I'm working with the complex function $f(z) = \frac{1}{{z^2 + 5iz - 4}}$, and I'm trying to determine where this function is not analytic. I've been trying to compute its domain of analyticity, but I'm encountering some difficulties.
I'm a bit confused because it seems like there might be two regions where the function becomes indeterminate. However, in my class, I was taught that everything should be understood within a single neighborhood. My professor described the neighborhood as a single inequality that forms a circular region centered at the origin.
Could someone please help me clarify whether there are indeed two separate non-analytic regions for this function, or if it should be comprehended within a single neighborhood? Any insights, methods, or explanations would be greatly appreciated.
If you need any additional information or details about my approach, please let me know. I'm eager to gain a better understanding of this function's behavior.
Thank you in advance for your assistance!
We can rewrite the function as $f(z) = \frac{1}{(z + 4i)(z + i)}$, and notice that the denominator has two zeros at $z = -4i$ and $z = -i$. These zeros are poles of the function, and so the function will be non-analytic in any neighbourhood containing either of the poles. We would say that the domain of the function's analyticity is the entire complex plane minus these points.
If we are focusing on the origin, which we might be doing because we're interested in something like the radius of convergence of a Taylor series, then we would consider the largest circle we can draw around the origin that doesn't capture either of the poles, and in that case the radius will be 1 since the circle $|z| = 1$ captures the pole at $-i$.