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....

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....