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If $f_0, \ldots, f_n$ are functions from a closed interval $[w, z]$ to $\mathbb{C}$ such that $f_i$ is the $i$th derivative of $f_{i-1}$ for $i = 1, \ldots, n$, then there exists a point $u \in [w, z]$ such that $f_0(z) = \sum_{i=0}^n f_i(w) \frac{(z-w)^i}{i!} + f_{n+1}(u) \frac{(z-u)^n}{n!} (z-w)$. |
If two predicates are equal, then the set of all elements satisfying the predicates are equal. |
If two functions $f$ and $g$ are equal in a neighborhood of a point $z$, then their residues at $z$ are equal. |
If $f$ is holomorphic on the punctured disk $D(z,r)$, then the integral of $f$ over the circle of radius $r$ is equal to the integral of $f$ over the circle of radius $r'$ for any $0 < r' < r$. |
If $f$ is holomorphic on a punctured neighborhood of $z$, then the contour integral of $f$ around a circle of radius $r$ centered at $z$ is $2\pi i$ times the residue of $f$ at $z$. |
If $f$ is holomorphic on an open set $s$ and $z \in s$, then the residue of $f$ at $z$ is zero. |
The residue of a constant function is zero. |
If $f$ and $g$ are holomorphic functions on a punctured open set $s - \{z\}$, then the residue of $f + g$ at $z$ is the sum of the residues of $f$ and $g$ at $z$. |
If $f$ is holomorphic on a punctured neighborhood of $z$, then the residue of $f$ at $z$ is equal to the residue of $cf$ at $z$. |
If $f$ is holomorphic on a punctured neighborhood of $z$, then the residue of $f$ at $z$ is equal to the residue of $cf$ at $z$. |
If $f$ is holomorphic on a punctured neighborhood of $z$, then the residue of $f(z)/c$ at $z$ is equal to the residue of $f$ at $z$ divided by $c$. |
If $f$ is holomorphic on a punctured neighborhood of $z$, then the residue of $-f$ at $z$ is the negative of the residue of $f$ at $z$. |
If $f$ and $g$ are holomorphic functions on a domain $D$ except at a point $z \in D$, then the residue of $f - g$ at $z$ is equal to the residue of $f$ at $z$ minus the residue of $g$ at $z$. |
If $f$ is holomorphic on an open set $s$ and $z \in s$, then the residue of $f(w)/(w-z)$ at $z$ is $f(z)$. |
If $f$ is holomorphic on $s - \{z\}$ and $\lim_{w \to z} f(w) (w - z) = c$, then the residue of $f$ at $z$ is $c$. |
If $f$ is holomorphic on an open set $A$ and $z_0 \in A$, then the residue of $f(z)/(z-z_0)^{n+1}$ at $z_0$ is equal to $f^{(n)}(z_0)/n!$. |
If $f$ is holomorphic on an open set $A$ containing $0$, then the residue of $f(z)/z^{n+1}$ at $0$ is equal to the $n$th derivative of $f$ at $0$ divided by $n!$. |
If $f$ has a power series expansion $F$, then the $n$th coefficient of $F$ is equal to the residue of $f/z^{n+1}$ at $0$. |
If $f$ has a pole of order $n$ at $z$, then the residue of $f$ at $z$ is equal to the coefficient of $1/z^n$ in the Laurent series of $f$ at $z$. |
If $f$ has a simple pole at $z_0$, then the residue of $f$ at $z_0$ is equal to the value of $f$ at $z_0$. |
If $f$ has a simple pole at $z_0$, then the residue of $f$ at $z_0$ is equal to the limit of $f(g(x))(g(x) - z_0)$ as $x$ approaches $z_0$ along any filter $F$ such that $g(x)$ approaches $z_0$ along $F$. |
If $f$ and $g$ are holomorphic functions on a connected open set $S$, and $g$ has a simple zero at $z \in S$, then $f/g$ has a simple pole at $z$ and its residue is $f(z)/g'(z)$. |
The Gamma function has a pole at $-n$ for every non-negative integer $n$. |
The residue of the Gamma function at $-n$ is $(-1)^n/n!$. |
If two functions are equal in a neighborhood of a point, then they have the same pole at that point. |
If $f$ has a pole at $a$, and $f$ and $g$ agree in a neighborhood of $a$, then $g$ has a pole at $a$. |
If $f$ has a pole at $x$, then $\frac{1}{f}$ tends to $0$ at $x$. |
If $f$ is holomorphic on $s - \{z\}$ and $z$ is a pole of $f$, then $1/f$ is holomorphic on $s$. |
If $f$ is holomorphic on an open set $A$ and $x \in A$, then $f$ is not a pole at $x$. |
If $n > 0$, then $1 / (z - a)^n$ has a pole at $a$. |
The function $f(z) = 1 / (z - a)$ has a pole at $a$. |
