Error functions

Symbol: Erf —

\operatorname{erf}\!\left(z\right)

— Error function

Definitions:

Fungrim symbol	Notation	Short description
`Erf`	$\operatorname{erf}\!\left(z\right)$	Error function

Source code for this entry:

Entry(ID("e46223"),
    SymbolDefinition(Erf, Erf(z), "Error function"))

7375c0

Symbol: Erfc —

\operatorname{erfc}\!\left(z\right)

— Complementary error function

Definitions:

Fungrim symbol	Notation	Short description
`Erfc`	$\operatorname{erfc}\!\left(z\right)$	Complementary error function

Source code for this entry:

Entry(ID("7375c0"),
    SymbolDefinition(Erfc, Erfc(z), "Complementary error function"))

d2914b

Symbol: Erfi —

\operatorname{erfi}\!\left(z\right)

— Imaginary error function

Definitions:

Fungrim symbol	Notation	Short description
`Erfi`	$\operatorname{erfi}\!\left(z\right)$	Imaginary error function

Source code for this entry:

Entry(ID("d2914b"),
    SymbolDefinition(Erfi, Erfi(z), "Imaginary error function"))

3be335

Image: X-ray of

\operatorname{erf}\!\left(z\right)

on

z \in \left[-4, 4\right] + \left[-4, 4\right] i

Download: png (small) — png (medium) — png (large) — pdf (small) — pdf (medium/large) — svg (small) — svg (medium/large)

An X-ray plot illustrates the geometry of a complex analytic function

f\!\left(z\right)

. Thick black curves show where

\operatorname{Im}\!\left(f\!\left(z\right)\right) = 0

(the function is pure real). Thick red curves show where

\operatorname{Re}\!\left(f\!\left(z\right)\right) = 0

(the function is pure imaginary). Points where black and red curves intersect are zeros or poles. Magnitude level curves

\left|f\!\left(z\right)\right| = C

are rendered as thin gray curves, with brighter shades corresponding to larger

C

. Blue lines show branch cuts. The value of the function is continuous with the branch cut on the side indicated with a solid line, and discontinuous on the side indicated with a dashed line. Yellow is used to highlight important regions.

Definitions:

Fungrim symbol	Notation	Short description
`Erf`	$\operatorname{erf}\!\left(z\right)$	Error function
`ClosedInterval`	$\left[a, b\right]$	Closed interval
`ConstI`	$i$	Imaginary unit
`Im`	$\operatorname{Im}\!\left(z\right)$	Imaginary part
`Re`	$\operatorname{Re}\!\left(z\right)$	Real part
`Abs`	$\left\|z\right\|$	Absolute value

Source code for this entry:

Entry(ID("3be335"),
    Image(Description("X-ray of", Erf(z), "on", Element(z, Add(ClosedInterval(-4, 4), Mul(ClosedInterval(-4, 4), ConstI)))), ImageSource("xray_erf")),
    Description("An X-ray plot illustrates the geometry of a complex analytic function", f(z), ".", "Thick black curves show where", Equal(Im(f(z)), 0), "(the function is pure real).", "Thick red curves show where", Equal(Re(f(z)), 0), "(the function is pure imaginary).", "Points where black and red curves intersect are zeros or poles.", "Magnitude level curves", Equal(Abs(f(z)), C), "are rendered as thin gray curves, with brighter shades corresponding to larger", C, ".", "Blue lines show branch cuts.", "The value of the function is continuous with the branch cut on the side indicated with a solid line, and discontinuous on the side indicated with a dashed line.", "Yellow is used to highlight important regions."))

2aaba8

\operatorname{erf}\!\left(z\right) = \frac{2}{\sqrt{\pi}} \int_{0}^{z} {e}^{-{t}^{2}} \, dt

Assumptions:

z \in \mathbb{C}

TeX:

\operatorname{erf}\!\left(z\right) = \frac{2}{\sqrt{\pi}} \int_{0}^{z} {e}^{-{t}^{2}} \, dt

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Erf`	$\operatorname{erf}\!\left(z\right)$	Error function
`Sqrt`	$\sqrt{z}$	Principal square root
`ConstPi`	$\pi$	The constant pi (3.14...)
`Exp`	${e}^{z}$	Exponential function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("2aaba8"),
    Formula(Equal(Erf(z), Mul(Div(2, Sqrt(ConstPi)), Integral(Exp(Neg(Pow(t, 2))), Tuple(t, 0, z))))),
    Variables(z),
    Assumptions(Element(z, CC)))

