Solving Analytic Function: F(x) Expression

[asi123]

Homework Statement

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

Homework Equations

The Attempt at a Solution

 

[HallsofIvy]

Integrate by letting u= z²+ 4.

 

[asi123]

Integrate by letting u= z²+ 4.

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

Thanks a lot.

 

[Gregg]

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

Thanks a lot.

Integration by substitution?

 

[asi123]

Integration by substitution?

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

Thanks a lot.

 

[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?

 

[asi123]

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?

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.

 

[Dick]

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.

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.

 

[asi123]

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.

Ok, Thanks.