query: At $z=0$, the function $f(z)=\exp\left(\frac{z}{1-\cos(z)}\right)$ has infinitely many positive and negative powers of $z$.
positive_passages.docid: 20047
positive_passages.text: At $z=0$ the function $f(z)=\exp({z\over 1-\cos z})$ has
negative_passages.docid: 139168
negative_passages.text: Prove that there is an unique $z$ s.t. $f(z) = z$ where $z$ is a complex number
negative_passages.docid: 55138
negative_passages.text: An analytic function $f$ on $\{z \in\mathbb{C} : |z|>1 \}$
negative_passages.docid: 78008
negative_passages.text: A conjectured closed form of $\int\limits_0^\infty\frac{x-1}{\sqrt{2^x-1}\ \ln\left(2^x-1\right)}dx$
negative_passages.docid: 71071
negative_passages.text: Nonzero $f \in C([0, 1])$ for which $\int_0^1 f(x)x^n dx = 0$ for all $n$
negative_passages.docid: 34308
negative_passages.text: Given $|z|<1$ prove $\lim_{n \to \infty} z^n=0$.
negative_passages.docid: 66867
negative_passages.text: Question regarding the function $z=a+bi \mapsto f(z)=a+i(a+b)$.
negative_passages.docid: 102553
negative_passages.text: How to show that $f(z) = f(i)$
negative_passages.docid: 76441
negative_passages.text: Max. and Min. value of $|z|$ in $\left|z+\frac{2}{z}\right| = 2\,$
negative_passages.docid: 116257
negative_passages.text: How to show $\lim_{k\rightarrow \infty} \left(1 + \frac{z}{k}\right)^k=e^z$
negative_passages.docid: 27347
negative_passages.text: Verify that $\left|\int_{\gamma} \exp(iz^2)dz\right| \leq \frac{\pi\big(1-\exp(-r^2)\big)}{4r}$ where $\gamma(t)=re^{it}$, for $0\leq t \leq \pi/4$.
negative_passages.docid: 28148
negative_passages.text: Are there any simple ways to see that $e^z-z=0$ has infinitely many solutions?
negative_passages.docid: 103706
negative_passages.text: If $\{f(0)\}^{2} + \{f'(0)\}^{2} = 4$ then there is a $c$ with $f(c) + f''(c) = 0$
negative_passages.docid: 78319
negative_passages.text: Show that $x \mapsto \frac{f(x)}{x}$ is strictly increasing on (0,1) given that f '(x) is strictly increasing on (0,1) and that f(0)=0
negative_passages.docid: 145265
negative_passages.text: A non-zero function satisfying $g(x+y) = g(x)g(y)$ must be positive everywhere
negative_passages.docid: 6722
negative_passages.text: to show $f$ is a polynomial in $z$
negative_passages.docid: 83080
negative_passages.text: Question regarding $f(n)=\cot^2\left(\frac\pi n\right)+\cot^2\left(\frac{2\pi}n\right)+\cdots+\cot^2\left(\frac{(n-1)\pi}n\right)$