Laurent expansion on border of annulus

Question

$$ f(z)= \frac{1}{(z-2)(z^2+i)} $$

I want to find the laurent expansion of this function at z=2 in the region

$$ 1 < |z| < 2 $$

if someone could help me with this not just telling me what to do but help me visualizing what this means.

Answer 1

As stated, this is nonsensical, since $2$ is not in the region $1<|z|<2,$ so we won't be able to help you visualize what it means. Probably what happened is that your professor copied and pasted from another problem, but didn't quite finish editing out the parts that weren't supposed to be there.

The best I can do for you, then, is speculate about what your professor may have intended to say, and see what sense the alternatives might make. Off the top of my head, the following could be what was intended:

1. Find the laurent expansion in the region $1<|z|<2.$
2. Find the laurent expansions at $z=2$ and in the region $1<|z|<2.$
3. Find the laurent expansion at $z=2.$
4. Find the laurent expansions at $z=2.$
5. Find the laurent expansion at $z=2$ in the region $1<|z-2|<2.$

Of course, it could be that something wildly different was intended, but those are the only obvious alternatives I can see.

I suspect that options 2-4 are incorrect (and know that option 5 is incorrect), for several reasons. For one, we typically specify regions for laurent expansions, unless the function is entire, which is not the case, here. Thus, asking for a laurent expansion at a point (a singular point, no less) is nonsensical. It could be that "at $z=2$" should be interpreted as "centered at $z=2,$" but even then, option 3 is nonsensical, as there are three annuli about $z=2$ on which a laurent series for $f(z)$ can be obtained, and it is not specified which of the three to use. Also, option 5 is impossible even under that interpretation, because the annulus $1<|z-2|<2$ contains the pole $z=-\sqrt{2}+i\sqrt{2}.$ Thus, we're left with options 1, 2, and 4.

If we want to fully cover our options, we should assume that option 2 is what was meant, but that means finding four different laurent expansions, so in a testing situation, that would perhaps be inadvisable. However, it only differs from the posted problem by the single short word "and," so it could well be what was intended. It seems unlikely that option 4 was intended, because that would mean that the professor would have to misspell a word and fail to notice a fairly long phrase that wasn't supposed to be there, especially if "$1<|z|<2$" was typeset in the center, as it is in your post. It would be relatively easy to miss the short phrase "at $z=2,$" though, so I could easily see it being option 1.

The best advice I can give, if you encounter such a situation again, is to get up, take the exam copy to your professor, and ask for clarification. Everyone makes mistakes, and hopefully, your professor isn't the sort who won't admit to them.

If your professor is unwilling to provide such clarification, or if you are for some reason unwilling or unable to ask for it, there are really only two ways to go that I can see. You could assume that option 2 was meant, which would cover all the obvious options. Alternatively, you could (and in your place, I would) assume that option 1 was meant, then move on to the rest of the exam, and return to cover the other expansions if you have time to spare after completing the rest.