36ef64

\operatorname{erfc}\!\left(z\right) = \frac{2}{\sqrt{\pi}} \int_{z}^{\infty} {e}^{-{t}^{2}} \, dt

Assumptions:

z \in \mathbb{C}

TeX:

\operatorname{erfc}\!\left(z\right) = \frac{2}{\sqrt{\pi}} \int_{z}^{\infty} {e}^{-{t}^{2}} \, dt

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Erfc`	$\operatorname{erfc}\!\left(z\right)$	Complementary error function
`Sqrt`	$\sqrt{z}$	Principal square root
`ConstPi`	$\pi$	The constant pi (3.14...)
`Exp`	${e}^{z}$	Exponential function
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("36ef64"),
    Formula(Equal(Erfc(z), Mul(Div(2, Sqrt(ConstPi)), Integral(Exp(Neg(Pow(t, 2))), Tuple(t, z, Infinity))))),
    Variables(z),
    Assumptions(Element(z, CC)))

622772

\operatorname{erfi}\!\left(z\right) = \frac{2}{\sqrt{\pi}} \int_{0}^{z} {e}^{{t}^{2}} \, dt

Assumptions:

z \in \mathbb{C}

TeX:

\operatorname{erfi}\!\left(z\right) = \frac{2}{\sqrt{\pi}} \int_{0}^{z} {e}^{{t}^{2}} \, dt

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Erfi`	$\operatorname{erfi}\!\left(z\right)$	Imaginary error function
`Sqrt`	$\sqrt{z}$	Principal square root
`ConstPi`	$\pi$	The constant pi (3.14...)
`Exp`	${e}^{z}$	Exponential function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("622772"),
    Formula(Equal(Erfi(z), Mul(Div(2, Sqrt(ConstPi)), Integral(Exp(Pow(t, 2)), Tuple(t, 0, z))))),
    Variables(z),
    Assumptions(Element(z, CC)))

7f355d

\operatorname{erf}\!\left(z\right) + \operatorname{erfc}\!\left(z\right) = 1

Assumptions:

z \in \mathbb{C}

TeX:

\operatorname{erf}\!\left(z\right) + \operatorname{erfc}\!\left(z\right) = 1

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Erf`	$\operatorname{erf}\!\left(z\right)$	Error function
`Erfc`	$\operatorname{erfc}\!\left(z\right)$	Complementary error function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("7f355d"),
    Formula(Equal(Add(Erf(z), Erfc(z)), 1)),
    Variables(z),
    Assumptions(Element(z, CC)))

bfc86e

\operatorname{erfc}\!\left(z\right) = 1 - \operatorname{erf}\!\left(z\right)

Assumptions:

z \in \mathbb{C}

TeX:

\operatorname{erfc}\!\left(z\right) = 1 - \operatorname{erf}\!\left(z\right)

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Erfc`	$\operatorname{erfc}\!\left(z\right)$	Complementary error function
`Erf`	$\operatorname{erf}\!\left(z\right)$	Error function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("bfc86e"),
    Formula(Equal(Erfc(z), Sub(1, Erf(z)))),
    Variables(z),
    Assumptions(Element(z, CC)))

01440f

\operatorname{erfi}\!\left(z\right) = -i \operatorname{erf}\!\left(i z\right)

Assumptions:

z \in \mathbb{C}

TeX:

\operatorname{erfi}\!\left(z\right) = -i \operatorname{erf}\!\left(i z\right)

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Erfi`	$\operatorname{erfi}\!\left(z\right)$	Imaginary error function
`ConstI`	$i$	Imaginary unit
`Erf`	$\operatorname{erf}\!\left(z\right)$	Error function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("01440f"),
    Formula(Equal(Erfi(z), Neg(Mul(ConstI, Erf(Mul(ConstI, z)))))),
    Variables(z),
    Assumptions(Element(z, CC)))

94db18

\operatorname{erf}\!\left(-z\right) = -\operatorname{erf}\!\left(z\right)

Assumptions:

z \in \mathbb{C}

TeX:

\operatorname{erf}\!\left(-z\right) = -\operatorname{erf}\!\left(z\right)

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Erf`	$\operatorname{erf}\!\left(z\right)$	Error function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("94db18"),
    Formula(Equal(Erf(Neg(z)), Neg(Erf(z)))),
    Variables(z),
    Assumptions(Element(z, CC)))

