Imaginary unit

Symbol: ConstI —

i

— Imaginary unit

Represents the constant

i

, the imaginary unit.

Definitions:

Fungrim symbol	Notation	Short description
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("be8e05"),
    SymbolDefinition(ConstI, ConstI, "Imaginary unit"),
    Description("Represents the constant", i, ", the imaginary unit."))

88ad6f

i \in \mathbb{C}

TeX:

i \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("88ad6f"),
    Formula(Element(ConstI, CC)))

cd8a07

i \in \overline{\mathbb{Q}}

TeX:

i \in \overline{\mathbb{Q}}

Definitions:

Fungrim symbol	Notation	Short description
`ConstI`	$i$	Imaginary unit
`AlgebraicNumbers`	$\overline{\mathbb{Q}}$	Algebraic numbers

Source code for this entry:

Entry(ID("cd8a07"),
    Formula(Element(ConstI, AlgebraicNumbers)))

a08fb9

i \notin \mathbb{R}

TeX:

i \notin \mathbb{R}

Definitions:

Fungrim symbol	Notation	Short description
`ConstI`	$i$	Imaginary unit
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("a08fb9"),
    Formula(NotElement(ConstI, RR)))

08ad28

\mathop{\operatorname{solutions}\,}\limits_{x \in \mathbb{C}} \left[{x}^{2} + 1 = 0\right] = \left\{i, -i\right\}

TeX:

\mathop{\operatorname{solutions}\,}\limits_{x \in \mathbb{C}} \left[{x}^{2} + 1 = 0\right] = \left\{i, -i\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Solutions`	$\mathop{\operatorname{solutions}\,}\limits_{x \in S} Q(x)$	Solution set
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("08ad28"),
    Formula(Equal(Solutions(Brackets(Equal(Add(Pow(x, 2), 1), 0)), ForElement(x, CC)), Set(ConstI, Neg(ConstI)))))

72cef9

i = \sqrt{-1}

TeX:

i = \sqrt{-1}

Definitions:

Fungrim symbol	Notation	Short description
`ConstI`	$i$	Imaginary unit
`Sqrt`	$\sqrt{z}$	Principal square root

Source code for this entry:

Entry(ID("72cef9"),
    Formula(Equal(ConstI, Sqrt(-1))))

27586f

i = {\left(-1\right)}^{1 / 2}

TeX:

i = {\left(-1\right)}^{1 / 2}

Definitions:

Fungrim symbol	Notation	Short description
`ConstI`	$i$	Imaginary unit
`Pow`	${a}^{b}$	Power

Source code for this entry:

Entry(ID("27586f"),
    Formula(Equal(ConstI, Pow(-1, Div(1, 2)))))

65bbd6

\left|i\right| = 1

TeX:

\left|i\right| = 1

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("65bbd6"),
    Formula(Equal(Abs(ConstI), 1)))

249fd6

\operatorname{Re}(i) = 0

TeX:

\operatorname{Re}(i) = 0

Definitions:

Fungrim symbol	Notation	Short description
`Re`	$\operatorname{Re}(z)$	Real part
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("249fd6"),
    Formula(Equal(Re(ConstI), 0)))

61784f

\operatorname{Im}(i) = 1

TeX:

\operatorname{Im}(i) = 1

Definitions:

Fungrim symbol	Notation	Short description
`Im`	$\operatorname{Im}(z)$	Imaginary part
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("61784f"),
    Formula(Equal(Im(ConstI), 1)))

735409

\arg(i) = \frac{\pi}{2}

TeX:

\arg(i) = \frac{\pi}{2}

Definitions:

Fungrim symbol	Notation	Short description
`Arg`	$\arg(z)$	Complex argument
`ConstI`	$i$	Imaginary unit
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("735409"),
    Formula(Equal(Arg(ConstI), Div(Pi, 2))))

089f85

\arg\!\left(-i\right) = -\frac{\pi}{2}

TeX:

\arg\!\left(-i\right) = -\frac{\pi}{2}

Definitions:

Fungrim symbol	Notation	Short description
`Arg`	$\arg(z)$	Complex argument
`ConstI`	$i$	Imaginary unit
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("089f85"),
    Formula(Equal(Arg(Neg(ConstI)), Neg(Div(Pi, 2)))))

09c107

\operatorname{sgn}(i) = i

TeX:

\operatorname{sgn}(i) = i

Definitions:

Fungrim symbol	Notation	Short description
`Sign`	$\operatorname{sgn}(z)$	Sign function
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("09c107"),
    Formula(Equal(Sign(ConstI), ConstI)))

31b0df

{i}^{2} = -1

TeX:

{i}^{2} = -1

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("31b0df"),
    Formula(Equal(Pow(ConstI, 2), -1)))

8be138

{i}^{3} = -i

TeX:

{i}^{3} = -i

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("8be138"),
    Formula(Equal(Pow(ConstI, 3), Neg(ConstI))))

e0425a

{i}^{4} = 1

TeX:

{i}^{4} = 1

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("e0425a"),
    Formula(Equal(Pow(ConstI, 4), 1)))

c12a41

{i}^{n} = \begin{cases} 1, & n \equiv 0 \pmod {4}\\i, & n \equiv 1 \pmod {4}\\-1, & n \equiv 2 \pmod {4}\\-i, & n \equiv 3 \pmod {4}\\ \end{cases}

Assumptions:

n \in \mathbb{Z}

TeX:

