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If $x \geq 0$, then $\sqrt{x} = \sqrt{x}$. |
$\sqrt{0} = 0$. |
$\sqrt{1} = 1$. |
$\sqrt{i} = \frac{1 + i}{\sqrt{2}}$. |
The square of the complex square root of $z$ is $z$. |
$\sqrt{z} = 0$ if and only if $z = 0$. |
$\sqrt{z} = 1$ if and only if $z = 1$. |
The real part of the principal square root of a complex number is nonnegative. |
The real part of the complex square root of a complex number is nonnegative. |
If $b$ is a complex number with nonnegative real part, then $\sqrt{b^2} = b$. |
If $w$ is a complex number such that $w^2 = z$ and $w$ is in the first or fourth quadrant, then $w$ is the principal square root of $z$. |
If $x$ is in the lower half-plane, then $\sqrt{-x} = i \sqrt{x}$. |
The complex modulus of a complex number $z$ is defined to be the real number $|z|$. |
The following are some basic properties of complex numbers. |
The complex number $x + 0i$ is a real number. |
If $z$ is a nonnegative real number, then $|z| = z$. |
For any real number $c$ and any natural number $n$, we have $\frac{(n+1)c}{(n+1)!} = \frac{c}{n!}$. |
If $f$ is differentiable at $x$ within $A$, then $\overline{f}$ is differentiable at $x$ within $A$, and the derivative of $\overline{f}$ at $x$ within $A$ is the complex conjugate of the derivative of $f$ at $x$ within $A$. |
If $f$ is differentiable at $x$, then the complex conjugate of $f$ is differentiable at $x$, and the derivative of the complex conjugate of $f$ is the complex conjugate of the derivative of $f$. |
The half-spaces $\{z \in \mathbb{C} : \Re(z) < b\}$, $\{z \in \mathbb{C} : \Re(z) > b\}$, $\{z \in \mathbb{C} : \Re(z) \geq b\}$, $\{z \in \mathbb{C} : \Re(z) \leq b\}$, $\{z \in \mathbb{C} : \Re(z) = b\}$, $\{z \in \mathbb{C} : \Im(z) < b\}$, $\{z \in \mathbb{C} : \Im(z) > b\}$, $\{z \in \mathbb{C} : \Im(z) \geq b\}$,... |
The set of real numbers is closed in the complex numbers. |
The set of complex numbers with real part less than or equal to $x$ is closed. |
The set of complex numbers with non-positive real part is closed. |
The set of complex numbers whose real part is greater than or equal to $x$ is closed. |
The set of nonnegative real numbers is closed in the complex numbers. |
The set of complex numbers whose real part has absolute value at most $r$ is closed. |
If $f$ converges to $l$ and $f$ is eventually real, then $l$ is real. |
If $f$ converges to $l$ and $f$ is eventually real, then $l$ is real. |
If a series of real numbers converges to a complex number, then the complex number is real. |
If $f$ is eventually bounded above by the real part of $g$, and $g$ converges to $0$, then $f$ converges to $0$. |
If $f$ is differentiable at every point in $s$, then $f$ is holomorphic on $s$. |
If $f$ is holomorphic on a set $S$, and $x \in S$, then $f$ is differentiable at $x$ on $S$. |
If $f$ is holomorphic on $s$, then $f$ is differentiable on $s$. |
If $f$ is holomorphic on an open set $s$ and $x \in s$, then $f$ is differentiable at $x$. |
The empty set is a domain. |
A function $f$ is holomorphic on an open set $S$ if and only if it is differentiable at every point of $S$. |
If $f$ is holomorphic on a set $s$, then $f$ is continuous on $s$. |
If $f$ is holomorphic on $s$, then $f$ is holomorphic on any subset $t$ of $s$. |
If $f$ is holomorphic on $s$ and $f(x) = g(x)$ for all $x \in s$, then $g$ is holomorphic on $s$. |
