# Find the largest subset of $\mathbb{C}$ where $g(z)$ is holomorphic

I’m struggling a lot with this question:

—- Fix $$a \in \mathbb{C}$$ where $$g(z) = \frac{1}{z^2-a^2}$$, Let $$\gamma = \gamma (0;|a|+1)$$. Find the largest subset of $$\mathbb{C}$$ where g(z) is holomorphic. Illustrate this with a sketch. —- I understand that a holomorphic function is a function is differentiable in the complex plane, I believe that the only time g(z) would not be holomorphic is when the function is invalid e.g $$\frac{1}{0}$$, however this leaves me with just assuming the only points g(z) is not holomorphic is where $$z^2 = a^2$$ however this then leaves just the subset $$\mathbb{C}$$ not including positive or negative a, which doesn’t seem right. Any help help would be fab.

• You are correct. $g$ is holomorphic everywhere except at the poles $\pm a$ (which coincide if $a=0$).
– MPW
Mar 5, 2019 at 16:35
• “postive or negative $a$” doesn't make sense when $a$ is complex, but yes, the only points where $g$ is not defined are $\pm a$. $g$ is holomorphic on its whole domain. Mar 5, 2019 at 16:35

You are right : The function $$g$$ is holomorphic on $$\mathbb{C} - \{a, -a\}$$ as you said. when $$z$$ is in this set, you can say

$$g(z) = - \frac{1}{a^2} \frac{1}{1 - z^2}$$

which you can develop into a power series near $$z$$ using the development of $$\frac{1}{1 + z}$$

(which does prove $$g$$ is holomorphic on $$z$$)

You could also calculate the derivative $$g'(z)$$ using the formula $$\lim_{z' \rightarrow z} \frac{g(z') - g(z)}{z' - z}$$ which will converge for each z in this set

• Oh alright it’s good I’m on the right track, the thing that made me unsure however is that the question asks for the ‘largest subset’ and then to illustrate this with a sketch. This makes me assume there are smaller subsets and that I’m unsure about how to sketch the full complex set, not including {a,-a}. That’s why I believed I was wrong.
– L G
Mar 5, 2019 at 16:45
• well $g$ is also holomorphic on every open non-empty subset of this one, which is probably why they ask for the largest one. Mar 5, 2019 at 16:48
• As to what kind of sketch they are asking for, I honestly have no idea… If you want a sketch of the full complex set, not including these two points, you could just draw two axes and these two points, but I’m not sure whether that’s what they want Mar 5, 2019 at 16:52
• Yeah that’s what I was thinking, it’s a bit confusing thanks a lot though, just curious the question does include ‘let $\gamma = /gamma (0;|a|+1)$ ‘ I assumed this path had nothing to do with the question but I’m probably wrong
– L G
Mar 5, 2019 at 16:56
• I don’t know that $\gamma(0; |a| + 1)$ notation. Is it perhaps the circle of center 0 and radius $|a| + 1$ ? Mar 5, 2019 at 17:05