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If $f$ is big-O of $g$ and $h$ is big-O of $1$, then $f \cdot h$ is big-O of $g$. |
The following lemmas are all equivalent: big_mult_1 big_mult_1' small_mult_1 small_mult_1' small_mult_1'' small_mult_1''' big_1_mult big_1_mult' small_1_mult small_1_mult' small_1_mult'' small_1_mult''' |
$\mathcal{O}(f) = \mathcal{O}(g)$ if and only if $f \in \Theta(g)$. |
If $f(x)$ is in $L_F(g(x))$ for all $x$, then $\prod_{x \in A} f(x)$ is in $L_F(\prod_{x \in A} g(x))$. |
If $f$ is a function from $A$ to $L^1$, then the product of the functions in $f$ is in $L^1$. |
If $f(x) = o(g(x))$, then $\lim_{x \to \infty} (f(x) + g(x)) = \lim_{x \to \infty} g(x)$. |
If $g$ is $o(f)$, then $\lim_{x \to \infty} (f(x) + g(x)) = \lim_{x \to \infty} f(x)$. |
If $f$ is $o(g)$, then $\lim_{x \to a} (f(x) - g(x)) = \lim_{x \to a} g(x)$. |
If $g$ is $o(f)$, then $L(f - g) = L(f)$. |
The following identities hold: $x + 0 = x$, $0 + x = x$, $x - 0 = x$, $0 - x = -x$. |
If $f$ is $O(g)$ and $\Omega(g)$, then $f$ is $\Theta(g)$. |
If $f \in \Theta(g)$, then $f \in O(g)$ and $f \in \Omega(g)$. |
If $f$ is a function, then $f \in \Theta[F](f)$. |
If $f$ is big-Theta of $g$, then $g$ is big-Theta of $f$. |
The Landau symbols $\Omega$, $\omega$, and $\Theta$ are symmetric. |
The Landau symbols $o$, $\omega$, $\Omega$, $\Theta$ are defined. |
If $f$ is eventually bounded by $cg$ for some constant $c$, then $f$ is big-O of $g$. |
If $f$ is a small-omega function of $g$, then $f$ is eventually bounded below by a constant multiple of $g$. |
If $f$ and $g$ are functions such that $f(x)$ is eventually bounded above and below by $c_1 g(x)$ and $c_2 g(x)$ respectively, then $f(x) \in \Theta(g(x))$. |
If $f$ and $g$ are eventually equal, then $f \in \Theta(g)$. |
If $f$ is in $L(F, g(h))$, and $h$ and $h'$ are eventually equal, then $f$ is in $L(F, g(h'))$. |
If $f$ and $f'$ are eventually equal, and $g$ is a function such that $g(f')$ is in $L(F,h)$, then $g(f)$ is in $L(F,h)$. |
If $f \in L_F(g)$ and $h \in \Theta_F(g)$, then $f \in L_F(h)$. |
If $g$ is $\Theta(f)$ and $g$ is $O(h)$, then $f$ is $O(h)$. |
If $f \in O(g)$ and $h \in \Omega(g)$, then $f \in O(h)$. |
The Landau symbols $\mathcal{O}$, $\mathcal{o}$, $\Theta$, and $\Omega$ are transitive. |
If $f(x) = O(g(x))$ and $g(x) = O(h(x))$, then $f(x) = O(h(x))$. |
The Landau symbols are transitive. |
If $f \in \Theta[F](g)$, then $\frac{1}{f} \in \Theta[F](\frac{1}{g})$. |
If $f_1(x) \in \Theta(f_2(x))$ and $g_1(x) \in \Theta(g_2(x))$, then $\frac{f_1(x)}{g_1(x)} \in \Theta(\frac{f_2(x)}{g_2(x)})$. |
If $f \in \Theta(g)$, then $f$ is eventually nonzero if and only if $g$ is eventually nonzero. |
If $f$ and $g$ are functions such that $\lim_{x \to \infty} \frac{f(x)}{g(x)} = c$ and $g(x) \neq 0$ for all sufficiently large $x$, then $f \in O(g)$. |
If $f(x)/g(x)$ tends to a constant $c$ as $x$ tends to infinity, and $g(x)$ is eventually nonzero, then $f(x)$ is $O(g(x))$. |
If $\lim_{x \to \infty} \frac{f(x)}{g(x)} = c \neq 0$, then $f \in \Omega(g)$. |
If $f(x)/g(x)$ tends to $c \neq 0$ as $x$ tends to infinity, then $f(x)$ is big-Omega of $g(x)$. |
If $\lim_{x \to \infty} \frac{f(x)}{g(x)} = 0$, then $f \in \omega[F](g)$. |
If $f(x)/g(x)$ tends to infinity as $x$ tends to infinity, then $f(x)$ is small-o of $g(x)$. |
If $f$ is small-o of $g$ and $g$ is eventually nonzero, then $\frac{f}{g}$ tends to infinity. |