If two sets are equal and the functions defined on them are equal, then the functions are holomorphic on the sets if and only if the functions are holomorphic on the sets. |
The function $f(z) = cz$ is holomorphic on any set $s$. |
The constant function $f(z) = c$ is holomorphic on any set $s$. |
The identity function is holomorphic on any set. |
The identity function is holomorphic on any set. |
If $f$ is holomorphic on $s$ and $g$ is holomorphic on $f(s)$, then $g \circ f$ is holomorphic on $s$. |
If $f$ is holomorphic on $s$ and $g$ is holomorphic on $t$, and $f(s) \subseteq t$, then $g \circ f$ is holomorphic on $s$. |
If $f$ is holomorphic on every ball centered at $c$ and $A$ is unbounded above, then $f$ is entire. |
If $f$ is holomorphic on every ball centered at $c$, then $f$ is holomorphic on the whole complex plane. |
If $f$ is holomorphic on a set $s$, then $-f$ is holomorphic on $s$. |
If $f$ and $g$ are holomorphic on a set $S$, then $f + g$ is holomorphic on $S$. |
If $f$ and $g$ are holomorphic on a set $S$, then $f - g$ is holomorphic on $S$. |
If $f$ and $g$ are holomorphic on a set $S$, then $f \cdot g$ is holomorphic on $S$. |
If $f$ is holomorphic on a set $S$ and $f(z) \neq 0$ for all $z \in S$, then $1/f$ is holomorphic on $S$. |
If $f$ and $g$ are holomorphic functions on a set $S$ and $g$ is never zero on $S$, then the function $f/g$ is holomorphic on $S$. |
If $f$ is holomorphic on a set $S$, then $f^n$ is holomorphic on $S$. |
If each $f_i$ is holomorphic on $S$, then $\sum_{i \in I} f_i$ is holomorphic on $S$. |
If each $f_i$ is holomorphic on $S$, then the product function $\prod_{i \in I} f_i$ is holomorphic on $S$. |
If $f$ is holomorphic on a set $A$, then the function $s \mapsto (f(s))_n$ is holomorphic on $A$. |
If $f$ is holomorphic on a set $A$, then $c \cdot f$ is holomorphic on $A$ for any complex number $c$. |
If $f$ is holomorphic on two open sets $A$ and $B$, then $f$ is holomorphic on $A \cup B$. |
If $f$ and $g$ are holomorphic on open sets $A$ and $B$, respectively, and $f$ and $g$ agree on $A \cap B$, then the function $h$ defined by $h(z) = f(z)$ if $z \<in> A$ and $h(z) = g(z)$ if $z \<in> B$ is holomorphic on $A \cup B$. |
If $f$ is holomorphic on an open set $S$ and $x \in S$, then $f$ is differentiable at $x$ with derivative $f'(x)$. |
If $f$ and $g$ are holomorphic functions on an open set $S$ and $f(x) = g(x)$ for all $x \in S$, then $f'(x) = g'(x)$ for all $x \in S$. |
If $f$ is a holomorphic function on an open set $S$ and $f'(\xi) \neq 0$ for some $\xi \in S$, then $f$ is not constant on $S$. |
If $f$ is analytic on $S$, then $f$ is holomorphic on $S$. |
If $S$ is an open set, then $f$ is analytic on $S$ if and only if $f$ is holomorphic on $S$. |
If $f$ is analytic on a set $S$, then $f$ is differentiable at every point of $S$. |
If $f$ is analytic on a set $S$, then $f$ is analytic on any subset $T$ of $S$. |
A function is analytic on the union of two sets if and only if it is analytic on each of the sets. |
A function is analytic on a union of sets if and only if it is analytic on each of the sets. |
A function is analytic on the union of a collection of sets if and only if it is analytic on each set in the collection. |
A function $f$ is analytic on a set $S$ if and only if there exists an open set $T$ containing $S$ such that $f$ is holomorphic on $T$. |
The function $f(z) = cz$ is analytic on any set $S$. |
The constant function $f(z) = c$ is analytic on any set $S$. |
