Solved: TEAM PROJECT. Square Root. (a) Show that w = ~ has

Chapter 13, Problem 13.1.46

(choose chapter or problem)

Square Root. (a) Show that \(w=\sqrt{z}\) has the values

\(w_{1}=\sqrt{r}\left[\cos \frac{\theta}{2}+i \sin \frac{\theta}{2}\right]\)

(18) \(w_{2}=\sqrt{r}\left[\cos \left(\frac{\theta}{2}+\pi\right)+i \sin \left(\frac{\theta}{2}+\pi\right)\right]\)

\(=-w_{1}\)

(b) Obtain from (18) the often more practical formula

(19) \(\sqrt{z}=\pm\left[\sqrt{\frac{1}{2}(|z|+x)}+(\operatorname{sign} y) i \sqrt{\frac{1}{2}(|z|+x)}\right]\)

where sign y = 1 if \(y \geqq 0\), sign y = - 1 if y < 0, and all square roots of positive numbers are taken with positive sign. Hint: Use (10) in App. A3.1 with \(x=\theta / 2\).

(c) Find the square roots of 4i, 16 - 30i, and \(9+8 \sqrt{7} i\) by both (18) and (19) and comment on the work involved.

(d) Do some further examples of your own and apply a method of checking your results.

Text Transcription:

w = sqrt{z}

w_{1} = sqrt{r} [cos theta / 2 + i sin theta / 2]

w_2 = sqrt{r} [cos (theta / 2 + pi) + i sin (theta / 2 + pi)]

= -w_1

sqrt{z} = pm[sqrt{1 / 2 (|z| + x) + sign y) i sqrt{1 / 2(|z| + x)}]

y geqq 0

x = theta/2