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The complex conjugate of the factorial of a natural number is the factorial of that natural number. |
The complex conjugate of the Pochhammer symbol is the Pochhammer symbol of the complex conjugate. |
The complex conjugation map is a bounded linear operator. |
The complex conjugation map is linear. |
The complex conjugate function is continuous. |
The limit of the complex conjugate of a function is the complex conjugate of the limit of the function. |
If the complex conjugate of a series converges, then the series itself converges. |
The complex conjugate of a differentiable function is differentiable. |
If $f$ has a derivative at $z$, then $\overline{f}$ has a derivative at $z$. |
A complex number $z$ is equal to $0$ if and only if its real part squared plus its imaginary part squared is equal to $0$. |
A complex number $z$ is nonzero if and only if $z$ has nonzero norm. |
The square of the norm of a complex number is equal to the product of the complex number with its complex conjugate. |
The complex conjugate of the denominator of a complex fraction is the denominator of the complex conjugate of the fraction. |
The real part of a complex number $z$ is zero if and only if the real part of $z$ times the complex conjugate of $z$ is zero. |
The imaginary part of a complex number $z$ is zero if and only if the imaginary part of $z$ times the complex conjugate of $z$ is zero. |
The real and imaginary parts of a complex number $a/b$ are positive if and only if the real and imaginary parts of $a \overline{b}$ are positive. |
The real part of a complex number $z$ is positive if and only if the real part of $z \overline{z}$ is positive. The imaginary part of a complex number $z$ is positive if and only if the imaginary part of $z \overline{z}$ is positive. |
The real part of a complex number $a$ divided by a complex number $b$ is nonnegative if and only if the real part of $a$ times the complex conjugate of $b$ is nonnegative. |
The imaginary part of a complex number $a/b$ is non-negative if and only if the imaginary part of $a \overline{b}$ is non-negative. |
The real part of a complex number $a / b$ is negative if and only if the real part of $a \overline{b}$ is negative. |
The imaginary part of a complex number $a/b$ is negative if and only if the imaginary part of $a \overline{b}$ is negative. |
$\Re(\frac{a}{b}) \leq 0$ if and only if $\Re(a \overline{b}) \leq 0$. |
The imaginary part of a complex number $a$ divided by a complex number $b$ is less than or equal to $0$ if and only if the imaginary part of $a$ times the complex conjugate of $b$ is less than or equal to $0$. |
The real part of a complex number divided by a real number is the real part of the complex number divided by the real number. |
The imaginary part of a complex number divided by a real number is the imaginary part of the complex number divided by the real number. |
If $r$ is a real number, then $\frac{\text{Re}(z)}{\text{Re}(r)} = \text{Re}(\frac{z}{r})$. |
If $r$ is a real number, then $\frac{z}{r}$ is a complex number whose imaginary part is $\frac{\text{Im}(z)}{\text{Re}(r)}$. |
The real part of a sum of complex numbers is the sum of the real parts. |
The imaginary part of the sum of a function $f$ over a set $s$ is equal to the sum of the imaginary parts of $f$ over $s$. |
A complex series $\sum_{n=0}^\infty a_n$ converges if and only if both the real series $\sum_{n=0}^\infty \Re(a_n)$ and the imaginary series $\sum_{n=0}^\infty \Im(a_n)$ converge. |
A complex-valued function is summable if and only if its real and imaginary parts are summable. |
A real-valued sequence is summable if and only if the corresponding complex-valued sequence is summable. |
If a complex-valued function is summable, then its real part is also summable. |
If a complex-valued function is summable, then its imaginary part is also summable. |
A complex number $z$ is a natural number if and only if $z$ is real and $z$ is an integer. |
A complex number $z$ is an integer if and only if its imaginary part is zero and its real part is an integer. |
A complex number $z$ is real if and only if its imaginary part is zero. |
A complex number $z$ is real if and only if $z = \overline{z}$. |
If $z$ is a real number, then $|z| = |\Re(z)|$. |
