Plotting with complex numbers

Hi, I'm trying to plot a function like (2/3)Z + 1/(3Z^2), where Z is complex.

I understand the basic idea of Z^2 = (x+iy)*(x+iy)= x^2+ 2xyi - y^2 is represented as x^2 - y^2 in the real direction, and 2xy in the imaginary direction.

For the function I'm concerned with, however, there does not seem to be a way to break it into real and complex directions.

Please let me know if there needs to be further clarification.

Thank you!

Re: Plotting with complex numbers

Quote:

Originally Posted by

**Aesun** Hi, I'm trying to plot a function like (2/3)Z + 1/(3Z^2), where Z is complex.

I understand the basic idea of Z^2 = (x+iy)*(x+iy)= x^2+ 2xyi - y^2 is represented as x^2 - y^2 in the real direction, and 2xy in the imaginary direction.

For the function I'm concerned with, however, there does not seem to be a way to break it into real and complex directions.

Please let me know if there needs to be further clarification.

Thank you!

$\displaystyle (2/3)Z + \frac{1}{3Z^2}=(2/3)Z + \frac{\overline{Z}^2}{3|Z|^2}$

Now split the right hand side into real and imaginary parts and plot.

CB

Re: Plotting with complex numbers

CaptainBlack, thank you for your help! Unfortunately I still don't see how to continue. Would you mind showing me a few more algebraic steps? I still can't seem to separate the real and imaginary parts.

Thank you!

Re: Plotting with complex numbers

Quote:

Originally Posted by

**Aesun** CaptainBlack, thank you for your help! Unfortunately I still don't see how to continue. Would you mind showing me a few more algebraic steps? I still can't seem to separate the real and imaginary parts.

Thank you!

$\displaystyle (2/3)Z + \frac{1}{3Z^2}=(2/3)Z + \frac{\overline{Z}^2}{3|Z|^2}$

Put:

$\displaystyle Z=x+iy$

then:

$\displaystyle \overline{Z}=x-iy$

and:

$\displaystyle \overline{Z}^2=x^2-2ixy-y^2=(x^2-y^2)-2ixy$

and:

$\displaystyle |Z|^2 = x^2+y^2$

Hence:

$\displaystyle (2/3)Z + \frac{1}{3Z^2}=(2/3)Z + \frac{\overline{Z}^2}{3|Z|^2} \\ \\ \\ \phantom{SSSSS}=\left[(2/3)x+\frac{x^2-y^2}{3(x^2+y^2)}\right]+\left[(2/3)y-\frac{2xy}{3(x^2+y^2)}\right]i$

Now check the algebra for mistakes.

CB

Re: Plotting with complex numbers

Thank you so much! I'm new to working with complex numbers, so this was extremely helpful.