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If $r \geq 0$, then the image of the path $t \mapsto z + r e^{it}$ for $s \leq t \leq t$ is contained in the sphere of radius $r$ centered at $z$. |
The complex conjugate of a part of a circle path is the part of the circle path with the same center, radius, and endpoints, but with the opposite orientation. |
If $f$ is a continuous function on the image of the path $c + re^{i\theta}$ for $\theta \in [a,b]$, then the contour integral of $f$ along this path is bounded by $B \cdot r \cdot |b-a|$, where $B$ is an upper bound for $|f(z)|$ on the image of the path. |
If $a < b$, then the contour integral of $f$ along the arc of the circle with center $c$ and radius $r$ from $a$ to $b$ is equal to the integral of $f(c + r e^{i\theta}) r i e^{i\theta}$ from $a$ to $b$. |
If $a < b$, then $f$ is contour-integrable on the part of the circle path from $a$ to $b$ if and only if the function $g(t) = f(c + r \cdot e^{it}) \cdot r \cdot i \cdot e^{it}$ is integrable on the interval $[a, b]$. |
If $a < b$, then the contour integral of $f$ along the arc of the circle with center $c$ and radius $r$ from $a$ to $b$ is equal to the integral of $f(c + r e^{i t}) r i e^{i t}$ from $a$ to $b$. |
The contour integral of a function $f$ along a part of a circle is equal to the negative of the contour integral of $f$ along the same part of the circle in the opposite direction. |
If $b < a$, then the contour integral of $f$ along the part of the circle of radius $r$ centered at $c$ from $a$ to $b$ is equal to the negative of the contour integral of $f$ along the part of the circle of radius $r$ centered at $c$ from $b$ to $a$. |
The set of complex numbers $z$ such that $|z| \leq b$ and $e^z = w$ is finite. |
If $a \neq 0$, then the set of all complex numbers $z$ such that $|z| \leq b$ and $e^{az} = w$ is finite. |
If $f$ is a function with a contour integral along a part of a circle, and if $f$ is bounded on the path, then the contour integral is bounded by the length of the path times the bound on $f$. |
If $f$ is a function with a bounded contour integral along a part of a circle, then the integral is bounded by the length of the path times the bound on the function. |
If $f$ is continuous on the image of the path $\gamma(t) = z + r e^{it}$ for $s \leq t \leq t$, then $f$ is contour integrable on $\gamma$. |
A path is simple if and only if it is a part of a circle path with nonzero radius, distinct start and end points, and the angle between the start and end points is less than or equal to $2\pi$. |
If $r \neq 0$ and $s \neq t$, then the arc length of the part of the circle with radius $r$ and center $z$ from $s$ to $t$ is less than $2\pi$. |
The circlepath function is defined as $z + r \exp(2 \pi i x)$. |
The starting point of the circle path with center $z$ and radius $r$ is $z + r$. |
The endpoint of a circle path is the starting point plus the radius. |
The circlepath $z$ with radius $-r$ at $x$ is the same as the circlepath $z$ with radius $r$ at $x + 1/2$. |
The circle path $z + re^{2\pi i t}$ is the same as the circle path $z + re^{2\pi i (t+1)}$. |
The circlepath function is periodic with period 1. |
The image of the circle path $z - r$ is a subset of the image of the circle path $z + r$. |
The path image of a circle path with negative radius is the same as the path image of a circle path with positive radius. |
The derivative of the circlepath function is given by the formula $2 \pi i r e^{2 \pi i x}$. |
The derivative of the circlepath function is given by the formula above. |
If $0 \leq x \leq 1$, then the derivative of the circle path $z$ with radius $r$ at $x$ is $2 \pi i r e^{2 \pi i x}$. |
The path $z + re^{it}$ is a valid path. |
The function $f(t) = z + r e^{2 \pi i t}$ is a path. |
