[NT] Extending a Jacobi function using Riemann Surfaces JmsNxn Ultimate Fellow     Posts: 1,214 Threads: 126 Joined: Dec 2010 02/26/2023, 06:10 PM (This post was last modified: 02/26/2023, 06:28 PM by JmsNxn.) As Caleb is on the search for these types of functions. I thought I'd use some of my history with Jacobi/modular functions, and what I understand by Caleb's approach--and cook a stupid thread. Caleb has opened a wonderful path here. I'd just like to add my two cents. So let's take a Jacobi function--which is similar to a Jacobi $\vartheta$ function: $\theta(z) = \prod_{j=1}^\infty 1-z^j\\$ Jacobi's $\vartheta$ function is discoverable through this function. We just must alternate products of linear transformations. We are going to ignore the modular relation at this point. The function $\theta(z)$ is holomorphic for $|z| < 1$. And almost all modular relations relevant to this post, are related to this product function, or small variations. This function has a zero at $z=1$: $\theta(1) = 0\\$ And additionally we have that: $\theta'(z) = -\sum_{n=1}^\infty nz^{n-1}\prod_{j=1\,\,j\neq n}^\infty (1-z^j)\\$ And evaluating at $z=1$ we have: $\theta'(1) = -1\\$ Therefore $\theta(z)$ is holomorphic for $|z| < 1$, but it is differentiable at $z=1$... This continues for every rational point on the unit circle. Through the exact same mechanics. I suggest reading my replies on Caleb's LONG post. We can write this as: $\theta(e^{2\pi i \frac{m}{k}}+t) = \theta'(e^{2\pi i \frac{m}{k}}) t + O(t^2)\\$ For all $m,k \in \mathbb{Z}$. Where by now we have that: $\frac{d}{dz} \theta(z) = \text{FINITE VALUE FOR}\,z^n = 1\\$ This does not mean that $\theta$ is holomorphic at these points. It means in real analysis it is differentiable at these points along the boundary. And it is differentiable in both variables $z = x+iy$. Thing is, this is a closed boundary with a dense amount of differential points... If we try to fold it onto itself; the same way Caleb is suggesting with: $f(z) = \sum_{n=0}^\infty \frac{z^n}{1+z^n}\frac{1}{2^n}\\$ (The values with $|z| <1$ fold into the values $|z|>1$ at the value $z=1$). We must ask for a much deeper result. Which ultimately, resides in modular theory. We can't apply a quadratic Riemann surface. We must apply an uncountable Riemann surface for $\theta(z)$. I am only posting this, because Caleb mentioned modular/Jacobi functions, and their extensions. The discussion of their extensions is very fucking deep. And remember: $\theta(e^{2\pi i \frac{m}{k}}+t) = \theta'(e^{2\pi i \frac{m}{k}}) t + O(t^2)\\$ And this is what makes the wall of singularities so deep and complex. Choosing $\xi \neq \frac{m}{q}$, an irrational number: $\theta(e^{2\pi i \xi}+t) = \infty\\$ Discussing growth in $t$ or in behaviour on $\xi$--requires Modular discussion. Zero ifs, ands, or buts. This is how universities could throw away Fermat Last Theorem solutions from anonymous sources. If you didn't talk about the modular effect. It is worthless.... tommy1729 Ultimate Fellow     Posts: 1,906 Threads: 409 Joined: Feb 2009 02/26/2023, 08:16 PM (02/26/2023, 06:10 PM)JmsNxn Wrote: As Caleb is on the search for these types of functions. I thought I'd use some of my history with Jacobi/modular functions, and what I understand by Caleb's approach--and cook a stupid thread. Caleb has opened a wonderful path here. I'd just like to add my two cents. So let's take a Jacobi function--which is similar to a Jacobi $\vartheta$ function: $\theta(z) = \prod_{j=1}^\infty 1-z^j\\$ Jacobi's $\vartheta$ function is discoverable through this function. We just must alternate products of linear transformations. We are going to ignore the modular relation at this point. The function $\theta(z)$ is holomorphic for $|z| < 1$. And almost all modular relations relevant to this post, are related to this product function, or small variations. This function has a zero at $z=1$: $\theta(1) = 0\\$ And additionally we have that: $\theta'(z) = -\sum_{n=1}^\infty nz^{n-1}\prod_{j=1\,\,j\neq n}^\infty (1-z^j)\\$ And evaluating at $z=1$ we have: $\theta'(1) = -1\\$ Therefore $\theta(z)$ is holomorphic for $|z| < 1$, but it is differentiable at $z=1$... This continues for every rational point on the unit