# Homework Help: Analytic Complex Functions

1. Sep 27, 2010

### Matterwave

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

The following are pieces of analytic functions of the form f(z)=u(x,y)+iv(x,y). Find the missing pieces.

a) $$u(x,y)=x^2-y^2+x+2$$

Find v, and f.

b) $$v(x,y)=\sqrt{\frac{1}{2}(\sqrt{x^2+y^2}-x)}$$

Find u, and f.

c) $$f(z)=tan^{-1}(z)$$

Find u, v.

2. Relevant equations

3. The attempt at a solution

Ok, I have no idea what to even start with...I don't understand why if u is given, then v is somehow constricted... (or vice versa). What's to say that v can't be any random function? I get that if f is given, then obviously u and v are given (although even the third part completely eludes me).

Someone help me start this problem please?

BTW, the professor has posted solutions online, but I don't understand them at all, so I was hoping you guys could help.

2. Sep 27, 2010

### ╔(σ_σ)╝

Do you know what a harmonic conjugate is ? You need to know what it is to solve your problem.

A function is analytic on a domain iff u is the harmonic conjugate of v.

3. Sep 27, 2010

### Matterwave

Uh...I know what a complex conjugate is...I don't think I've ever heard of a harmonic conjugate...

4. Sep 27, 2010

### ╔(σ_σ)╝

Well open up your book or look it up online. If you still need help after that then post again.

5. Sep 27, 2010

### Matterwave

The professor said that the books were only recommended and not mandatory...so I thought that his lecture notes would cover everything...apparently they do not? >__> Ok wikipedia here I come...

6. Sep 27, 2010

### Quinzio

Go read Wikipedia on Complex Analisys.
In the first 15 rows or so there is a strong hint for you, which will bring you to another page. There you have the solution.

7. Sep 27, 2010

### ╔(σ_σ)╝

Well you could also use the cauchy-riemann equations to find v when given u.

8. Sep 27, 2010

### Quinzio

What if you read a book that states that "book are mandatory and professors are not ?"

9. Sep 27, 2010

### Matterwave

Ok, no need to be hostile...

I didn't get a book and it's now too late to buy a book for this problem haha. I do own a book on this stuff, but it's inaccessible to me at the moment. I will try to get a book soon.

10. Sep 27, 2010

### Quinzio

No intention to be hostile.... :)

11. Sep 27, 2010

### ╔(σ_σ)╝

Haha...I like you and your humor :)

If you know the cauchy riemann equations, simply equate the derievatives and integrate to solve for u or v.

12. Sep 27, 2010

### Matterwave

Ok, I get how they will help me find v from u or u from v. But what about the last one? Any hints on that? I can't simply split off the real and imaginary parts of that function in any method that I know of...

13. Sep 27, 2010

### ╔(σ_σ)╝

For the last one artan (z) is defined in terms of the logarithm and you may be able to split the fuction into real and imaginary parts but I am not sure if it will work out.

You could always differential arctan(z) and write down u and v for the derievative and then integrate back but there might be some issues with points where the functions are not defined.

14. Sep 28, 2010

### Quinzio

I still don't see any solution... :)

$$u(x,y)=x^2-y^2+x+2$$

$$u_x= 2x+1$$
$$u_y= -2y$$

$$v_y = u_x = 2x+1$$
$$v_x = -u_y= 2y$$

mmmmmm.... feels like $$v(x,y) = 2xy+y$$

15. Sep 28, 2010

### Matterwave

Yea I got part 1 and 2...(sort of cheated on 2...looked at the solutions to see the hint of transforming to polar coordinates haha)

Thanks for the help. =)