If $r \geq 0$, then the image of the circle path $z$ with radius $r$ is the sphere of radius $r$ centered at $z$. |
The image of a circle path is a sphere. |
If $f$ has a contour integral around a circle of radius $r$ centered at $z$, and $f$ is bounded by $B$ on the circle, then the contour integral is bounded by $B \cdot 2 \pi r$. |
If $f$ has a contour integral along a circle of radius $r$ centered at $z$, and $f$ is bounded by $B$ on the circle, then the contour integral is bounded by $B \cdot 2 \pi r$. |
If $f$ is continuous on the image of the circle path $z$ with radius $r$, then $f$ is contour integrable on the circle path $z$ with radius $r$. |
A circle path is simple if and only if the radius is nonzero. |
If $w$ is inside the circle of radius $r$ centered at $z$, then $w$ is not on the circle. |
The contour integral of $1/(w - z)$ along the circle of radius $r$ centered at $z$ is $2\pi i$. |
If $f_n$ is a sequence of functions that are all integrable over a path $\gamma$, and if $f_n$ converges uniformly to $f$ over the image of $\gamma$, then $f$ is integrable over $\gamma$, and the sequence of integrals of $f_n$ converges to the integral of $f$. |
If $f_n$ is a sequence of functions that are all integrable over the circle of radius $r$ centered at $z$, and if $f_n$ converges uniformly to $f$ on the circle of radius $r$ centered at $z$, then $f$ is integrable over the circle of radius $r$ centered at $z$, and the integral of $f$ over the circle of radius $r$ cent... |
If the convex combination of any two points in a set is also in the set, then the set is convex. |
If $s$ is a convex set and $x, y \in s$, then $u x + v y \in s$ for all $u, v \geq 0$ such that $u + v = 1$. |
A set $S$ is convex if and only if for all $x, y \in S$ and $0 \leq u \leq 1$, the point $(1-u)x + uy$ is in $S$. |
If $s$ is a convex set and $a, b \in s$, then for any $u \in [0, 1]$, $(1 - u)a + ub \in s$. |
If $S$ is a convex set, $x, y \in S$, and $u, v \geq 0$ with $u + v > 0$, then $(u/(u+v))x + (v/(u+v))y \in S$. |
The empty set is convex. |
The singleton set $\{a\}$ is convex. |
The whole space is convex. |
The intersection of a collection of convex sets is convex. |
The intersection of two convex sets is convex. |
If $B_i$ is convex for each $i \in A$, then $\bigcap_{i \in A} B_i$ is convex. |
If $s$ and $t$ are convex sets, then $s \times t$ is convex. |
The set of points $x$ such that $a \cdot x \leq b$ is convex. |
The set of points $x$ such that $a \cdot x \geq b$ is convex. |
The set of points $x$ such that $|\langle a, x \rangle| \leq b$ is convex. |
The set of points satisfying $a \cdot x = b$ is convex. |
The set of points $x$ such that $a \cdot x < b$ is convex. |
The set of points $x$ such that $a \cdot x > b$ is convex. |
The set of complex numbers with real part greater than or equal to $b$ is convex. |
The set of complex numbers with real part less than or equal to $b$ is convex. |
The set of complex numbers with imaginary part greater than or equal to $b$ is convex. |
The set of complex numbers with imaginary part less than or equal to $b$ is convex. |
The set of complex numbers with real part greater than $b$ is convex. |
The set of complex numbers with real part less than $b$ is convex. |
The set of complex numbers with imaginary part greater than $b$ is convex. |
The set of complex numbers with imaginary part less than $b$ is convex. |
The intervals $[a, \infty)$, $(-\infty, b]$, $(a, \infty)$, $(-\infty, b)$, $[a, b]$, $(a, b]$, $[a, b)$, and $(a, b)$ are all convex. |
The set of real numbers is convex. |
If $C$ is a convex set, $S$ is a finite set, and $y_i \in C$ for all $i \in S$, then $\sum_{i \in S} a_i y_i \in C$ for any $a_i \geq 0$ such that $\sum_{i \in S} a_i = 1$. |