If $f$ is small-o of $g$ and $g$ is eventually nonzero, then $f/g$ tends to infinity. |
A function $f$ is small $\omega$ if and only if it converges to infinity with respect to the filter $F$. |
If $f(x)/g(x) \to 0$ as $x \to \infty$, and $g(x) \neq 0$ for all sufficiently large $x$, then $f(x) = o(g(x))$ as $x \to \infty$. |
If $f(x) = o(g(x))$, then $\frac{f(x)}{g(x)} \to 0$. |
If $f(x)/g(x)$ tends to a nonzero constant $c$ as $x$ tends to infinity, then $f(x)$ is asymptotically equivalent to $g(x)$. |
If $f(x)/g(x)$ tends to a nonzero constant $c$ as $x$ tends to infinity, then $f(x)$ is big-Theta of $g(x)$. |
If $f_1$ converges to $a$ and $f_2$ is $o(f_1)$, then $f_1 + f_2$ converges to $a$. |
If $f_1$ converges to $a$ and $f_2$ is $o(f_1)$, then $f_1 - f_2$ converges to $a$. |
If $f_2$ is $o(f_1)$, then $f_1$ tends to $a$ if and only if $f_1 + f_2$ tends to $a$. |
If $f_2$ is $o(f_1)$, then $f_1$ tends to $a$ if and only if $f_1 - f_2$ tends to $a$. |
If $f_1(x)/g_1(x)$ tends to $a$ as $x$ tends to $x_0$, and $f_2(x)$ and $g_2(x)$ are both small compared to $f_1(x)$ and $g_1(x)$, respectively, then $(f_1(x) + f_2(x))/(g_1(x) + g_2(x))$ tends to $a$ as $x$ tends to $x_0$. |
If $f(x) = O(g(x))$, then $|f(x)|^p = O(|g(x)|^p)$ for any $p \geq 0$. |
If $f(x) = o(g(x))$, then $|f(x)|^p = o(|g(x)|^p)$ for any $p > 0$. |
If $f(x)$ is small compared to $g(x)$ and both $f(x)$ and $g(x)$ are eventually nonnegative, then $f(x)^p$ is small compared to $g(x)^p$. |
If $f$ is big-Theta of $g$, then $|f|^p$ is big-Theta of $|g|^p$. |
If $f$ is $O(g)$, then $f^p$ is $O(g^p)$ for any nonnegative real number $p$. |
The function $f(x) = 0$ is in the little-o of $f$. |
The function $f(x) = 0$ is in $O(f)$. |
If $f$ is a function, then $f \in \Omega[F](0)$. |
The zero function is in the space of small-omega functions. |
$f \in o[F](\lambda x. 0)$ if and only if $f(x) = 0$ for all $x$ in $F$. |
A function $f$ is in $O[F](\lambda x. 0)$ if and only if $f$ is eventually zero on $F$. |
A function is in $\omega[F](f)$ if and only if it is eventually zero. |
A function is in $\Omega[F](f)$ if and only if it is eventually zero. |
A function is in $\Theta(f)$ if and only if it is eventually equal to $0$. |
$f \in \Theta[F](0)$ if and only if $f$ is eventually $0$ with respect to $F$. |
If $f(x) = cg(x)$ for some constant $c$, then $f(x)$ is $O(g(x))$ if and only if $c = 0$ or $f(x)$ is $O(g(x))$. |
The constant function $c$ is $O(1)$. |
A constant function is big-O of another constant function if and only if the first constant is zero or the second constant is nonzero. |
A constant function is big-Omega of another constant function if and only if the first constant is zero or the second constant is zero. |
If $f(x)$ is $o(g(x))$ as $x \to \infty$, then $f(n)$ is $o(g(n))$ as $n \to \infty$. |
If $f(x)$ is $O(g(x))$ as $x \to \infty$, then $f(x)$ is $O(g(x))$ as $x \to \infty$ for $x \in \mathbb{N}$. |
If $f$ is $\omega$-bigger than $g$, then the function $x \mapsto f(x)$ is $\omega$-bigger than the function $x \mapsto g(x)$. |
If $f$ is big-Omega of $g$, then the function $x \mapsto f(x)$ is big-Omega of the function $x \mapsto g(x)$. |
If $f(x)$ is asymptotically equivalent to $g(x)$ as $x \to \infty$, then $f(x)$ is asymptotically equivalent to $g(x)$ as $x \to \infty$ for $x$ a natural number. |