If $f$ is a continuous function at $z$ and $g$ converges to $0$ at $z$, then $f/g$ has a pole at $z$. |
If $f$ is a holomorphic function on an open set $A$ and $z \in A$ is a zero of $f$ of order $n$, then $f(w)/(w-z)^n$ has a pole at $z$. |
If $f$ is holomorphic on an open set $A$ containing $0$, $f(0) \neq 0$, and $n > 0$, then $\frac{f(w)}{w^n}$ has a pole at $0$. |
If $f$ is holomorphic on a punctured ball $B(z,r)$ and can be written as $g(w)(w-z)^n$ and $h(w)(w-z)^m$ for some holomorphic functions $g$ and $h$ on $B(z,r)$, then $n=m$. |
If $f$ has an isolated singularity at $z$ and is not identically zero in a neighbourhood of $z$, then there exists a unique integer $n$ and a holomorphic function $g$ such that $f(w) = g(w)(w-z)^n$ for all $w$ in a punctured neighbourhood of $z$. |
If $g$ is not an essential singularity of $f$ at $z$, and $g$ converges to $f$ at $z$, then $f$ is not an essential singularity of $f$ at $z$. |
If $g$ has an isolated singularity at $z$ and $g$ and $f$ agree on a neighborhood of $z$, then $f$ has an isolated singularity at $z$. |
If $f$ has a limit at $z$, then $f^n$ is not essential at $z$. |
If $f$ has an isolated singularity at $z$ and $f$ is nonzero in a neighborhood of $z$, then $f^n$ has an isolated singularity at $z$. |
If $f$ has an isolated singularity at $z$ and $f$ is not identically zero in any neighborhood of $z$, then $f$ is not zero in any neighborhood of $z$. |
If $f$ has a pole at $z$, then $f$ is non-zero in a neighbourhood of $z$. |
If $f$ is holomorphic on an open connected set $S$ and $f(\beta) \neq 0$, then there is a sequence of points $z_n \in S$ converging to $z$ such that $f(z_n) \neq 0$ for all $n$. |
If $f$ and $g$ are holomorphic functions with isolated singularities at $z$, then $f \cdot g$ has an isolated singularity at $z$. |
If $f$ is not an essential singularity at $z$ and $f$ has an isolated singularity at $z$, then $1/f$ is not an essential singularity at $z$. |
If $f$ has an isolated singularity at $z$ and $f$ is not essential at $z$, then the inverse of $f$ has an isolated singularity at $z$. |
If $f$ and $g$ are holomorphic functions with isolated singularities at $z$, then $f/g$ has an isolated singularity at $z$. |
If $f$ and $g$ have isolated singularities at $z$, then $f + g$ and $f \cdot g$ also have isolated singularities at $z$. |
If $f$ has an isolated singularity at $z$, then $-f$ has an isolated singularity at $z$. |
The identity function has an isolated singularity at $z$. |
If $f$ and $g$ have isolated singularities at $z$, then $f - g$ has an isolated singularity at $z$. |
If $f$ and $g$ have isolated singularities at $z$ and $g$ is not essential at $z$, then $\frac{f}{g}$ has an isolated singularity at $z$. |
The function $f(w) = c$ has an isolated singularity at $z$. |
If $f$ is holomorphic on a punctured neighborhood of $z$, then $z$ is an isolated singularity of $f$. |
If $f$ is holomorphic at $z$ and $f$ is not identically zero in any neighborhood of $z$, then there exists a holomorphic function $g$ and a real number $n$ such that $g(z) \neq 0$ and $f(w) = g(w) (w-z)^n$ for all $w$ in some neighborhood of $z$. |
Suppose $f$ is a complex-valued function with an isolated singularity at $z$. If $f$ is not an essential singularity at $z$, then the zeroes of $f$ are isolated at $z$. |
If $f$ and $g$ are holomorphic functions with isolated singularities at $z$, and if $f$ and $g$ are not essential singularities at $z$, then the order of the product $f \cdot g$ at $z$ is the sum of the orders of $f$ and $g$ at $z$. |
Suppose $f$ and $g$ are holomorphic functions defined on a neighborhood of $z$ such that $f$ and $g$ have isolated singularities at $z$ and are not essential singularities at $z$. Suppose also that $f$ and $g$ are not identically zero on any neighborhood of $z$. Then the order of the function $f/g$ at $z$ is equal to t... |
If $f$ is a holomorphic function on an open connected set $S$ and $z \in S$, then there exists a positive integer $n$ and a holomorphic function $g$ on a neighborhood of $z$ such that $f(w) = g(w)(w-z)^n$ for all $w$ in that neighborhood. If $f(z) \neq 0$, then $n = 0$. |
