# Homework Help: Analytic function

1. Feb 23, 2009

### asi123

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

Hey guys.
I'm pretty much stuck on this one.
The question is, is there an analytic function F(x) which fulfill the expression in the pic? (I'm sure I messed up the English, sorry )

Thanks

2. Relevant equations

3. The attempt at a solution

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2. Feb 23, 2009

### HallsofIvy

Integrate by letting u= z2+ 4.

3. Feb 23, 2009

### asi123

I need to Integrate this thing in order to find the function?
What do you mean by "letting u= z2+ 4"?

Thanks a lot.

4. Feb 23, 2009

### Gregg

Integration by substitution?

5. Feb 23, 2009

### asi123

Oh, ok so I got half Ln of something, is that the function? seems too easy...

Thanks a lot.

6. Feb 23, 2009

### Dick

But does log(z^2+4) define an analytic function on the plane minus those two points? Can you pick values of the log that make it continuous everywhere? What happens with the 'function' log(z) as you move in a circle around the origin?

7. Feb 23, 2009

### asi123

Yeah right, it's not analytic unless you take off the entire x negative axis.
So does that mean that such analytic function does not exist?

Thanks a lot.

8. Feb 23, 2009

### Dick

That's the general picture, yes. You can write the antiderivative as log(z+2i)+log(z-2i). You have exactly the same kind of problem as log(z) has around the points +/-2i.

9. Feb 23, 2009

Ok, Thanks.