If x and y are real what is the complex conjugate z when: x+iy+1x+iy? I tried to tidy up the 1x+iy to get x−iyx2−y2, multiply it out etc but still just no cigar.

Usually one takes z=x+jy. Do you want an expression for z⋅+1z⋅? In Cartesian coordinates, z=x+jy and z⋅=x−jy;1z⋅=x+jyx2+y2 →z⋅+1z⋅=x+x+xx2+y2+j(yx2+y2−y) In Polar coordinates, z=rejt=r(cos⁡(t)+jsin⁡(t)); z⋅=re−jt=r(cos⁡(t)−jsin⁡(t)); 1z⋅=(1r)ejt=(1r)(cos⁡(t)+jsin⁡(t)) and z⋅+1z⋅=(r+1r)cos⁡(t)+j(1r−r)sin⁡(t)

1x+iy≠x−iyx2−y2. Instead, 1x+iy=x−iyx2−(iy)2 =x−iyx2−i2y2 =x−iyx2+y2

Usually one takes z=x+jy. Do you want an expression for z⋅+1z⋅? In Cartesian coordinates, z=x+jy and z⋅=x−jy;1z⋅=x+jyx2+y2 →z⋅+1z⋅=x+x+xx2+y2+j(yx2+y2−y) In Polar coordinates, z=rejt=r(cos⁡(t)+jsin⁡(t)); z⋅=re−jt=r(cos⁡(t)−jsin⁡(t)); 1z⋅=(1r)ejt=(1r)(cos⁡(t)+jsin⁡(t)) and z⋅+1z⋅=(r+1r)cos⁡(t)+j(1r−r)sin⁡(t)

1x+iy≠x−iyx2−y2. Instead, 1x+iy=x−iyx2−(iy)2 =x−iyx2−i2y2 =x−iyx2+y2

Find out the answer to "If x and y are real what is..."

Carol Valentine

Answered question

2022-01-18

If x and y are real what is the complex conjugate z when: $ \frac{x + i y + 1}{x + i y}$ ? I tried to tidy up the $\frac{1}{x + i y}$ to get $\frac{x - i y}{x^{2} - y^{2}}$ , multiply it out etc but still just no cigar.

Answer & Explanation

Robert Pina

Beginner2022-01-19Added 42 answers

Usually one takes

z = x + j y

. Do you want an expression for

z \cdot + \frac{1}{z \cdot}

?
In Cartesian coordinates,

z = x + j y and z \cdot = x - j y; \frac{1}{z \cdot} = \frac{x + j y}{x^{2} + y^{2}}

\to z \cdot + \frac{1}{z \cdot} = x + \frac{x + x}{x^{2} + y^{2}} + j (\frac{y}{x^{2} + y^{2}} - y)

In Polar coordinates,

z = r e^{j t} = r (\cos (t) + j \sin (t))

;

z \cdot = r e^{- j t} = r (\cos (t) - j \sin (t))

;

\frac{1}{z} \cdot = (\frac{1}{r}) e^{j t} = (\frac{1}{r}) (\cos (t) + j \sin (t))

and

z \cdot + \frac{1}{z \cdot} = (r + \frac{1}{r}) \cos (t) + j (\frac{1}{r} - r) \sin (t)

Esta Hurtado

Beginner2022-01-20Added 39 answers

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

.
Instead,

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

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

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

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