Question

How to find real part of complex number of complex number α ?

If `k>0`,`|z|=|w|=k` and `alpha=(z-bar w)/(k^2+zbarw)`, then find the value of `Re(alpha)`.

Answer 1

The other two people did a much better job of answering this question than I. Any attempt to fix this answer would result in plagiarism of their answers. Therefore, I recommend that you read their answers.

Answer 2

See below.

Explanation:

Given a complex number `z` and being `bar z` it's conjugate then

`"Re"(z) = 1/2(z+bar z)`
`"Im"(z)=1/(2i)(z-bar z)`

then

`"Re"(alpha) = 1/2((z-bar w)/(k^2+z bar w)+(bar z-w)/(k^2+bar z w))=0`

`"Im"(alpha) = 1/(2i)((z-bar w)/(k^2+z bar w)-(bar z-w)/(k^2+bar z w)) = 1/(2i)((2k^2(z-barz+w-barw))/(2k^4+k^2(z bar w+bar z w))) = 1/(2i)((2(z-barz+w-barw))/(2k^2+(z bar w+bar z w)))`

now assuming that `z = k e^(i phi)` and `w = k e^(i psi)` we have

`"Im"(alpha) = 1/(2i)((2k(e^(i phi)-e^(-i phi)+e^(i phi)-e^(-ipsi)))/(2k^2+k^2(e^(i(phi-psi))-e^(-i(phi-psi)))))=`

`=1/(ik)((e^(i phi)-e^(-i phi)+e^(i phi)-e^(-ipsi))/(2+(e^(i(phi-psi))+e^(-i(phi-psi))))) = 1/(ik)((2i(sin phi+sin psi))/(2+2cos (phi-psi))) = 1/k((sin phi+sin psi)/(1+cos(phi-psi)))`

Answer 3

`Re(alpha)=0`

Explanation:

Suppose `z=x+iy` and `w=u+iv` where `x,y,u,v in RR`

`\ |z|=k => x^2+y^2=k^2` ..... [A]
`|w|=k => u^2+v^2=k^2`

Then, given the definition of `alpha`, we have:

`alpha = (z - bar w)/(k^2+zbarw)`

`\ \ \ = ((x+iy) - (u-iv))/(k^2+(x+iy)(u-iv))`

`\ \ \ = (x+iy - u+iv) / (k^2+(xu-ivx+iuy-i^2vy))`

`\ \ \ = (x-u+(v+y)i) / ( (k^2+xu+vy)+(uy-vx)i )`

Now we multiply numerator and denominator by the complex conjugate of the denominator to rationalize it:

`alpha = (x-u+(v+y)i) / ( (k^2+xu+vy)+(uy-vx)i ) * ( (k^2+xu+vy)-(uy-vx)i ) / ( (k^2+xu+vy)-(uy-vx)i )`

And if we expand we get:

`alpha = (-k^2u + i k^2 v + k^2 x + i k^2 y - u^2 x + i u^2 y + u x^2 + u y^2 - v^2 x + i v^2 y + i v x^2 + i v y^2) / (k^4 + 2 k^2 u x + 2 k^2 v y + u^2 x^2 + u^2 y^2 + v^2 x^2 + v^2 y^2)`

`\ \ \ = (-k^2u + i k^2 v + k^2 x + i k^2 y - u^2 x + i u^2 y + u x^2 + u y^2 - v^2 x + i v^2 y + i v x^2 + i v y^2) / (k^4 + 2 k^2 u x + 2 k^2 v y + u^2 x^2 + u^2 y^2 + v^2 x^2 + v^2 y^2)`

So if we separate out the real component of this expression we have:

`Re(alpha) = (-k^2u + k^2 x - u^2 x + u x^2 + u y^2 - v^2 x) / (k^4 + 2 k^2 u x + 2 k^2 v y + u^2 x^2 + u^2 y^2 + v^2 x^2 + v^2 y^2)`

Now we use the earlier result [A]:

`Re(alpha) = (-k^2u + k^2 x - x(u^2+v^2) + u( x^2 + u y^2) ) / (k^4 + 2 k^2 u x + 2 k^2 v y + u^2( x^2 + y^2) + v^2 (x^2 + y^2))`

`\ \ \ \ \ \ \ \ \ \ = (-k^2u + k^2 x - k^2 x + k^2u ) / (k^4 + 2 k^2 u x + 2 k^2 v y + k^2u^2 + k^2v^2 )`

`\ \ \ \ \ \ \ \ \ \ = (0) / (k^4 + 2 k^2 u x + 2 k^2 v y + k^2u^2 + k^2v^2 )`

`\ \ \ \ \ \ \ \ \ \ = 0`