If $r$ is a real number, then $\frac{\mathrm{Re}(r)}{z} = \frac{\mathrm{Re}(r) \cdot \mathrm{Re}(z)}{|z|^2}$. |
If $r$ is a real number and $z$ is a complex number, then the imaginary part of $r/z$ is equal to $-r \cdot \text{Im}(z) / |z|^2$. |
If $g$ is a summable sequence of non-negative real numbers, and $f$ is a sequence of complex numbers such that $|f_n| \leq g_n$ for all $n \geq N$, then $f$ is summable. |
If $z$ is a complex number with norm $1$, then there exists a real number $t$ such that $0 \leq t < 2\pi$ and $z = \cos(t) + i \sin(t)$. |
$\cos(0) + i \sin(0) = 1$. |
The norm of the complex number $e^{ia}$ is $1$. |
The sign of a complex number $z$ is $z$ itself. |
$\cos(2\pi) + i\sin(2\pi) = 1$. |
The complex number $e^{ia}$ is not equal to zero. |
The complex conjugate of $e^{it}$ is $e^{-it}$. |
$\mathrm{cis}(a) \mathrm{cis}(b) = \mathrm{cis}(a + b)$. |
$(e^{ia})^n = e^{in a}$. |
The inverse of $e^{ia}$ is $e^{-ia}$. |
$\frac{\text{cis}(a)}{\text{cis}(b)} = \text{cis}(a - b)$. |
$\cos(n\theta) = \Re(e^{i\theta})^n$ |
$\sin(na) = \Im(e^{ia}^n)$. |
$\cos(\pi) + i \sin(\pi) = -1$. |
$\cos(\pi/2) + i\sin(\pi/2) = i$. |
$\cos(-\pi/2) = -i$. |
For any integer $n$, $\cos(2\pi n) = 1$ and $\sin(2\pi n) = 0$. |
The real part of $e^{ia}$ is $cos(a)$. |
The imaginary part of $rcis(r, a)$ is $r \sin(a)$. |
Every complex number can be written in the form $re^{ia}$ for some real numbers $r$ and $a$. |
The modulus of a complex number of the form $re^{ia}$ is equal to the absolute value of $r$. |
$\mathrm{cis}(a) = \mathrm{rcis}(1, a)$. |
If $r_1$ and $r_2$ are real numbers and $a$ and $b$ are complex numbers, then $r_1 e^{ia} \cdot r_2 e^{ib} = (r_1 r_2) e^{i(a+b)}$. |
$\operatorname{rcis}(0, a) = 0$. |
The complex number $rcis(r, 0)$ is equal to the real number $r$. |
The real-valued complex exponential function $e^{ir}$ is zero if and only if $r = 0$. |
$(e^{i\theta})^n = e^{in\theta}$. |
The inverse of $z \mapsto r \exp(i a) z$ is $z \mapsto \frac{1}{r} \exp(-i a) z$. |
If $r_1$ and $r_2$ are positive real numbers and $a$ and $b$ are real numbers, then $\frac{e^{i r_1 a}}{e^{i r_2 b}} = e^{i \frac{r_1}{r_2} (a - b)}$. |
If $z$ is a real number, then $e^z = e^{\Re(z)}$. |
$\mathrm{cis}(b) = \exp(\mathrm{i} b)$. |
$\exp(z) = \exp(\Re(z)) \cdot \mathrm{cis}(\Im(z))$. |
The real part of $e^z$ is equal to $e^{\Re(z)} \cos(\Im(z))$. |
The imaginary part of $e^z$ is $e^{\Re(z)} \sin(\Im(z))$. |
$\|\cos(t) + i\sin(t)\| = 1$. |
The norm of the exponential function is the exponential function. |
Every complex number $z$ can be written as $r e^a$ for some real number $r$ and some complex number $a$. |
$e^{i * \pi} = -1$. |
$e^{i\pi} = -1$. |
$e^{2\pi i} = 1$. |
$e^{2\pi i} = 1$. |
If $f$ is a continuous function from $A$ to $\mathbb{R}$, then the function $x \mapsto \exp(i f(x))$ is continuous from $A$ to $\mathbb{C}$. |
The argument of $0$ is $0$. |
If $z$ is a complex number with $|z| = 1$ and $-\pi < x \leq \pi$, then $z = \cos(x) + i \sin(x)$ if and only if $x = \arg(z)$. |
If $z \neq 0$, then $z = \operatorname{sgn}(z) \cdot \exp(i \cdot \operatorname{arg}(z))$, and $-\pi < \operatorname{arg}(z) \leq \pi$. |
The argument of a complex number is bounded by $\pi$. |
If $z \neq 0$, then $\text{cis}(\text{arg}(z)) = \text{sgn}(z)$. |
The complex number $z$ can be written as $re^{i\theta}$, where $r = |z|$ and $\theta = \arg(z)$. |
If $y$ is a nonzero complex number with real part $0$, then $\cos(\arg(y)) = 0$. |
For any positive integer $n$, the map $k \mapsto e^{2 \pi i k / n}$ is a bijection from $\{0, 1, \ldots, n-1\}$ to the set of $n$th roots of unity. |
The number of $n$th roots of unity is $n$. |
If $c$ is a nonzero complex number and $n$ is a positive integer, then the map $z \mapsto c' z$ is a bijection from the set of $n$th roots of unity to the set of $n$th roots of $c$, where $c' = \sqrt[n]{|c|} e^{i \arg(c)/n}$. |
If $n > 0$, then the set of $n$th roots of $c$ is finite. |
If $c \neq 0$ and $n > 0$, then the number of $n$th roots of $c$ is $n$. |
If $n > 1$, then the sum of all the $n$th roots of unity is zero. |
If $n > 1$, then the sum of the $n$th roots of $c$ is $0$. |
If $x$ is a real number, then $\sqrt{x}$ is the square root of $x$. |
If $x$ is a real number, then $\sqrt{x}$ is a real number. |
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