angle of rotation

[logistic_guy]

Determine the angle of rotation at the point $\displaystyle z_{0} = 2+i$ when $\displaystyle w = z^2$, and illustrate it for some particular curve. Show that the scale factor at that point is $\displaystyle 2\sqrt{5}$.

 

[khansaheb]

Determine the angle of rotation at the point $\displaystyle z_{0} = 2+i$ when $\displaystyle w = z^2$, and illustrate it for some particular curve. Show that the scale factor at that point is $\displaystyle 2\sqrt{5}$.

show us your effort/s to solve this problem.

 

[logistic_guy]

show us your effort/s to solve this problem.

I will let $\displaystyle f(z) = w = z^2$

then, the angle of rotation is:

$\displaystyle \theta = \text{arg}\left(\frac{df}{dz}\bigg|_{z_0}\right) = \text{arg}\left(\frac{d}{dz}z^2\bigg|_{z_0}\right) = \text{arg}\left(2z\bigg|_{z_0}\right) = \text{arg}(2z_0) = \text{arg}[ \ 2(2 + i) \ ]$

$\displaystyle = \tan^{-1}\frac{2}{4} = \tan^{-1}\frac{1}{2} \approx 0.4636 \ \text{rad} \approx 26.6^{\circ}$

And the factor is:

$\displaystyle \left|\left(\frac{df}{dz}\bigg|_{z_0}\right)\right| = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}$

I will continue in the next post.