If $f$ is holomorphic on a neighborhood of $z$ and $f$ has a pole at $z$, then there exists a positive real number $r$ such that $f$ is holomorphic on the open ball of radius $r$ centered at $z$ and $f$ has a zero of order $n$ at $z$, where $n$ is the order of the pole of $f$ at $z$. |
If $f$ and $g$ are holomorphic functions on an open set $s$ containing $z$, and $g(z) \neq 0$, and $f(w) = g(w) (w - z)^n$ for all $w \in s$ with $w \neq z$, then the order of $f$ at $z$ is $n$. |
If $f$ is holomorphic on an open set $s$ and $f(z) = g(z)(z - z_0)$ for all $z \in s$, then $z_0$ is a simple zero of $f$. |
The $j$th derivative of $(w - z)^n$ is $\binom{n}{j}(w - z)^{n - j}$. |
If $f$ is holomorphic on an open set $s$, and if $z \in s$ and $f$ has a zero of order $n$ at $z$, then $z$ is a zero of order $n$ of $f$. |
If $f$ and $g$ are analytic functions that agree on a neighborhood of $z$, then the order of $f$ at $z$ is equal to the order of $g$ at $z$. |
If $f$ is a holomorphic function on an open set $s$ and $z$ is a point in $s$ such that $f(z) \neq 0$, then the order of $f(w)/(w-z)^n$ at $z$ is $-n$. |
If $f$ has an isolated singularity at $z$ and $f$ is not essential at $z$, then there exists a function $g$ such that $g$ is analytic at $z$ and $g(z) = f(z)$. |
Suppose $f$ is a holomorphic function on an open connected set $S$ and $z \in S$. If $f$ is not identically zero on $S$, then the zero-order coefficient of the Taylor series of $f$ at $z$ is equal to $f(z) / (w - z)^n$, where $n$ is the order of the zero of $f$ at $z$. |
If $f$ has an isolated singularity at $z$ and is a pole at $z$, then the zero-order regular part of $f$ at $z$ is equal to $f(w) (w - z)^{-\operatorname{order}_z f}$ for all $w$ sufficiently close to $z$. |
If $f$ is a holomorphic function with an isolated singularity at $z_0$, and $g$ is a sequence converging to $z_0$, then the limit of $f(g(x))(g(x) - z_0)^{-n}$ is equal to the zero order coefficient of $f$ at $z_0$. |
Suppose $f$ is a holomorphic function on an open connected set $A$ containing $z_0$. Suppose $f$ is not identically zero on $A$. Let $n$ be the order of $f$ at $z_0$. Suppose $g$ is a function such that $g(x) \to z_0$ as $x \to z_0$. Suppose $f(g(x)) / (g(x) - z_0)^n \to c$ as $x \to z_0$. Then $zor\_poly(f, z_0, z_0) ... |
If $f$ has an isolated singularity at $z_0$ and $f$ has a pole at $z_0$, then the zero-order residue of $f$ at $z_0$ is equal to the limit of $f(g(x))(g(x) - z_0)^{-n}$ as $x$ approaches $z_0$ along any filter $F$ such that $g(x)$ approaches $z_0$ along $F$. |
If $f$ is a holomorphic function on an open connected set $S$ and $f(\xi) = 0$ for some $\xi \in S$, then there exists a neighborhood of $\xi$ in $S$ on which $f$ is nonzero. |
If $f$ is a holomorphic function on an open connected set $S$, and $U$ is a subset of $S$ such that $f$ is zero on $U$ and $\xi$ is a limit point of $U$ in $S$, then $f$ is zero at $\xi$. |
If $f$ and $g$ are holomorphic functions on an open set $S'$ and agree on an open subset $S$ of $S'$, then $f$ and $g$ agree on all of $S'$. |
Suppose $f$ is a holomorphic function on the open ball $B(\xi, r)$ and continuous on the closed ball $\overline{B}(\xi, r)$. If $f(\xi)$ is smaller than $f(z)$ for all $z$ on the boundary of $\overline{B}(\xi, r)$, then $f$ has a zero in $B(\xi, r)$. |
If $f$ is a non-constant holomorphic function on an open connected set $S$, then $f$ is an open map. |
If $f$ is a holomorphic function on an open set $S$, and $U$ is an open subset of $S$ such that $f$ is not constant on any nonempty open subset of $U$, then $f(U)$ is open. |
If $f$ is a holomorphic function on an open set $S$ and $f$ is injective on $S$, then $f(S)$ is open. |
If $f$ is a holomorphic function on an open connected set $S$ and $U$ is an open subset of $S$ such that $f$ attains its maximum modulus on $U$, then $f$ is constant on $S$. |
