# State the domain where the function is analytic

• Jun 20th 2009, 08:14 PM
noppawit
State the domain where the function is analytic
State the domain where the function is analytic.
$\displaystyle f(z) = \frac{1}{1-z^4}$

From what I have studied, f(z) is analytic on a domain D, if f(z) is differentiable at all points in D.

However, I don't know where I should begin. And I think that $\displaystyle f(z)$ is not analytic at $\displaystyle 1-z^4=0$.

Anyway, how to find the domain, how to show in details to solve this.

Thank you.
• Jun 20th 2009, 08:27 PM
Chris L T521
Quote:

Originally Posted by noppawit
State the domain where the function is analytic.
$\displaystyle f(z) = \frac{1}{1-z^4}$

From what I have studied, f(z) is analytic on a domain D, if f(z) is differentiable at all points in D.

However, I don't know where I should begin. And I think that $\displaystyle f(z)$ is not analytic at $\displaystyle 1-z^4=0$.

Anyway, how to find the domain, how to show in details to solve this.

Thank you.

As you have said, consider when $\displaystyle 1-z^4=0$.

This would construct the 4th roots of unity $\displaystyle z=i,\,z=-i,\,z=1,\,z=-1$: $\displaystyle f\!\left(z\right)$ is not differentiable at these values.

Thus, our domain would be $\displaystyle D=\mathbb{C}\backslash\{1,i,-1,-i\}$.