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If $f$ is holomorphic on the open ball of radius $r$ centered at $0$ and $f(0) = 0$, then there exists a holomorphic function $h$ on the open ball of radius $r$ centered at $0$ such that for all $z$ with $|z| < r$, we have $f(z) = zh(z)$ and $f'(0) = h(0)$. |
If $f$ is a holomorphic function on the unit disk such that $f(0) = 0$ and $|f(z)| < 1$ for all $z$ in the disk, then $|f(z)| \leq |z|$ for all $z$ in the disk. Furthermore, $|f'(0)| \leq 1$, and if $|f(z)| = |z|$ for some $z$ in the disk or $|f'(0)| = 1$, then $f(z) = \alpha z$ for some $|\alpha| = 1$. |
If $f$ is a holomorphic function on the unit disk such that $f(0) = 0$ and $|f(z)| < 1$ for all $z$ in the unit disk, then either $|f(z)| \leq |z|$ for all $z$ in the unit disk, or $|f(z)| = |z|$ for some $z$ in the unit disk. |
If $d \cdot a \leq k \leq d \cdot b$, then there exists a point $c$ on the line segment from $a$ to $b$ such that $d \cdot c = k$ and $d \cdot z \leq k$ for all $z$ on the line segment from $a$ to $c$, and $k \leq d \cdot z$ for all $z$ on the line segment from $c$ to $b$. |
If $f$ is holomorphic on a convex open set $S$ and continuous on $S$, then the integral of $f$ around any triangle in $S$ is zero. |
If $f$ is holomorphic on the interior of a convex open set $S$ and continuous on $S$, then the integral of $f$ around any triangle in $S$ is zero. |
If $f$ is a continuous function on a convex open set $S$ and $f$ is holomorphic on the two open sets $S_1$ and $S_2$ where $S = S_1 \cup S_2$, then the contour integral of $f$ along the boundary of $S$ is zero. |
Suppose $S$ is a convex open set in $\mathbb{R}^n$, $a, b, c \in S$, $d \in \mathbb{R}^n$ is a nonzero vector, and $f$ is a continuous function on $S$ that is holomorphic on the two sets $\{z \in S : d \cdot z < k\}$ and $\{z \in S : k < d \cdot z\}$ for some $k \in \mathbb{R}$. Then the contour integral of $f$ along t... |
If $f$ is holomorphic on the upper and lower half-planes and continuous on the real line, then $f$ is holomorphic on the whole plane. |
If $f$ is holomorphic on the upper half-plane and continuous on the real line, then the Schwarz reflection of $f$ is holomorphic on the whole plane. |
If $f$ is a holomorphic function on a ball of radius $r$ centered at $0$, and $f(0) = 0$, and the derivative of $f$ is bounded by $2$ times the derivative of $f$ at $0$, then the image of the ball of radius $r$ under $f$ contains a ball of radius $(3 - 2 \sqrt{2}) r \|f'(0)\|$. |
If $f$ is holomorphic on a ball $B(a,r)$ and the norm of the derivative of $f$ is bounded by $2$ times the norm of the derivative of $f$ at $a$, then the image of $B(a,r)$ under $f$ contains a ball of radius $(3 - 2 \sqrt{2}) r \|f'(a)\|$. |
If $f$ is holomorphic on the unit ball around $a$ and $f'(a) = 1$, then there exists a ball $B$ of radius at least $1/12$ such that $f(B)$ contains the unit ball around $f(a)$. |
If $f$ is holomorphic on a ball of radius $r$ centered at $a$, then $f$ maps a ball of radius $r' \leq r \frac{\|f'(a)\|}{12}$ centered at $a$ onto a ball of radius $r'$. |
Suppose $f$ is holomorphic on a set $S$ and $a \in S$. If $t$ is a positive real number such that $t \leq \text{dist}(a, z)$ for all $z \in \partial S$, then there exists a ball $B$ of radius $r \leq t \cdot \|f'(a)\| / 12$ such that $B \subseteq f(S)$. |
A set $S$ is connected if and only if it is not the union of two nonempty open sets that are disjoint. |
For any property $P$ of sets, the statement $\exists S. P(-S)$ is equivalent to the statement $\exists S. P(S)$. |
A set $S$ is connected if and only if every clopen subset of $S$ is either empty or $S$ itself. |
If $T$ is a connected subset of $S$ and $x, y \in T$, then $x$ and $y$ are in the same connected component of $S$. |
If $x$ and $y$ are in the same connected component of $S$, then $x$ and $y$ are in $S$. |
The connected component of a point $x$ in a set $S$ containing $x$ is $x$ itself. |
The connected component of $x$ in $S$ is $x$ if and only if $x \<in> S$. |
If $x$ and $y$ are in the same connected component of $S$, then $y$ and $x$ are in the same connected component of $S$. |
If $x$ and $y$ are in the same connected component of $S$, and $y$ and $z$ are in the same connected component of $S$, then $x$ and $z$ are in the same connected component of $S$. |
If $x$ and $y$ are in the same connected component of $S$, and $S$ is a subset of $T$, then $x$ and $y$ are in the same connected component of $T$. |
