query: How does $\exp(x+y) = \exp(x)\exp(y)$ imply $\exp(x) = [\exp(1)]^x$?
positive_passages.docid: 22303
positive_passages.text: How do I prove that $f(x)f(y)=f(x+y)$ implies that $f(x)=e^{cx}$, assuming f is continuous and not zero?
negative_passages.docid: 12083
negative_passages.text: How to prove $\limsup (x_{n}+y_{n})=\lim x_{n}+\limsup y_{n}$?
negative_passages.docid: 95145
negative_passages.text: Does $\exp(2ir\pi)$ equal $1$? What's wrong?
negative_passages.docid: 154270
negative_passages.text: Why is $ \overline{e^z} = e^\overline{z} $?
negative_passages.docid: 101394
negative_passages.text: Please explain the logic behind $d(xy) = y(dx) + x(dy)$
negative_passages.docid: 102777
negative_passages.text: Is $G^{(X, Y)} = (G^X)^Y?$ ($A^B$ just means that $B$ is mapped to $A$)
negative_passages.docid: 81720
negative_passages.text: $f(x) + f(x+f(y)) = y + f(f(x) + f(f(y)))$
negative_passages.docid: 2315
negative_passages.text: How to prove that $\lVert x + y\rVert = \lVert x\rVert + \lVert y \rVert \implies \lVert tx + (1-t)y\rVert = t\lVert x\rVert + (1-t)\lVert y \rVert$?
negative_passages.docid: 98271
negative_passages.text: Why does $(x \to a) ≠ (x = a)$ , But $f(x \to a) = f(x = a)$
negative_passages.docid: 155266
negative_passages.text: Is $\exp:\overline{\mathbb{M}}_n\to\mathbb{M}_n$ injective?
negative_passages.docid: 5350
negative_passages.text: Explain why $E(|X|^p)=p\int_{0}^{\infty}y^{p-1}P(|X|>y) dy$
negative_passages.docid: 158551
negative_passages.text: If $X \perp Y$ and $X+Y \in L^1$ then $X,Y \in L^1$
negative_passages.docid: 109295
negative_passages.text: Proving $1-\exp(-4x^2/ \pi) \ge \text{erf}(x)^2$
negative_passages.docid: 5595
negative_passages.text: $x^y = y^x$ for integers $x$ and $y$
negative_passages.docid: 113411
negative_passages.text: If $E$ is generated by $E_1$ and $E_2$, then $[E:F]\leq [E_1:F][E_2:F]$?
negative_passages.docid: 147892
negative_passages.text: Proving if $x<y$ then $\sqrt{x} < \sqrt{y}$
negative_passages.docid: 662
negative_passages.text: If $E[X|Y]=Y$ almost surely and $E[Y|X]=X$ almost surely then $X=Y$ almost surely