query: If integral of square of modulus of $f$ over $\mathbb C$ is finite, then $f \equiv 0$
positive_passages.docid: 87188
positive_passages.text: Prove that a square-integrable entire function is identically zero
negative_passages.docid: 111076
negative_passages.text: If $f$ is non-constant and entire, prove that there exists a $z_0 \in C$ such that $f(z_0)$ is a positive real number
negative_passages.docid: 5180
negative_passages.text: How to show $\alpha\int_a^cf(x)dx+\beta \int_c^b f(x)dx =0$ means $f(x)=0$ everywhere $?$
negative_passages.docid: 42853
negative_passages.text: If $f \geq 0$ is continuous and $\int_{a}^{b} f(x) \, dx = 0$, then $f =0$
negative_passages.docid: 143354
negative_passages.text: Prove that there exists $c\in\mathbb{R}$ such that $f'(c)=f(c)$
negative_passages.docid: 82588
negative_passages.text: If $\int_0^x f \ dm$ is zero everywhere then $f$ is zero almost everywhere
negative_passages.docid: 95469
negative_passages.text: Suppose $f:[0,1] \Rightarrow \mathbb{R}$ is continuous and $\int_0^x f(x)dx = \int_x^1 f(x)dx$. Prove that $f(x) = 0$ for all $x$
negative_passages.docid: 63590
negative_passages.text: Let $g$ be a function from $(0, 1)$ to $\mathbb{R}$ and $\int_0^1 g^2(x)dx$ finite. Does this imply that $\int_0^1 g(x)dx$ is finite?
negative_passages.docid: 145573
negative_passages.text: How to show that $f$ is a zero function?
negative_passages.docid: 14759
negative_passages.text: Prove that $ f(z) = 0 $ for all $ z \in \overline{D}(0,1) $.
negative_passages.docid: 140430
negative_passages.text: Let $E = \mathbb{Q}\big(\left\{\sqrt[n]{2} \mid n \in \mathbb{N} \setminus \left\{0\right\} \right\} \big)$. Prove $[E: \mathbb{Q}]$ is not finite
negative_passages.docid: 149328
negative_passages.text: A function such that $f:\mathbb C \to \mathbb C$ such that $f(z)f(iz)=z^2$ satisfies $f(z)+f(-z)=0$.(MADHAVA-2020)
negative_passages.docid: 16984
negative_passages.text: Finite extension of domains : preimage of non-zero ideal is non-zero?
negative_passages.docid: 139463
negative_passages.text: Commutative integral domain does not finitely generate its field of fractions
negative_passages.docid: 62568
negative_passages.text: Does there exists an entire function $f: \mathbb C \to \mathbb C$ which is bounded on real line and imaginary line?
negative_passages.docid: 88690
negative_passages.text: $\int_a^b f(x)\mathrm dx=0$ for all $-\infty<a<b<\infty$ a,b rational implies $f=0$ a.e
negative_passages.docid: 109034
negative_passages.text: Does a finite $[E:K]_s$ imply that $[E_s:K]$ is finite?