{2} + b ^{2} a ^{2}} i

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

\frac{7}{s ( s + 3 ) ( s + 2 )}

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

\frac{1}{s ( s + a ) ^{2}}

(\frac{3 i}{n})^{3}