The connected component of $x$ in $S$ is the union of all connected subsets of $S$ that contain $x$. |
The connected component of a point $x$ in a topological space $S$ is connected. |
A set $S$ is connected if and only if for every $x \in S$, the connected component of $x$ in $S$ is $S$ itself. |
The connected component of a point $x$ in a topological space $S$ is a subset of $S$. |
If $S$ is a connected set and $x \in S$, then the connected component of $x$ in $S$ is $S$ itself. |
A topological space is connected if and only if any two points in the space are in the same connected component. |
If $T$ is a connected subset of $S$ containing $x$, then $T$ is contained in the connected component of $S$ containing $x$. |
If $S \subseteq T$, then the connected component of $x$ in $S$ is a subset of the connected component of $x$ in $T$. |
The connected component of $x$ in $S$ is empty if and only if $x$ is not in $S$. |
The connected component of the empty set is the empty set. |
If $y$ is in the connected component of $x$, then the connected component of $y$ is the same as the connected component of $x$. |
If $S$ is a closed set, then the connected component of $S$ containing $x$ is closed. |
The connected component of $a$ is disjoint from the connected component of $b$ if and only if $a$ is not in the connected component of $b$. |
The connected components of a set $S$ are pairwise disjoint if and only if they are distinct. |
The intersection of two connected components is nonempty if and only if they are the same connected component. |
The connected component of $x$ in $S$ is the same as the connected component of $y$ in $S$. |
The connected component of $x$ is equal to the connected component of $y$ if and only if $x$ and $y$ are either both not in $S$ or both in $S$ and $x$ and $y$ are in the same connected component of $S$. |
A set $S$ is connected if and only if all of its connected components are equal. |
The connected component of a point $x$ in a set $S$ is the same as the connected component of $x$ in the connected component of $x$ in $S$. |
If $x$ is in a connected set $c$ and $c$ is contained in $S$, and if $c$ is the only connected set containing $x$ that is contained in $S$, then $c$ is the connected component of $x$ in $S$. |
If $T$ is a connected subset of a topological space $S$, and $x$ and $y$ are points of $T$ that belong to the same connected component of $S$, then $x$ and $y$ belong to the same connected component of $T$. |
The union of all connected components of a set $S$ is $S$. |
The complement of a connected component of a set $S$ is the union of all the other connected components of $S$. |
If $T$ is a subset of $U$ and $a$ is a point in $U$, then the connected component of $a$ in $T$ is the same as the connected component of $a$ in $U$. |
A set $S$ is a component of $U$ if and only if there exists $x \in U$ such that $S$ is the connected component of $U$ containing $x$. |
If $x \in U$, then the connected component of $x$ in $U$ is a component of $U$. |
If $S$ is a component of $U$, then there exists $x \in U$ such that $S$ is the connected component of $U$ containing $x$. |
The union of the components of a topological space is the space itself. |
The components of a topological space are pairwise disjoint. |
The components of a topological space are nonempty. |
If $c$ is a component of $s$, then $c$ is a subset of $s$. |
If $c$ is a component of $s$, then $c$ is connected. |
A set $c$ is a component of $s$ if and only if $c$ is nonempty, $c$ is a subset of $s$, $c$ is connected, and $c$ is maximal with respect to these properties. |
If $t$ is a connected subset of $s$ and $c_1$ and $c_2$ are components of $s$ that intersect $t$, then $c_1 = c_2$. |
If $s$ is a closed set and $c$ is a component of $s$, then $c$ is closed. |
If $c$ and $c'$ are components of $s$, then $c$ and $c'$ are disjoint if and only if $c \neq c'$. |
Two components of a topological space are equal if and only if they intersect. |
The components of a set are empty if and only if the set is empty. |
The components of the empty set are the empty set. |
A set is connected if and only if it has exactly one connected component. |
A set $s$ is connected if and only if the set of components of $s$ is equal to $\{s\}$. |
A set $s$ has exactly one component if and only if $s$ is connected and nonempty. |
