A complex rational equation I'm having trouble with

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October 18th 2007, 09:41 PM
saywhat

A complex rational equation I'm having trouble with

I've posted before about my fractal project, and this is something else that is related. The function:

$f(z)=\frac{1-z}{(2-z)^{\frac{3}{2}}}$ where $|z|=1$ .
The derivative along this line:

$f'(z)=-\frac{ie^{it}(e^{it}+1)\sqrt{2-e^{it}}}{2(2-e^{it})^{3}}$ for a new real variable $t$ .
I'm trying to find when the real and/or imaginary parts equal zero.

$\Re\{f'\}=0$
$\Im\{f'\}=0$
Is there any other way to solve these equations without splitting $f'$ into $x'$ and $iy'$ just because it is so messy and long? I know one method of removing the imaginary terms from the bottom by multiplying top and bottom by $(\frac{1}{2}-e^{it})^3$ , factoring $e^{3it}$ out of the denominator, and multiplying numerator and denominator by $\frac{1}{e^{3it}}$ resulting in a denominator of $4\cos t-5$ . It's still messy with a numerator of $-\frac{i}{e^{2it}}(e^{it}+1)(\frac{1}{2}-e^{it})^3\sqrt{2-e^{it}}$ but now, algebraically, the worst part is $\sqrt{2-e^{it}}$ . I'd appreciate any suggestions.
October 19th 2007, 04:46 AM
topsquark

Quote:

Originally Posted by saywhat

I've posted before about my fractal project, and this is something else that is related. The function:
$f(z)=\frac{1-z}{(2-z)^{\frac{3}{2}}}$ where $|z|=1$ .
The derivative along this line:
$f'(z)=-\frac{ie^{it}(e^{it}+1)\sqrt{2-e^{it}}}{2(2-e^{it})^{3}}$ for a new real variable $t$ .
I'm trying to find when the real and/or imaginary parts equal zero.
$\Re\{f'\}=0$
$\Im\{f'\}=0$
Is there any other way to solve these equations without splitting $f'$ into $x'$ and $iy'$ just because it is so messy and long? I know one method of removing the imaginary terms from the bottom by multiplying top and bottom by $(\frac{1}{2}-e^{it})^3$ , factoring $e^{3it}$ out of the denominator, and multiplying numerator and denominator by $\frac{1}{e^{3it}}$ resulting in a denominator of $4\cos t-5$ . It's still messy with a numerator of $-\frac{i}{e^{2it}}(e^{it}+1)(\frac{1}{2}-e^{it})^3\sqrt{2-e^{it}}$ but now, algebraically, the worst part is $\sqrt{2-e^{it}}$ . I'd appreciate any suggestions.

The only thing I can think of is this:
$\sqrt{2 - e^{it}} = \sqrt{2 - cos(t) - i~sin(t)} = \sqrt{2}\sqrt{1 - \frac{1}{2}cos(t) - \frac{i}{2}sin(t)}$ $= \sqrt{2} \left ( 1 - \frac{1}{2}cos(t) - \frac{i}{2}sin(t) \right )^{1/2}$

And then apply
$(cos(\theta) + i~sin(\theta))^n = cos(n\theta) + i~sin(n \theta)$

But it's still going to be pretty messy.

-Dan
October 19th 2007, 08:37 AM
CaptainBlack

Quote:

Originally Posted by saywhat

I've posted before about my fractal project, and this is something else that is related. The function:
$f(z)=\frac{1-z}{(2-z)^{\frac{3}{2}}}$ where $|z|=1$ .
The derivative along this line:
$f'(z)=-\frac{ie^{it}(e^{it}+1)\sqrt{2-e^{it}}}{2(2-e^{it})^{3}}$ for a new real variable $t$ .
I'm trying to find when the real and/or imaginary parts equal zero.
$\Re\{f'\}=0$
$\Im\{f'\}=0$
Is there any other way to solve these equations without splitting $f'$ into $x'$ and $iy'$ just because it is so messy and long? I know one method of removing the imaginary terms from the bottom by multiplying top and bottom by $(\frac{1}{2}-e^{it})^3$ , factoring $e^{3it}$ out of the denominator, and multiplying numerator and denominator by $\frac{1}{e^{3it}}$ resulting in a denominator of $4\cos t-5$ . It's still messy with a numerator of $-\frac{i}{e^{2it}}(e^{it}+1)(\frac{1}{2}-e^{it})^3\sqrt{2-e^{it}}$ but now, algebraically, the worst part is $\sqrt{2-e^{it}}$ . I'd appreciate any suggestions.

As when $e^{it}$ goes to $2$ , $\left| f'(z)|_{z=t} \right|$ goes to $\infty$ , we know that this
des not correspond to a root, so we can can multiply through by
$2(2-e^{it})^{2.5}$ to get for a root $t$ :

$<br /> ie^{it}(e^{it}+1)=0<br />$

so we need: $e^{it}=-1$ , or $t=\pi$ (single value as t is parameterizing the unit circle).

RonL
October 19th 2007, 02:59 PM
saywhat

To Dan, thanks. I have gotten thus far before in other equations and have used a formula to separate into x and y, but it is so large and not worth it.
To RonL, I should have specified, I'm looking for the intercept of the real axis and the intercept of the imaginary axis. That point is the cusp of the curve $f(z)$ . Thanks anyway.

I stumbled upon the identity

$\Im \{z\}=\frac{z-\bar z}{2}$
and because $f'(t)$ is periodic

$\overline{f'(t)}=-f'(-t)$
leading to $\Im \{f'(t)\}=\frac{f'(t)+f'(-t)}{2}=0$ and thus I need to solve $f'(t)=-f'(-t)$ , which hopefully won't call for me to find x and y.
The other equation $\Re \{f'\}=0$ can also be done in the same manor. I'll tell you how this goes and thanks again.

Oh and a note, I substituted $z=e^{it}$ . I should have said that in the original post.