The identity function is analytic on any set. |
The identity function is analytic on any set. |
If $f$ is analytic on $S$ and $g$ is analytic on $f(S)$, then $g \circ f$ is analytic on $S$. |
If $f$ is analytic on $S$ and $g$ is analytic on $T$, and $f(S) \subseteq T$, then $g \circ f$ is analytic on $S$. |
If $f$ is analytic on $S$, then $-f$ is analytic on $S$. |
If $f$ and $g$ are analytic on a set $S$, then $f + g$ is analytic on $S$. |
If $f$ and $g$ are analytic on a set $S$, then $f - g$ is analytic on $S$. |
If $f$ and $g$ are analytic on a set $S$, then $f \cdot g$ is analytic on $S$. |
If $f$ is analytic on a set $S$ and $f(z) \neq 0$ for all $z \in S$, then the function $1/f$ is analytic on $S$. |
If $f$ and $g$ are analytic on a set $S$, and $g$ is nonzero on $S$, then $f/g$ is analytic on $S$. |
If $f$ is analytic on a set $S$, then $f^n$ is analytic on $S$. |
If each $f_i$ is analytic on $S$, then the sum $\sum_{i \in I} f_i$ is analytic on $S$. |
If $f$ and $g$ are holomorphic functions on open sets $S$ and $T$ respectively, and $f$ maps $S$ into $T$, and $g$ is the left inverse of $f$ on $S$, then $f$ and $g$ are differentiable at $w \in S$ and $f'(w)g'(f(w)) = 1$. |
A function $f$ is analytic at a point $z$ if and only if there exists an $\epsilon > 0$ such that $f$ is holomorphic on the open ball of radius $\epsilon$ centered at $z$. |
A function $f$ is analytic at a point $z$ if and only if there exists an open set $S$ containing $z$ such that $f$ is holomorphic on $S$. |
A function is analytic on a set $S$ if and only if it is analytic at every point of $S$. |
A function $f$ is analytic at a point $z$ if and only if there exists an open set $S$ containing $z$ such that $f$ is holomorphic on $S$. |
If $f$ and $g$ are analytic at $z$, then the derivatives of $f + g$, $f - g$, and $f \cdot g$ at $z$ are given by the usual formulas. |
If $f$ is analytic at $z$, then the derivative of $c f$ at $z$ is $c$ times the derivative of $f$ at $z$. |
If $f$ is analytic at $z$, then the derivative of $f(w) \cdot c$ at $z$ is $f'(z) \cdot c$. |
If a sequence of functions $f_n$ defined on a convex set $S$ has a pointwise limit $f$ and the derivatives $f'_n$ converge uniformly to a function $g'$, then $f$ is differentiable and $f' = g'$. |
If $f_n$ is a sequence of functions that are differentiable on a convex set $S$, and if the sequence of derivatives $f'_n$ converges uniformly to a function $g'$, then the sequence of functions $f_n$ converges uniformly to a function $g$ whose derivative is $g'$. |
For any function $f$ from the natural numbers to an abelian group, we have $\sum_{i=0}^n f(i) = f(0) - f(n+1) + \sum_{i=0}^n f(i+1)$. |
If $f$ is a function with $n$ continuous derivatives on a convex set $S$, and if $f^{(n+1)}$ is bounded on $S$, then $f$ is approximated by its Taylor polynomial of degree $n$ on $S$. |
Suppose $f$ is a function from a convex set $S$ to the complex numbers, and suppose that $f$ has continuous derivatives up to order $n$. Suppose also that the $n+1$st derivative of $f$ is bounded by $B$. Then the Taylor polynomial of order $n$ for $f$ at $w$ is a good approximation to $f$ at $z$. |
Suppose $f$ is a complex-valued function defined on a line segment $[w, z]$ and $f$ is differentiable at every point of $[w, z]$. Then there exists a point $u$ in $[w, z]$ such that $f(z) - f(w) = f'(u) (z - w)$. |
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