circle. Through the exact same mechanics. I suggest reading my replies on Caleb's LONG post. We can write this as: $\theta(e^{2\pi i \frac{m}{k}}+t) = \theta'(e^{2\pi i \frac{m}{k}}) t + O(t^2)\\$ For all $m,k \in \mathbb{Z}$. Where by now we have that: $\frac{d}{dz} \theta(z) = \text{FINITE VALUE FOR}\,z^n = 1\\$ This does not mean that $\theta$ is holomorphic at these points. It means in real analysis it is differentiable at these points along the boundary. And it is differentiable in both variables $z = x+iy$. Thing is, this is a closed boundary with a dense amount of differential points... If we try to fold it onto itself; the same way Caleb is suggesting with: $f(z) = \sum_{n=0}^\infty \frac{z^n}{1+z^n}\frac{1}{2^n}\\$ (The values with $|z| <1$ fold into the values $|z|>1$ at the value $z=1$). We must ask for a much deeper result. Which ultimately, resides in modular theory. We can't apply a quadratic Riemann surface. We must apply an uncountable Riemann surface for $\theta(z)$. I am only posting this, because Caleb mentioned modular/Jacobi functions, and their extensions. The discussion of their extensions is very fucking deep. And remember: $\theta(e^{2\pi i \frac{m}{k}}+t) = \theta'(e^{2\pi i \frac{m}{k}}) t + O(t^2)\\$ And this is what makes the wall of singularities so deep and complex. Choosing $\xi \neq \frac{m}{q}$, an irrational number: $\theta(e^{2\pi i \xi}+t) = \infty\\$ Discussing growth in $t$ or in behaviour on $\xi$--requires Modular discussion. Zero ifs, ands, or buts. This is how universities could throw away Fermat Last Theorem solutions from anonymous sources. If you didn't talk about the modular effect. It is worthless.... hmmm uncountable riemann surfaces. Well if a function f(z) is analytic for |z| < 1 then its inverse is only defined for range within |z| =< 1. say f(0.5) = f( - 0.5 ) = 0.25. then the riemann surface at 0.25 gives 0.5 and - 0.5 on 2 sheets. I do not see it going to |z| > 1. I think a natural boundary is the boundary of the riemann surface. This makes sense because otherwise we could use analytic continuation to get there ( via the riemann surface or normally * that should be equivalent ! * ) Afterall the way to move on a riemann surface is by analytic continuation. What you probably MEANT TO SAY was riemann surfaces with uncountable sheets. But the problem remains. Then we get 2 or more riemann surfaces with an uncountable number of sheets , where uncountable is at least the amount of max( non-differentiable or differentiable points ) . something like that. I do not see the uncountable sheets connecting to other riemann surfaces isomorphic to going beyond that natural boundary. That is one of the reasons why i mentioned unfalsifiable in the other thread. YEAH , you could just plug in values from beyond the boundary and if it converges be happy. or collect residue and make fubini's head explode. but seems arbitrary. And arbitrary choices will not result in NUMBER THEORY PROOFS. Sorry for being skeptical. regards tommy1729 tommy1729 Ultimate Fellow     Posts: 1,906 Threads: 409 Joined: Feb 2009 02/26/2023, 08:22 PM but thanks for using the tag « Next Oldest | Next Newest »

 Possibly Related Threads… Thread Author Replies Views Last Post Anyone have any ideas on how to generate this function? JmsNxn 3 195 05/21/2023, 03:30 PM Last Post: Ember Edison [MSE] Mick's function Caleb 1 232 03/08/2023, 02:33 AM Last Post: Caleb [special] binary partition zeta function tommy1729 1 190 02/27/2023, 01:23 PM Last Post: tommy1729 toy zeta function tommy1729 0 194 01/20/2023, 11:02 PM Last Post: tommy1729 geometric function theory ideas tommy1729 0 239 12/31/2022, 12:19 AM Last Post: tommy1729 Iterated function convergence Daniel 1 367 12/18/2022, 01:40 AM Last Post: JmsNxn Fibonacci as iteration of fractional linear function bo198214 48 7,320 09/14/2022, 08:05 AM Last Post: Gottfried Constructing an analytic repelling Abel function JmsNxn 0 538 07/11/2022, 10:30 PM Last Post: JmsNxn Galidakis & Extending tetration to non-integers Daniel 4 2,431 05/31/2021, 01:26 AM Last Post: JmsNxn Math.Stackexchange.com question on extending tetration Daniel 3 2,743 03/31/2021, 12:28 AM Last Post: JmsNxn

Users browsing this thread: 1 Guest(s) 