{i}^{n} = \begin{cases} 1, & n \equiv 0 \pmod {4}\\i, & n \equiv 1 \pmod {4}\\-1, & n \equiv 2 \pmod {4}\\-i, & n \equiv 3 \pmod {4}\\ \end{cases}

n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`ConstI`	$i$	Imaginary unit
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("c12a41"),
    Formula(Equal(Pow(ConstI, n), Cases(Tuple(1, CongruentMod(n, 0, 4)), Tuple(ConstI, CongruentMod(n, 1, 4)), Tuple(-1, CongruentMod(n, 2, 4)), Tuple(Neg(ConstI), CongruentMod(n, 3, 4))))),
    Variables(n),
    Assumptions(Element(n, ZZ)))

44ae4a

\overline{i} = -i

TeX:

\overline{i} = -i

Definitions:

Fungrim symbol	Notation	Short description
`Conjugate`	$\overline{z}$	Complex conjugate
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("44ae4a"),
    Formula(Equal(Conjugate(ConstI), Neg(ConstI))))

67c262

\frac{1}{i} = -i

TeX:

\frac{1}{i} = -i

Definitions:

Fungrim symbol	Notation	Short description
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("67c262"),
    Formula(Equal(Div(1, ConstI), Neg(ConstI))))

f8a56f

{i}^{z} = {e}^{\pi i z / 2}

Assumptions:

z \in \mathbb{C}

TeX:

{i}^{z} = {e}^{\pi i z / 2}

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`ConstI`	$i$	Imaginary unit
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("f8a56f"),
    Formula(Equal(Pow(ConstI, z), Exp(Div(Mul(Mul(Pi, ConstI), z), 2)))),
    Variables(z),
    Assumptions(Element(z, CC)))

15f92d

{i}^{z} = \cos\!\left(\frac{\pi}{2} z\right) + \sin\!\left(\frac{\pi}{2} z\right) i

Assumptions:

z \in \mathbb{C}

TeX:

{i}^{z} = \cos\!\left(\frac{\pi}{2} z\right) + \sin\!\left(\frac{\pi}{2} z\right) i

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`ConstI`	$i$	Imaginary unit
`Cos`	$\cos(z)$	Cosine
`Pi`	$\pi$	The constant pi (3.14...)
`Sin`	$\sin(z)$	Sine
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("15f92d"),
    Formula(Equal(Pow(ConstI, z), Add(Cos(Mul(Div(Pi, 2), z)), Mul(Sin(Mul(Div(Pi, 2), z)), ConstI)))),
    Variables(z),
    Assumptions(Element(z, CC)))

0ad836

\sqrt{i} = \frac{1}{\sqrt{2}} \left(1 + i\right)

TeX:

\sqrt{i} = \frac{1}{\sqrt{2}} \left(1 + i\right)

Definitions:

Fungrim symbol	Notation	Short description
`Sqrt`	$\sqrt{z}$	Principal square root
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("0ad836"),
    Formula(Equal(Sqrt(ConstI), Mul(Div(1, Sqrt(2)), Add(1, ConstI)))))

a39534

{i}^{i} = {e}^{-\pi / 2}

TeX:

{i}^{i} = {e}^{-\pi / 2}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`ConstI`	$i$	Imaginary unit
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("a39534"),
    Formula(Equal(Pow(ConstI, ConstI), Exp(Neg(Div(Pi, 2))))))

c331da

\log(i) = \frac{\pi i}{2}

TeX:

\log(i) = \frac{\pi i}{2}

Definitions:

Fungrim symbol	Notation	Short description
`Log`	$\log(z)$	Natural logarithm
`ConstI`	$i$	Imaginary unit
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("c331da"),
    Formula(Equal(Log(ConstI), Div(Mul(Pi, ConstI), 2))))

9c93bb

\left|\Gamma(i)\right| = \sqrt{\frac{\pi}{\sinh(\pi)}}

TeX:

\left|\Gamma(i)\right| = \sqrt{\frac{\pi}{\sinh(\pi)}}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`Gamma`	$\Gamma(z)$	Gamma function
`ConstI`	$i$	Imaginary unit
`Sqrt`	$\sqrt{z}$	Principal square root
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("9c93bb"),
    Formula(Equal(Abs(Gamma(ConstI)), Sqrt(Div(Pi, Sinh(Pi))))))

3ac0ce

\operatorname{Im}\!\left(\psi\!\left(i\right)\right) = \frac{1}{2} \left(\pi \coth(\pi) + 1\right)

TeX:

\operatorname{Im}\!\left(\psi\!\left(i\right)\right) = \frac{1}{2} \left(\pi \coth(\pi) + 1\right)

Definitions:

Fungrim symbol	Notation	Short description
`Im`	$\operatorname{Im}(z)$	Imaginary part
`DigammaFunction`	$\psi\!\left(z\right)$	Digamma function
`ConstI`	$i$	Imaginary unit
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("3ac0ce"),
    Formula(Equal(Im(DigammaFunction(ConstI)), Mul(Div(1, 2), Add(Mul(Pi, Coth(Pi)), 1)))))

208da7

\operatorname{Li}_{2}\!\left(i\right) = -\frac{{\pi}^{2}}{48} + G i

TeX:

\operatorname{Li}_{2}\!\left(i\right) = -\frac{{\pi}^{2}}{48} + G i

Definitions:

Fungrim symbol	Notation	Short description
`ConstI`	$i$	Imaginary unit
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`ConstCatalan`	$G$	Catalan's constant

Source code for this entry:

Entry(ID("208da7"),
    Formula(Equal(PolyLog(2, ConstI), Add(Neg(Div(Pow(Pi, 2), 48)), Mul(ConstCatalan, ConstI)))))

Definitions

Domain

Quadratic equations

Numerical value

Complex parts

Transformations

Special functions at this value