A set is connected if and only if it has no proper connected subsets. |
A set is connected if and only if all of its components are equal. |
A set $s$ is a component of itself if and only if it is connected and nonempty. |
If $c$ is a component of $s$, $t$ is a connected subset of $s$, and $t$ intersects $c$, then $t$ is contained in $c$. |
If $t$ is a connected subset of $s$ and $s$ is nonempty, then there exists a component of $s$ that contains $t$. |
If $s$ is a component of $u$ and $s \subseteq t \subseteq u$, then $s$ is a component of $t$. |
If $c$ is a component of $s$, then $s - c$ is the union of all components of $s$ other than $c$. |
If $s$ is connected and $t$ is contained in the closure of $s$, then $t$ is connected. |
The connected component of a point $x$ in a set $s$ is closed in the subspace topology on $s$. |
If $C$ is a component of a set $s$, then $C$ is closed in the topology of $s$. |
If $f$ is a continuous function on a connected set $S$, and the level set $\{x \in S : f(x) = a\}$ is open in $S$, then either $f$ is never equal to $a$ on $S$, or $f$ is always equal to $a$ on $S$. |
If $f$ is a continuous function on a connected set $S$, and the level set $\{x \in S \mid f(x) = a\}$ is open in $S$, then either $f$ is identically equal to $a$ on $S$, or $f$ is never equal to $a$ on $S$. |
If $f$ is a continuous function on a connected set $S$ and $f$ has an open level set, then $f$ is constant. |
If two topological spaces are homeomorphic, then they are connected if and only if they are connected. |
If $f$ is a continuous function from a topological space $S$ to a connected topological space $T$, and if the preimage of every point in $T$ is connected, then $S$ is connected. |
If $f$ is a continuous function from a set $S$ to a set $T$, and $f$ maps open sets in $S$ to open sets in $T$, and the preimage of any point in $T$ is connected, then the preimage of any connected set in $T$ is connected. |
If $f$ is a continuous function from a set $S$ to a set $T$, and if $f$ maps closed sets in $S$ to closed sets in $T$, and if the preimage of every point in $T$ is connected, then the preimage of every connected set in $T$ is connected. |
If $S$ and $U$ are connected sets with $S \subseteq U$, and $T$ is a clopen subset of $U - S$, then $S \cup T$ is connected. |
If $S$ and $U$ are connected sets with $S \subseteq U$, and $C$ is a component of $U - S$, then $U - C$ is connected. |
If $f$ is a continuous function from a connected set $S$ to a set $t$ such that for every $y \in t$, the connected component of $y$ in $t$ is just $\{y\}$, then $f$ is constant. |
If every continuous function from a set $S$ to a normed vector space $V$ with finite range is constant, then $S$ is connected. |
If $f$ and $g$ are differentiable at $z$, $f(z) = g(z) = 0$, $g'(z) \neq 0$, and $\frac{f'(z)}{g'(z)} = c$, then $\lim_{w \to z} \frac{f(w)}{g(w)} = c$. |
If a function is not integrable, then its integral is zero. |
If $f$ has a contour integral $i$ along $g$, then the contour integral of $f$ along $g$ is $i$. |
If $f$ has a contour integral along a path $p$ and $f$ is contour integrable along a path $\gamma$, and the contour integral of $f$ along $p$ is equal to the contour integral of $f$ along $\gamma$, then $f$ has a contour integral along $\gamma$. |
If $f$ is integrable on a contour $i$, then $f$ has a contour integral on $i$ equal to the value of the contour integral of $f$ on $i$. |
If a function $f$ has a contour integral $i$ along a contour $g$, and $f$ has a contour integral $j$ along the same contour $g$, then $i = j$. |
If a function $f$ has a contour integral along a contour $g$, then $f$ is contour integrable along $g$. |
The integral of $f(g(x)) \cdot g'(x)$ over $[a,b]$ is the same as the integral of $f(g(x)) \cdot g'(x)$ over $[a,b]$ with respect to the localized vector derivative. |
The integral of $f(g(x)) \cdot g'(x)$ is the same as the integral of $f(g(x)) \cdot g'(x)$ over the interval $[a,b]$. |
The contour integral of $f$ along a curve $g$ is equal to the integral of $f \circ g$ times the derivative of $g$. |
A function $f$ is contour-integrable on a path $g$ if and only if the function $t \mapsto f(g(t)) \cdot g'(t)$ is integrable on the interval $[0,1]$. |
If $f$ has a contour integral along a path $g$, then $f$ has a contour integral along the reverse path $g^{-1}$ with the opposite sign. |
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