603a49

\operatorname{erfi}\!\left(-z\right) = -\operatorname{erfi}\!\left(z\right)

Assumptions:

z \in \mathbb{C}

TeX:

\operatorname{erfi}\!\left(-z\right) = -\operatorname{erfi}\!\left(z\right)

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Erfi`	$\operatorname{erfi}\!\left(z\right)$	Imaginary error function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("603a49"),
    Formula(Equal(Erfi(Neg(z)), Neg(Erfi(z)))),
    Variables(z),
    Assumptions(Element(z, CC)))

ec0205

\operatorname{erfc}\!\left(-z\right) = 2 - \operatorname{erfc}\!\left(z\right)

Assumptions:

z \in \mathbb{C}

TeX:

\operatorname{erfc}\!\left(-z\right) = 2 - \operatorname{erfc}\!\left(z\right)

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Erfc`	$\operatorname{erfc}\!\left(z\right)$	Complementary error function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("ec0205"),
    Formula(Equal(Erfc(Neg(z)), Sub(2, Erfc(z)))),
    Variables(z),
    Assumptions(Element(z, CC)))

abadc7

\operatorname{erf}\!\left(z\right) = \frac{2 z}{\sqrt{\pi}} \,{}_1F_1\!\left(\frac{1}{2}, \frac{3}{2}, -{z}^{2}\right)

Assumptions:

z \in \mathbb{C}

TeX:

\operatorname{erf}\!\left(z\right) = \frac{2 z}{\sqrt{\pi}} \,{}_1F_1\!\left(\frac{1}{2}, \frac{3}{2}, -{z}^{2}\right)

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Erf`	$\operatorname{erf}\!\left(z\right)$	Error function
`Sqrt`	$\sqrt{z}$	Principal square root
`ConstPi`	$\pi$	The constant pi (3.14...)
`Hypergeometric1F1`	$\,{}_1F_1\!\left(a, b, z\right)$	Kummer confluent hypergeometric function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("abadc7"),
    Formula(Equal(Erf(z), Mul(Div(Mul(2, z), Sqrt(ConstPi)), Hypergeometric1F1(Div(1, 2), Div(3, 2), Neg(Pow(z, 2)))))),
    Variables(z),
    Assumptions(Element(z, CC)))

98688d

\operatorname{erf}\!\left(z\right) = \frac{2 z {e}^{-{z}^{2}}}{\sqrt{\pi}} \,{}_1F_1\!\left(1, \frac{3}{2}, {z}^{2}\right)

Assumptions:

z \in \mathbb{C}

TeX:

\operatorname{erf}\!\left(z\right) = \frac{2 z {e}^{-{z}^{2}}}{\sqrt{\pi}} \,{}_1F_1\!\left(1, \frac{3}{2}, {z}^{2}\right)

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Erf`	$\operatorname{erf}\!\left(z\right)$	Error function
`Exp`	${e}^{z}$	Exponential function
`Pow`	${a}^{b}$	Power
`Sqrt`	$\sqrt{z}$	Principal square root
`ConstPi`	$\pi$	The constant pi (3.14...)
`Hypergeometric1F1`	$\,{}_1F_1\!\left(a, b, z\right)$	Kummer confluent hypergeometric function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("98688d"),
    Formula(Equal(Erf(z), Mul(Div(Mul(Mul(2, z), Exp(Neg(Pow(z, 2)))), Sqrt(ConstPi)), Hypergeometric1F1(1, Div(3, 2), Pow(z, 2))))),
    Variables(z),
    Assumptions(Element(z, CC)))

cb93ea

\operatorname{erf}\!\left(z\right) = \frac{z}{\sqrt{{z}^{2}}} - \frac{{e}^{-{z}^{2}}}{z \sqrt{\pi}} U^{*}\!\left(\frac{1}{2}, \frac{1}{2}, {z}^{2}\right)

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \ne 0

TeX:

\operatorname{erf}\!\left(z\right) = \frac{z}{\sqrt{{z}^{2}}} - \frac{{e}^{-{z}^{2}}}{z \sqrt{\pi}} U^{*}\!\left(\frac{1}{2}, \frac{1}{2}, {z}^{2}\right)

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \ne 0

Definitions:

