# Complex analysis

1. Jun 6, 2014

### DotKite

1. The problem statement, all variables and given/known data

Let f(z) = sqrt(z) be the branch of the square root function with sqrt(z) = (r^1/2) (e^iΘ/2),
0≤Θ<2$\pi$, r > 0

(a) for what values of z is sqrt(z^2) = z?

(b) Which part of the complex plane stretches, and which part shrinks under this transformation?

2. Relevant equations

3. The attempt at a solution

Ok so for this branch i believe the function will map all points within 0≤Θ<2$\pi$ to the upper half plane (ie Im(f) > 0).

I do not really understand what part a is asking and for part b it seems everything is shrunk.

2. Jun 7, 2014

### Simon Bridge

For part (a), it is asking for $z:f(z^2)=z$
For part (b) please show your reasoning. What does it mean to say that the complex plane has shrunk or stretched? How would you tell?

3. Jun 7, 2014

### DotKite

For part a do we have to consider the principal nth root? Or particular branches of the square root function?

In otherwords where this function is not multivalued?

4. Jun 7, 2014

### micromass

Staff Emeritus
No, the function is not multivalued. Your OP has given the branch which you should consider:

$$f(re^{i\theta}) = \sqrt{r} e^{i\theta/2}$$

where $0\leq \theta< 2\pi$. That last restriction on $\theta$ makes sure it's not multivalued.

5. Jun 7, 2014

### DotKite

then wouldnt part a be true for all z then? It seems obvious that sqrt(z^2) = z for all z. Why would that not be the case?

6. Jun 7, 2014

### micromass

Staff Emeritus
Can you prove it?

7. Jun 7, 2014

### pasmith

Let $z = e^{i3\pi/2}$. Then
$$z^2 = (e^{i3\pi/2})^2 = e^{i3\pi} = e^{i\pi}$$ since we need $0 \leq \arg(z^2) < 2\pi$. But then
$$f(z^2) = (e^{i\pi})^{1/2} = e^{i\pi/2} \neq z.$$
That's one $z$ for which $f(z^2) \neq z$. Are there others?

8. Jun 7, 2014

### DotKite

It seems the equation fails for values of z where when you square them the argument is outside of 0 to 2pi

9. Jun 7, 2014

### lurflurf

^Good which z are those? What can you say about their real and imaginary parts?

For (b) Suppose we have two nearby points so that d(P1,P2)=h
what can we say about d(sqrt(P1),sqrt(P2))?
which is bigger? What is the formula for distance? (you could also consider areas)

10. Jun 7, 2014

### DotKite

For part a it would be all z such that 0≤arg(z)≤pi. Therefore im(z) > 0.

Is the distance formula just the standard Euklidian distance formula?

11. Jun 8, 2014

### lurflurf

^yes consider a short segment in the complex plane
If the length is h say what is the length after taking square root
Where in the plane does a segment stretch and where does it shrink?
Hint find where the length does not change. That region will separate the other two.

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