Continuum sums -- a big problem and some interesting observations

[mike3]

The same is true for integrals: if you take

\(
\int \frac1x \,dx
\)

you can get different functions wich differ not only by a constant. The convention here is to count the solution as \( \ln |x|+C \) but this is only one of possible solutions. Another for example is \( \ln |x|+ \operatorname{sgn}(x)+C \)

However, I have an eye toward the complex plane, with holomorphic functions (or multi-functions). It is true that, say, \( \int_{-1}^{x} \frac{1}{t} dt \) could equal \( \log(-x) \), which is akin to the whole \( \psi \) thing for the sum, but this function analytically continues in the complex plane to what is, essentially, just another branch of \( \log \), just shifted by a constant shift of \( i \pi \). When continued to a multifunction, there is no difference between \( \int_{-1}^{z} \frac{1}{t} dt \) and \( \int_{1}^{z} \frac{1}{t} dt \) except a constant shift.