What are the real and imaginary parts of f(z) = z + (1/z) where z is a complex number expressed in rectangular form as x + iy ?
Ron.
Ron.
Real or Imaginary?
- Thread starter SydneyRoverP6B
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darth sidious
New Member
SydneyRoverP6B said:
1/(x + jy) = [1/(x + jy)] x [(x - jy)/(x - jy)] = (x - jy)/(x^2 + y^2)
(x -jy) is the complex conjugate of (x + jy), and (x + jy) x (x - jy) = (x^2 + y^2) ==> ... a purely real number.
(x + jy) + (x - jy)/(x^2 + y^2) = [x + x/(x^2 + y^2)] + j[y - y/(x^2 + y^2)] = x[1 + 1/(x^2 + y^2)] + jy[1 - 1/(x^2 + y^2)]
= x[(x^2 + y^2 + 1)/(x^2 + y^2)] + jy ((x^2 + y^2 - 1)/(x^2 + y^2)]
The real part is x[(x^2 + y^2 + 1)/(x^2 + y^2)] = [(x^3 + xy^2 + x)/(x^2 + y^2)]
The imaginary part is y [(x^2 + y^2 - 1)/(x^2 + y^2)] = [(yx^2 + y^3 - y)/(x^2 + y^2)]
Hello Darth,
A nice and easy one...
Your answer is correct, although just one thing,...when you expanded your factored answers, in the case of the real part you left the x off the y^2 term, and the imaginary part, the y on the x^2 term is missing. No doubt testing me....
Glad you enjoyed it!!
Ron.
A nice and easy one...
Your answer is correct, although just one thing,...when you expanded your factored answers, in the case of the real part you left the x off the y^2 term, and the imaginary part, the y on the x^2 term is missing. No doubt testing me....
Glad you enjoyed it!!
Ron.
darth sidious
New Member
Of course! I'm glad you spotted my deliberate mistake... cough!