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

The solution to x+iy+1/(x+iy) is (x+x^3+xy^2)/(x^2+y^2)+i(-y+y^3+x^2y)/(x^2+y^2)

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simplify x+iy+1/(x+iy)

simplify

x + i y + \frac{1}{x + i y}

\frac{x + x ^{3} + x y ^{2}}{x ^{2} + y ^{2}} + i \frac{- y + y ^{3} + x ^{2} y}{x ^{2} + y ^{2}}

Solution steps

x + i y + \frac{1}{x + i y}

Convert element to fraction:

x = \frac{x ( x + i y )}{x + i y}, i y = \frac{i y ( x + i y )}{x + i y}

= \frac{1}{x + i y} + \frac{x ( x + i y )}{x + i y} + \frac{i y ( x + i y )}{x + i y}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 + x ( x + i y ) + i y ( x + i y )}{x + i y}

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

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

Remove parentheses:

(- a) = - a

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

Expand

1 + x (x + i y) - y^{2} + y x i : x^{2} + 2 i x y - y^{2} + 1

= \frac{x ^{2} + 2 i x y - y ^{2} + 1}{x + i y}

Rationalize

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

= \frac{x ^{3} + x y ^{2} + x + i ( y ^{3} + x ^{2} y - y )}{x ^{2} + y ^{2}}

Rewrite

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

in standard complex form:

\frac{x ^{3} + x y ^{2} + x}{x ^{2} + y ^{2}} + \frac{y ^{3} + x ^{2} y - y}{x ^{2} + y ^{2}} i

= \frac{x ^{3} + x y ^{2} + x}{x ^{2} + y ^{2}} + \frac{y ^{3} + x ^{2} y - y}{x ^{2} + y ^{2}} i

2 + 2 s > 8

\frac{3 a ^{- 1} + 2 b ^{- 2}}{a ^{- 1} - b ^{- 1}}

- 6 q + 1 \geq 2 q - 10

3 m^{2} \leq 9 m

12 - 5 (x + 3) = 2 x - 5

What is x+iy+1/(x+iy) ?
The solution to x+iy+1/(x+iy) is (x+x^3+xy^2)/(x^2+y^2)+i(-y+y^3+x^2y)/(x^2+y^2)