>>12057497Everything in live is a meme.
Okay, with that out of the way, let's look at the question.
>Is analytic continuation a fucking meme?Given an expression that defines a function f(x) on the reals x in R, the question
>what's a complex differentiable function g(z) such that for for x in R, g(x)=f(x)? is clearly a completely natural one.
E.g. consider for x in positive reals with
f(1/2) = -0.6931
f(1) = 0
f(2) = 0.6931
f(2.3) = 0.832
etc.
Also
I.e. all values on the positive reals are attainable if you just integrate 1/t from t=1 to x.
But you can't walk into the negative reals.
At least not on the real line, since 1/0 isn't defined.
Now if you parametrize the circle in the complex plane as which has
and also
,
using
With this insight, the only thing you have to be willing to is to say that the log in the complex numbers should be defined in a way that agrees with its definition on the reals and is differentiable wherever it's defined (which is guaranteed by the integral representation)
Then
you find, on the half-circle path that goes from
ON THE PATH GAMMA
With this insight, the only thing you have to be willing to is to say that the log in the complex numbers should be defined in a way that agrees with its definition on the reals, is differentiable wherever it's defined (which is guaranteed by the integral representation) and where the values is computer on the shortest path
Then
which is what WolframAlpha will tell you.
