3
votes
3answers
178 views

Limit on a spiral

I was thinking about limits of functions along various spirals and this one stumped me a bit. The limit that needs to be found is ultimately: $$\lim_{\varphi\to\infty} \coth\varphi\csc\varphi$$ ...
3
votes
1answer
43 views

finding rational complex numbers in a disk with least denominators

Suppose that I have a disk of radius $r$ around some complex $\alpha\in\mathbb{C}$: How would one find a complex number $g$ in that disk besides $\alpha$ such that $\mathrm{Re}(g)\in\mathbb{Q}$ ...
1
vote
1answer
65 views

characterizing the boundary of the convergent region of $f(z)= \sum_{n=1}^{\infty} z^{(1/z)^{n}}$

Let $$f(z)= \sum_{n=1}^{\infty} z^{(1/z)^{n}}$$ A domain colored portrait (with artifacts) for $f(z)$ on the unit disk looks like: The gray and white regions are where the software package had ...
2
votes
2answers
71 views

finding all $z$ such that $f(z)=g(z)=h(z)$

Suppose I have three functions $f(z):\mathbb{C}\rightarrow\mathbb{C}$, $g(z):\mathbb{C}\rightarrow\mathbb{C}$, and $h(z):\mathbb{C}\rightarrow\mathbb{C}$. What methods work for finding all $z$ such ...
7
votes
1answer
278 views

interpolating the primorial $p_{n}\#$

The primorial $p_{n}\#$ is given by the product $p_n\# = \prod_{k=1}^n p_k$ (where $p_{k}$ is the $k$th prime) -- is there a natural (a la the gamma function $\Gamma(z)$) way of interpolating it for ...
12
votes
2answers
362 views

regularity of root spacing of $G(z)=\sum\limits_{n=1}^{\infty} \frac{e^{-n^{2}}}{n^{z}}$

Define, on $\mathbb{C}$: $$G(z)=\sum_{n=1}^{\infty} \frac{e^{-n^{2}}}{n^{z}}$$ A domain colored portrait of $G(z)$ (boxes are supposed to be negative signs): suggests that the roots of $G(z)$ are ...