Apologies for the bump. I updated the equivalent thread on the other FES site, and realized I should do the same thing here.
I have to recommend this 3Blue1Brown video which does a good job explaining analytic continuation (AC). It cuts through a lot of the mysticism that other explanations seem to encourage by glossing over details. AC essentially sidesteps our previous inability to deal with undefined infinities and more extrapolate the true behavior of functions regardless without caring whether or not they invoke infinity. Step by step, we can construct an
intuitive visual explanation for the problem and how AC approaches it.
Personally, the visual explanation of AC seems analogous to a function where there is a hole in a straight line.
Consider the function:
f(x) = 5x/xThat function's output is obviously a line, and it consistently returns f(x) = 5, with the exception of x = 0 where we get a "0/0" undefined error.
If we somehow didn't know how to pull common factors out of a numerator and denominator and change the function to be f(x) = 5, we could still look at the graph and see what the answer clearly would be 5 when x is 0, despite a calculator's inability to evaluate clashing infinities.
You cannot operate with highly divergent infinite series using the rules of convergent series: the results will be spurious.
Also, when using analytic continuation, you are actually substracting infinity from infinity to get something finite: again, an abstract mathematical operation with no connection to the real world.
I don't see how analytic continuation invokes conflicting infinities. The original problem does, but AC should dodge that problem, just like factoring my super simple example function doesn't actually divide 0 by 0. Let me know if you still perceive this as a problem after watching the video.
As for your concern that we could get spurious results, or lack real world applicability, please skip to 6:28 in fappenhosen's video:
http://www.numberphile.com/videos/analytical_continuation1.html
Additionally, I recall posting a minutephysics video explaining how analytic continuation predicted the fine-structure constant (α), which has been continually confirmed over and over as the accuracy of experimentation improved enough to measure more significant digits. I can hunt that video down if it's really important to you.
There is only one solution to the problem, it is the solution nature agrees on, and arguably it is more coherent than saying the solution is "infinity". Infinities don't seem to appear in nature, but they are inherently built into how we model nature. I firmly believe infinity is a degeneracy of our mathematics models.
Food for thought.