Fungrim symbol	Notation	Short description
`Erf`	$\operatorname{erf}\!\left(z\right)$	Error function
`Sqrt`	$\sqrt{z}$	Principal square root
`Pow`	${a}^{b}$	Power
`Exp`	${e}^{z}$	Exponential function
`ConstPi`	$\pi$	The constant pi (3.14...)
`HypergeometricUStar`	$U^{*}\!\left(a, b, z\right)$	Scaled Tricomi confluent hypergeometric function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("cb93ea"),
    Formula(Equal(Erf(z), Sub(Div(z, Sqrt(Pow(z, 2))), Mul(Div(Exp(Neg(Pow(z, 2))), Mul(z, Sqrt(ConstPi))), HypergeometricUStar(Div(1, 2), Div(1, 2), Pow(z, 2)))))),
    Variables(z),
    Assumptions(And(Element(z, CC), Unequal(z, 0))))

ae3110

\operatorname{erfc}\!\left(z\right) = \frac{{e}^{-{z}^{2}}}{z \sqrt{\pi}} U^{*}\!\left(\frac{1}{2}, \frac{1}{2}, {z}^{2}\right)

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(z\right) \gt 0

TeX:

\operatorname{erfc}\!\left(z\right) = \frac{{e}^{-{z}^{2}}}{z \sqrt{\pi}} U^{*}\!\left(\frac{1}{2}, \frac{1}{2}, {z}^{2}\right)

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(z\right) \gt 0

Definitions:

Fungrim symbol	Notation	Short description
`Erfc`	$\operatorname{erfc}\!\left(z\right)$	Complementary error function
`Exp`	${e}^{z}$	Exponential function
`Pow`	${a}^{b}$	Power
`Sqrt`	$\sqrt{z}$	Principal square root
`ConstPi`	$\pi$	The constant pi (3.14...)
`HypergeometricUStar`	$U^{*}\!\left(a, b, z\right)$	Scaled Tricomi confluent hypergeometric function
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}\!\left(z\right)$	Real part

Source code for this entry:

Entry(ID("ae3110"),
    Formula(Equal(Erfc(z), Mul(Div(Exp(Neg(Pow(z, 2))), Mul(z, Sqrt(ConstPi))), HypergeometricUStar(Div(1, 2), Div(1, 2), Pow(z, 2))))),
    Variables(z),
    Assumptions(And(Element(z, CC), Greater(Re(z), 0))))

b5bd5d

\operatorname{erf}'(z) = \frac{2}{\sqrt{\pi}} {e}^{-{z}^{2}}

Assumptions:

z \in \mathbb{C}

TeX:

\operatorname{erf}'(z) = \frac{2}{\sqrt{\pi}} {e}^{-{z}^{2}}

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Derivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Derivative
`Erf`	$\operatorname{erf}\!\left(z\right)$	Error function
`Sqrt`	$\sqrt{z}$	Principal square root
`ConstPi`	$\pi$	The constant pi (3.14...)
`Exp`	${e}^{z}$	Exponential function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("b5bd5d"),
    Formula(Equal(Derivative(Erf(z), Tuple(z, z, 1)), Mul(Div(2, Sqrt(ConstPi)), Exp(Neg(Pow(z, 2)))))),
    Variables(z),
    Assumptions(Element(z, CC)))

fae9d3

{\operatorname{erf}}^{(n)}(z) = \frac{2}{\sqrt{\pi}} {\left(-1\right)}^{n + 1} H_{n - 1}\!\left(z\right) {e}^{-{z}^{2}}

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}_{\ge 1}

TeX:

{\operatorname{erf}}^{(n)}(z) = \frac{2}{\sqrt{\pi}} {\left(-1\right)}^{n + 1} H_{n - 1}\!\left(z\right) {e}^{-{z}^{2}}

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`Derivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Derivative
`Erf`	$\operatorname{erf}\!\left(z\right)$	Error function
`Sqrt`	$\sqrt{z}$	Principal square root
`ConstPi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`HermitePolynomial`	$H_{n}\!\left(z\right)$	Hermite polynomial
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	Complex numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("fae9d3"),
    Formula(Equal(Derivative(Erf(z), Tuple(z, z, n)), Mul(Mul(Mul(Div(2, Sqrt(ConstPi)), Pow(-1, Add(n, 1))), HermitePolynomial(Sub(n, 1), z)), Exp(Neg(Pow(z, 2)))))),
    Variables(z, n),
    Assumptions(And(Element(z, CC), Element(n, ZZGreaterEqual(1)))))

Definitions

Illustrations

Integral representations

Connection formulas

Functional equations

Hypergeometric representations

Derivatives