Suppose you are standing up and you look at a green wall. Also, you are a ballerina or just feel like doing a full 360 degrees (or ) turn. Naturally after your turn, you should see the same green wall, yes?
The answer (generally) is no. It all depends on what you can see! It also depends on how you are looking.
Suppose that the wall takes two colours: green and blue. They are definitely not the same!
Mathematics can be bizarre and sometimes things that do not appear to be the same actually… are. Suppose we are not in such a universe. Green and blue are definitely different for us.
We get a blue wall by using blue paint to paint the wall blue. Similarly we get a green wall by using green paint.
Then we can say dependent on what paint we used, we get a different output.
What if in our seemingly normal mathematical universe we use green paint to paint the wall green, look at the wall standing up, then do a full turn and the wall is now blue?
And further suppose that this can happen an infinite amount of times?!
But we did not use blue paint!!
Painting something is a single act. You do something and you get one (and only one) thing out in return, which makes sense.
But looking back at what you paint is not the same! Because you can paint with two different colours, when you look back there are two possible colours! You could call this a multi-act.
Also suppose there is not a single way to just look. Maybe we can look with squinted eyes or with just one eye, the other closed, or enclosed: pirate style!
Now this is crucial: how we look could affect what colour we see.
Suppose there is some way of looking that, if we make the look, then go to the wall and “look” back at the look, we can only see one thing: the look.
This appears silly, of course that’s true. But also suppose we can look, go to the wall and “look” back and see the same and perhaps another look. You could call this a 2-look.
Generalise it further and you have a multi-look. A capability to look back at your projection and see something else.
Then if you can look forward, do a turn, say a 1-turn, generalised to multi-turns, you should always see a green wall provided that you painted it green initially.
But if the “look” that you are using is a multi-look, clearly when you look back, which if you think about it… is just the same as doing a turn, then you don’t have the green wall. You have a blue wall.
This is precisely what a branch point is: the “look” that you choose that after a “turn” gives you a different colour to the wall.
To fix this, we could ban certain looks (so all multi looks) and focus on the single looks: these are all continuous since no matter how many turns we do, we have the same green paint always appearing.
Then if consider the multi looks, we make these continuous by fixing your turns: where you start and where you finish. You should still do a full turn either way. You just change the position where you do it.
This means if we are in the room and the paints are at some position in the room, so the green paint at position A and blue paint at position B, consider the distance between them, call it C.
This C will be removed: from blue a turn gives us green and likewise from green to blue. Formally, we take a cut in the axis between the points A and B.
This is a branch cut.
This part is removed as travelling in this small part of the considered axis gives us a discontinuity! Everything else stays!
Now suppose we could transfer this to a different room with different paint colours and different representations of what gives us the same or different (so in our considered case, the representation is looking at the wall).
This is analytic continuation.
This may not generally work: other rooms and other representations may have a specific structure which mean what is discontinuous for one is not discontinuous for the other.
Then suppose when in some specific section (a contour) of the room, everything is well defined and after every single turn we have the same green wall.
Suppose we are a different section and given the “nice” behaviour of the first section, if we can somehow exhibit the same behaviour, we have connected two sections.
Then we may do this for the whole room and hence when we look and turn, we always see a green wall.
This is the Monodromy Theorem.
Then we understand the whole room, after our ballerina turns, anywhere in the room, we always see green,
Essentially, we have just explained a bunch of useful (and closely related) concepts and theorems in complex analysis.
We weren’t being very formal – far from it, but we have the idea in our mind.
Formally, we looked at multivalued functions and their relationship as the inverse of single-valued functions.
We see what happens as we travel through a full period: the multivalued function returns a different value at some specific points (the branch points).
Then we fix angles and remove parts of the imaginary or real axis: a branch cut.
Whether this cut holds in a different region and whether there is any difference to how we look at the wall or the colour of the wall is analytic continuation.
Finally, the Monodromy Theorem characterises this with singular points and analyticity.