query: Let $f(z)=\sum_{-\infty}^\infty e^{2\pi inz}e^{-\pi n^2}$. Show there is a unique $z_0$ in the unit square such that $f(z_0)=0$.
positive_passages.docid: 29511
positive_passages.text: How to prove $f(z) = \sum_{n=-\infty}^{\infty}e^{2\pi inz}e^{-\pi n^2}$ has a unique zero inside a unit square in the 1st quadrant.
negative_passages.docid: 147219
negative_passages.text: Prove that a power series $\sum_{n=0}^{\infty}{{a_n}{z^n}}$ which converges for any $z \in \mathbb{N}$, converges for any $z \in \mathbb{C}$.
negative_passages.docid: 133386
negative_passages.text: If $f$ is entire and $\lim_\limits{z\to\infty} \frac{f(z)}{z} = 0$ show that $f$ is constant
negative_passages.docid: 26353
negative_passages.text: Can any harmonic function on $\{z:0<|z|<1\}$ be extended to $z=0$?
negative_passages.docid: 137640
negative_passages.text: Proof that $\sum_{1}^{\infty} \frac{1}{n^2} <2$
negative_passages.docid: 134152
negative_passages.text: Prove $\sum_{n=1}^\infty(e-\sum_{k=0}^n\frac1{k!})=1$
negative_passages.docid: 34060
negative_passages.text: Laurent Series of $f(z) = \sin {\frac{1}{z}}$ at $z=0$
negative_passages.docid: 8705
negative_passages.text: Evaluation of $\sum_{x=0}^\infty e^{-x^2}$
negative_passages.docid: 44970
negative_passages.text: Show $\sum_{n=0}^\infty\frac{1}{a^2+n^2}=\frac{1+a\pi\coth a\pi}{2a^2}$
negative_passages.docid: 78473
negative_passages.text: Showing $ \sum_{n=0}^{\infty} \frac{1}{(3n+1)(3n+2)}=\frac{\pi}{3\sqrt{3}}$
negative_passages.docid: 12966
negative_passages.text: Riemann Sums proving function identically zero
negative_passages.docid: 131692
negative_passages.text: How to show that $f_{0}f_{1}+f_{1}f_{2}+...+f_{2n-1}f_{2n} = f^{2}_{2n}$
negative_passages.docid: 119133
negative_passages.text: there is $M<\infty$ such that $\sum_{n} |\hat{f}(n)|\le M\int_{0}^{2\pi}|f(t)|dt$ for each $f\in X$
negative_passages.docid: 84313
negative_passages.text: $|f(z)|\le 1-|z|\forall z\in D$, we need to show $f\equiv 0$
negative_passages.docid: 156419
negative_passages.text: Convergence of $\sum\limits_{k=0}^\infty {\binom{z}{k}} $ for complex $z$
negative_passages.docid: 69219
negative_passages.text: Prove there is a point $z\in[a,b]$ at which $f(z)=\frac{f(x_1)+f(x_2)+\cdots+f(x_k)}{k}$
negative_passages.docid: 38062
negative_passages.text: Proving the convergence of $\sum\limits_{n=1}^{\infty}\frac{1}{1+z^n}$ for $|z| > 1$