# Complex functions

1. Jul 23, 2007

### pivoxa15

1. The problem statement, all variables and given/known data
Find the complex function of z^(1/2))=(x+iy)^(1/2)

3. The attempt at a solution
The first step is z^(1/2)=e^((1/2)ln(z))=e^(1/2)[(ln|z|+i(theta)+2((pi)n)]

But the answers were not in this form.

2. Jul 23, 2007

### Gib Z

What is your question asking? What do you mean by, find the complex function...

3. Jul 23, 2007

### CompuChip

Do you want to extend the square root to the complex plane?
And what form were the answers in? Perhaps they are the same (just written down differently)?

Last edited: Jul 23, 2007
4. Jul 23, 2007

### Kummer

The definition of $$a^b$$ for $$a,b\in \mathbb{C}$$ and $$a\not =0$$ is defined as $$\exp (b\ln a)$$.

So, $$z^{1/2} = \exp \left( \frac{\log z}{2} \right) = \exp \left( \frac{\ln |z|}{2} + i\cdot \frac{\arg z}{2} \right) = \sqrt{|z|}\cdot e^{i\arg(z)/2}$$

5. Jul 24, 2007

### pivoxa15

Sorry, just to clarify the question is asking to find u(x,y) and v(x,y) where z^(1/2)=u(x,y)+iv(x,y) where z=x+iy.

6. Jul 24, 2007

### CompuChip

That's called the real and imaginary part.
Recall Euler's identity
$$e^{i \phi} = \cos \phi + i \sin\phi.$$
Will that do?

7. Jul 24, 2007

### Kummer

$$\sqrt{|z|}\cos \left( \frac{\arg z}{2} \right) + i \sqrt{|z|}\sin \left( \frac{\arg z}{2} \right)$$