Show this is analytic

1. Dec 9, 2008

Firepanda

This isn't my whole question, just part of the question I am trying to do to show the whole thing is analytic.

I can do the rest but showing this is analytic:

(1+z^3)/(-1+z)

Is trickey for me..

I am trying to show it is analytic by showing it satisfies the cauchy riemann equations.

I take z = x + iy

And my function turns into (after simplifying)

[x^3 - 3xy^2 + 1 + i(3yx^2 - y^3)] / (x + iy -1)

Now I can split the numerator in real and imaginary parts, but the denominator has an i in it which is in the way for me, hence I can't split the whole thing into real and imaginary parts. So I can't show it satisfies the CRE.

Anyone know, or should I not be using the CRE?

Thanks

2. Dec 9, 2008

PingPong

Try multiplying the numerator and denominator of your expression by the complex conjugate of the denominator (that is, x-1-i y). What happens?

3. Dec 9, 2008

Firepanda

Ah thanks :)

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

Determine the largest subset of C (complex numbers) on which the following function is analytic, and compute its derivative.

2. Relevant equations

exp[(z^3+1)/(z-1)]

I'm trying to compute it's derivative right now, but how would I find its largest subset?

4. Dec 9, 2008

mutton

First find where the function is not analytic.

The same question was posed at https://www.physicsforums.com/showthread.php?t=278123

5. Dec 9, 2008

Firepanda

so by that do u mean it is no analytic when z = 1?

6. Dec 9, 2008

Firepanda

Apparantly it can be dont the same way as with real numbers, but I don't know how to do that either lol..

7. Dec 9, 2008

Firepanda

bump before bed

8. Dec 9, 2008

Dick

The only point in question is z=1. The best outcome there would be that it's a removable singularity. It's not. Can you show that it's not?