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 Continuum sums -- a big problem and some interesting observations mike3 Long Time Fellow Posts: 368 Threads: 44 Joined: Sep 2009 10/10/2010, 12:58 AM (10/09/2010, 03:13 PM)Ansus Wrote: 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. « Next Oldest | Next Newest »

 Messages In This Thread Continuum sums -- a big problem and some interesting observations - by mike3 - 10/05/2010, 11:40 AM RE: Continuum sums -- a big problem and some interesting observations - by tommy1729 - 10/05/2010, 11:33 PM RE: Continuum sums -- a big problem and some interesting observations - by mike3 - 10/06/2010, 03:34 AM RE: Continuum sums -- a big problem and some interesting observations - by tommy1729 - 10/06/2010, 12:44 PM RE: Continuum sums -- a big problem and some interesting observations - by mike3 - 10/07/2010, 04:32 AM RE: Continuum sums -- a big problem and some interesting observations - by Ansus - 10/09/2010, 03:13 PM RE: Continuum sums -- a big problem and some interesting observations - by mike3 - 10/10/2010, 12:58 AM RE: Continuum sums -- a big problem and some interesting observations - by Ansus - 10/09/2010, 05:46 PM RE: Continuum sums -- a big problem and some interesting observations - by Ansus - 10/10/2010, 05:31 AM RE: Continuum sums -- a big problem and some interesting observations - by Ansus - 10/10/2010, 06:06 AM RE: Continuum sums -- a big problem and some interesting observations - by Ansus - 10/10/2010, 08:02 AM RE: Continuum sums -- a big problem and some interesting observations - by mike3 - 10/11/2010, 10:36 AM RE: Continuum sums -- a big problem and some interesting observations - by Ansus - 10/11/2010, 11:33 AM RE: Continuum sums -- a big problem and some interesting observations - by mike3 - 10/11/2010, 11:46 PM RE: Continuum sums -- a big problem and some interesting observations - by Ansus - 10/12/2010, 04:41 AM RE: Continuum sums -- a big problem and some interesting observations - by mike3 - 10/12/2010, 06:44 AM RE: Continuum sums -- a big problem and some interesting observations - by mike3 - 10/12/2010, 06:45 AM RE: Continuum sums -- a big problem and some interesting observations - by Ansus - 10/12/2010, 10:41 AM

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