# Tag Info

### Zeroes of entire function on $\mathbb C^n$

This follows immediatelly from the following paper: The Zero Set of a Real Analytic Function
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Accepted

### Conjectured closed form of $\int_0^1 \frac{\text{Li}_2\left(\frac{x}{4}\right)}{4-x}\,\log\left(\frac{1+\sqrt{1-x}}{1-\sqrt{1-x}}\right)\,dx$

As Henri Cohen remarked, the identity to prove is equivalent to $$\sum_{n=1}^\infty \frac{H_n^{(2)}}{(n+1)(2n+1)\binom{2n}{n}}=\frac{\pi^4}{972}.\tag{1}$$ In turn, this follows readily from the OP's ...
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• 115k
1 vote
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• 88.2k

### Entire function of finite order with deficient value

The theorem you stated, and its various versions and generalizations, are the only simple sufficient conditions for $\delta(0)>0$. For example, if $f$ is entire of genus $1$, and zeros lie on a ray,...
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### Best approximation of the modulus function

Let $\mathbb{D} \subset \mathbb{C}$ be the unit disc. Let $\phi : (\mathbb{D} \to \mathbb{C}) \to \mathbb{R}$ be the following functional: $$\phi = f \mapsto \sup_{z \in \mathbb{D}} |f(z) - |z||$$ ...
• 1,793
1 vote
Accepted

Actually, here's a very strong counterexample (to the idea that disjoint ROCs lead to divergent convolutions), riffing off of my earlier comment. Let $S = \{ (-2)^k : k \in \mathbb{N} \}$. Choose $0 &... • 36.5k 2 votes ### Interpolation by holomorphic functions of small exponential type on a half-plane This is not always possible under your conditions. For example, if$a_n=0$for$n\geq 2$, then any function of exponential type$<\pi$interpolating this sequence must be zero by Carlson's theorem,... • 88.2k 1 vote ### The inverse of the digamma function We want to solve for$x$the equation $$\psi ^{(0)}(x)=a$$ Just four months before your question, this paper gave tight bounds$\$x_{\text{min}}=\frac{1}{\log \left(1+e^{-x}\right)} \lt \psi^{-1} (x) \...
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