A set $S$ is convex if and only if for all $k \in \mathbb{N}$, for all $u_1, \ldots, u_k \in \mathbb{R}$, for all $x_1, \ldots, x_k \in S$, if $u_1 + \cdots + u_k = 1$ and $u_i \geq 0$ for all $i$, then $u_1 x_1 + \cdots + u_k x_k \in S$. |
A set $S$ is convex if and only if for every finite subset $t$ of $S$ and every family of nonnegative real numbers $(u_x)_{x \in t}$ such that $\sum_{x \in t} u_x = 1$, we have $\sum_{x \in t} u_x x \in S$. |
A finite set $S$ is convex if and only if for every function $u$ such that $u(x) \geq 0$ for all $x \in S$ and $\sum_{x \in S} u(x) = 1$, we have $\sum_{x \in S} u(x) x \in S$. |
If $f$ satisfies the convexity inequality for all $t \in (0,1)$, then $f$ is convex on $A$. |
If $f$ is a function from a linearly ordered set $A$ to the real numbers such that for all $t \in (0,1)$ and all $x,y \in A$ with $x < y$, we have $f((1-t)x + ty) \leq (1-t)f(x) + tf(y)$, then $f$ is convex on $A$. |
If $f$ is convex on $A$, then for all $t \in [0,1]$, $x,y \in A$, we have $f((1-t)x + ty) \leq (1-t)f(x) + tf(y)$. |
If $f$ is convex on the interval $[x,y]$, then for all $t \in [0,1]$, we have $f((1-t)x + ty) \leq (1-t)f(x) + tf(y)$. |
If $f$ is convex on $t$, and $S \subseteq t$, then $f$ is convex on $S$. |
If $f$ and $g$ are convex functions on a set $S$, then $f + g$ is convex on $S$. |
If $f$ is convex on $S$ and $c \geq 0$, then $c f$ is convex on $S$. |
If $f$ is convex on a set $S$, and $x, y \in S$, then $f(u x + v y) \leq \max(f(x), f(y))$ for all $u, v \geq 0$ such that $u + v = 1$. |
The function $x \mapsto \|x - a\|$ is convex on any set $S$. |
If $f$ is a linear map and $S$ is a convex set, then $f(S)$ is convex. |
If $f$ is a linear function and $S$ is a convex set, then $f^{-1}(S)$ is convex. |
If $S$ is a convex set, then the set of all scalar multiples of points in $S$ is also convex. |
If $S$ is a convex set, then the set of all scalar multiples of elements of $S$ is convex. |
If $S$ is a convex set, then so is $-S$. |
If $S$ and $T$ are convex sets, then the set of all sums $x + y$ where $x \in S$ and $y \in T$ is convex. |
If $S$ and $T$ are convex sets, then the set of differences $S - T = \{x - y \mid x \in S, y \in T\}$ is convex. |
If $S$ is a convex set, then $S + a$ is a convex set. |
If $S$ is a convex set, then the set $S - a$ is convex. |
If $S$ is a convex set, then the set of all points of the form $a + cx$ for $x \in S$ is also convex. |
If $f$ is convex on a convex set $C$, then $f$ is convex on the convex hull of $C$. |
A function $f$ is convex on a set $C$ if and only if for all $x, y \in C$ and $\mu \in [0, 1]$, we have $f(\mu x + (1 - \mu) y) \leq \mu f(x) + (1 - \mu) f(y)$. |
If $f$ is convex on an interval $I$, and $x, y \in I$ with $x < t < y$, then the slopes of the secant lines from $(x, f(x))$ to $(t, f(t))$ and from $(x, f(x))$ to $(y, f(y))$ are non-increasing, and the slopes of the secant lines from $(t, f(t))$ to $(y, f(y))$ and from $(x, f(x))$ to $(y, f(y))$ are non-decreasing. |
If $f$ is a convex function, then $f$ is convex on $C$. |
If $C$ is a convex set and $x, y \in C$ with $x < y$, then the interval $[x, y]$ is contained in $C$. |
If $f$ is convex, then $f'$ is monotone increasing. |
If $f$ is twice differentiable on a convex set $C$ and $f''(x) \geq 0$ for all $x \in C$, then $f$ is convex on $C$. |
The function $-\log_b(x)$ is convex on the interval $(0, \infty)$. |
If $f$ is a real-valued function defined on a connected set $A$ and $f$ is differentiable on $A$ and its derivative is monotone increasing, then $f$ is convex on $A$. |
The inverse function is convex on the positive reals. |
If $f$ is convex on the interval $[x,y]$, then for any $c \in [x,y]$, we have $f(c) \leq \frac{f(y) - f(x)}{y-x}(c-x) + f(x)$. |
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