If $f$ is holomorphic on the interior of a bounded set $S$ and continuous on the closure of $S$, and if $f$ is bounded on the boundary of $S$, then $f$ is bounded on $S$. |
If $f$ is holomorphic on the interior of a bounded set $S$ and continuous on the closure of $S$, and if $f$ is bounded above on the boundary of $S$, then $f$ is bounded above on $S$. |
Suppose $f$ is a holomorphic function on an open set $S$, and $\xi \in S$. If $(\xi, f(\xi))$ is a zero of order $n$ of $f$, then there exists a holomorphic function $g$ and a positive real number $r$ such that for all $w \in B(\xi, r)$, we have $f(w) - f(\xi) = (w - \xi)^n g(w)$ and $g(w) \neq 0$. |
Suppose $f$ is a holomorphic function on an open set $S$, and $\xi \in S$. If $f$ has a zero of order $n$ at $\xi$, then there exists a holomorphic function $g$ and a positive real number $r$ such that for all $w \in B(\xi, r)$, we have $f(w) - f(\xi) = (w - \xi)^n g(w)$ and $g(w) \neq 0$. |
If $a$ is defined to be the least element of a set $S$ and $k$ is an element of $S$, then $a$ is an element of $S$ and $a \leq k$. |
Suppose $f$ is a holomorphic function defined on an open connected set $S$, and $f$ has a zero of order $n$ at $\xi \in S$. Then there exists a holomorphic function $g$ defined on a neighborhood of $\xi$ such that $f(w) = (w - \xi)^n g(w)$ for all $w$ in that neighborhood. |
If $f$ is a holomorphic function on an open connected set $S$ and $\xi, \phi \in S$ with $f(\phi) \neq f(\xi)$, then there exists a positive constant $k$ and a positive integer $n$ such that for all $w \in S$ with $|w - \xi| < r$, we have $k|w - \xi|^n \leq |f(w) - f(\xi)|$. |
If $f$ is holomorphic on $S - \{ \xi \}$ and $\xi$ is an interior point of $S$, then the following are equivalent: $f$ can be extended to a holomorphic function on $S$. $\lim_{z \to \xi} (z - \xi) f(z) = 0$. $f$ is bounded in a neighborhood of $\xi$. |
If $f$ is a holomorphic function on $\mathbb{C}$ and $\lim_{z \to \infty} \frac{1}{f(z)} = l$, then $f$ is a polynomial. |
If $S$ is a closed set and $K$ is a compact set, then the set of points $z$ in $S$ such that $\sum_{i=1}^n c_i z^i \in K$ is compact. |
If $K$ is a compact set and $c_1, \ldots, c_n$ are nonzero real numbers, then the set of all $z$ such that $c_1 z + \cdots + c_n z^n \in K$ is compact. |
A holomorphic function $f$ is proper if and only if there exist $c_0, c_1, \ldots, c_n$ and $n \in \mathbb{N}$ such that $c_n \neq 0$ and $f(z) = c_0 + c_1 z + \cdots + c_n z^n$. |
If $f$ is a holomorphic function on an open set $S$ and $\xi \in S$, then there exists a neighborhood $U$ of $\xi$ such that $f$ is injective on $U$. |
If $f$ is holomorphic on a neighborhood of $\xi$ and $f'(\xi) \neq 0$, then there exists a neighborhood $U$ of $\xi$ such that $f(U)$ is open and $f$ is injective on $U$. |
If $f$ is a holomorphic function on an open set $S$ and $f$ is injective on $S$, then $f'(\xi) \neq 0$ for all $\xi \<in> S$. |
If $f$ is a function with a nonzero derivative at $x$, and $g$ is the inverse of $f$, then $g$ has a derivative at $f(x)$ equal to the reciprocal of the derivative of $f$ at $x$. |
If $f$ is a function with a nonzero derivative at $x$, and $f$ is a bijection from an open set $S$ to its image, then the inverse function $g$ has a derivative at $y = f(x)$, and the derivative of $g$ at $y$ is the reciprocal of the derivative of $f$ at $x$. |
If $f$ is a holomorphic function on an open set $S$ and $f$ is injective on $S$, then $f$ has a holomorphic inverse on $f(S)$. |
If $f$ is holomorphic on an open, bounded, connected set $S$ and continuous on the closure of $S$, then there exists a point $w$ on the boundary of $S$ such that $|f(z)| \leq |f(w)|$ for all $z$ in the closure of $S$. |
If $f$ is holomorphic on a ball of radius $r$ centered at the origin, and if $w$ is a point in the ball such that $f$ is bounded on the ball of radius $s$ centered at $w$, then $f$ is constant on the ball of radius $r$ centered at the origin. |
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