The Landau symbols $\mathcal{O}$, $\mathcal{o}$, $\Omega$, $\omega$, and $\Theta$ can be transferred from the real to the natural numbers. |
If $f(x)$ and $g(x)$ are two functions such that $f(x) = g(x)$ for all $x$ sufficiently large, then the asymptotic behavior of $f(x)$ is the same as the asymptotic behavior of $g(x)$. |
If $f(x)$ is a function of $x$ and $f(x) = O(g(x))$ as $x \to \infty$, then $f(x) = o(g(x))$ as $x \to \infty$. |
If $f$ and $g$ are both $o(h)$, then $f + g$ and $f - g$ are also $o(h)$. |
If $f(x) \in o(g(x))$ for all $x \in A$, then $\sum_{y \in A} f(y, x) \in o(g(x))$. |
If $f$ and $g$ are both $O(h)$, then $f + g$ and $f - g$ are also $O(h)$. |
If $f(x)$ is $O(g(x))$ for all $x \in A$, then $\sum_{y \in A} f(y, x)$ is $O(g(x))$. |
If $f_1 \in \Theta(g_1)$ and $g_1(x) \neq 0$ for all sufficiently large $x$, then $(f_1 f_2) \in \Theta(g_1 g_2)$ if and only if $f_2 \in \Theta(g_2)$. |
If $f_2$ is asymptotically equivalent to $g_2$ and $g_2$ is eventually nonzero, then $f_1 f_2$ is asymptotically equivalent to $g_1 g_2$ if and only if $f_1$ is asymptotically equivalent to $g_1$. |
If $f_2$ is asymptotically equivalent to $g_2$ and $g_2$ is eventually nonzero, then $f_1 / f_2$ is asymptotically equivalent to $g_1 / g_2$ if and only if $f_1$ is asymptotically equivalent to $g_1$. |
If $f_1$ is asymptotic to $g_1$ and $g_1$ is eventually nonzero, then $f_1/f_2$ is asymptotic to $g_1/g_2$ if and only if $1/f_2$ is asymptotic to $1/g_2$. |
If $g$ is a function that grows faster than any polynomial, then $g^p$ grows faster than $g^q$ if and only if $p < q$. |
If $g$ is a function that grows faster than any polynomial, then $g^p$ grows faster than $g^q$ if and only if $p \leq q$. |
If $g$ is a function that grows faster than any polynomial, then $g^p$ and $g^q$ are asymptotically equivalent if and only if $p = q$. |
If $f(x)$ and $g(x)$ are functions such that $\ln(f(x))$ is small-o of $\ln(g(x))$, then $f(x)^{p}$ is small-o of $g(x)^{q}$ for any $p, q > 0$. |
If $f(x)$ and $g(x)$ are functions such that $\ln(f(x))$ is small-o of $\ln(g(x))$, then $g(x)^{q}$ is small-o of $f(x)^{p}$ for any $p, q < 0$. |
If $f$ and $g$ are asymptotically equivalent, then $\lim_{x \to \infty} \frac{f(x)}{g(x)} = 1$. |
If $f$ and $g$ are asymptotically equivalent, then the ratio $f(x)/g(x)$ tends to $1$ as $x$ tends to infinity. |
$f \sim_{\mathcal{F}} g$ if and only if $f \circ h \sim_{\mathcal{F}} g \circ h$. |
A function is asymptotically equivalent to itself. |
If $f$ is asymptotically equivalent to $g$, then $g$ is asymptotically equivalent to $f$. |
If $f$ is asymptotically equivalent to $g$, then $g$ is asymptotically equivalent to $f$. |
If the ratio of two functions converges to 1, then the two functions are asymptotically equivalent. |
If two functions $f_1$ and $f_2$ are asymptotically equivalent, and $g_1$ and $g_2$ are asymptotically equivalent, then $f_1$ and $g_1$ are asymptotically equivalent if and only if $f_2$ and $g_2$ are asymptotically equivalent. |
If $f$ and $g$ are asymptotically equivalent, then eventually $f(x) = 0$ if and only if $g(x) = 0$. |
If two functions are asymptotically equivalent, and they eventually coincide, then the functions they coincide with are asymptotically equivalent. |
If two functions are asymptotically equivalent, then so are their compositions with any two functions that are eventually equal. |
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