Beta function

Symbol: BetaFunction —

\mathrm{B}\!\left(a, b\right)

— Beta function

Definitions:

Fungrim symbol	Notation	Short description
`BetaFunction`	$\mathrm{B}\!\left(a, b\right)$	Beta function

Source code for this entry:

Entry(ID("1c46fa"),
    SymbolDefinition(BetaFunction, BetaFunction(a, b), "Beta function"))

795fe5

Symbol: IncompleteBeta —

\mathrm{B}_{x}\!\left(a, b\right)

— Incomplete beta function

Definitions:

Fungrim symbol	Notation	Short description
`IncompleteBeta`	$\mathrm{B}_{x}\!\left(a, b\right)$	Incomplete beta function

Source code for this entry:

Entry(ID("795fe5"),
    SymbolDefinition(IncompleteBeta, IncompleteBeta(x, a, b), "Incomplete beta function"))

6bd011

Symbol: IncompleteBetaRegularized —

I_{x}\!\left(a, b\right)

— Regularized incomplete beta function

Definitions:

Fungrim symbol	Notation	Short description
`IncompleteBetaRegularized`	$I_{x}\!\left(a, b\right)$	Regularized incomplete beta function

Source code for this entry:

Entry(ID("6bd011"),
    SymbolDefinition(IncompleteBetaRegularized, IncompleteBetaRegularized(x, a, b), "Regularized incomplete beta function"))

888581

\mathrm{B}\!\left(a, b\right) = \frac{\Gamma(a) \Gamma(b)}{\Gamma\!\left(a + b\right)}

Assumptions:

a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \setminus \{0, -1, \ldots\}

TeX:

\mathrm{B}\!\left(a, b\right) = \frac{\Gamma(a) \Gamma(b)}{\Gamma\!\left(a + b\right)}

a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \setminus \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`BetaFunction`	$\mathrm{B}\!\left(a, b\right)$	Beta function
`Gamma`	$\Gamma(z)$	Gamma function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("888581"),
    Formula(Equal(BetaFunction(a, b), Div(Mul(Gamma(a), Gamma(b)), Gamma(Add(a, b))))),
    Variables(a, b),
    Assumptions(And(Element(a, SetMinus(CC, ZZLessEqual(0))), Element(b, SetMinus(CC, ZZLessEqual(0))))))

c92da4

I_{x}\!\left(a, b\right) = \frac{\mathrm{B}_{x}\!\left(a, b\right)}{\mathrm{B}\!\left(a, b\right)}

Assumptions:

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; a + b \notin \{0, -1, \ldots\}

TeX:

I_{x}\!\left(a, b\right) = \frac{\mathrm{B}_{x}\!\left(a, b\right)}{\mathrm{B}\!\left(a, b\right)}

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; a + b \notin \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`IncompleteBetaRegularized`	$I_{x}\!\left(a, b\right)$	Regularized incomplete beta function
`IncompleteBeta`	$\mathrm{B}_{x}\!\left(a, b\right)$	Incomplete beta function
`BetaFunction`	$\mathrm{B}\!\left(a, b\right)$	Beta function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("c92da4"),
    Formula(Equal(IncompleteBetaRegularized(x, a, b), Div(IncompleteBeta(x, a, b), BetaFunction(a, b)))),
    Variables(x, a, b),
    Assumptions(And(Element(x, CC), Element(a, SetMinus(CC, ZZLessEqual(0))), Element(b, SetMinus(CC, ZZLessEqual(0))), NotElement(Add(a, b), ZZLessEqual(0)))))

ba7baf

\mathrm{B}_{0}\!\left(a, b\right) = 0

Assumptions:

a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \setminus \{0, -1, \ldots\}

TeX:

\mathrm{B}_{0}\!\left(a, b\right) = 0

a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \setminus \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`IncompleteBeta`	$\mathrm{B}_{x}\!\left(a, b\right)$	Incomplete beta function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("ba7baf"),
    Formula(Equal(IncompleteBeta(0, a, b), 0)),
    Variables(a, b),
    Assumptions(And(Element(a, SetMinus(CC, ZZLessEqual(0))), Element(b, SetMinus(CC, ZZLessEqual(0))))))

3141e4

\mathrm{B}_{1}\!\left(a, b\right) = \mathrm{B}\!\left(a, b\right)

Assumptions:

a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \setminus \{0, -1, \ldots\}

TeX:

\mathrm{B}_{1}\!\left(a, b\right) = \mathrm{B}\!\left(a, b\right)

a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \setminus \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`IncompleteBeta`	$\mathrm{B}_{x}\!\left(a, b\right)$	Incomplete beta function
`BetaFunction`	$\mathrm{B}\!\left(a, b\right)$	Beta function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("3141e4"),
    Formula(Equal(IncompleteBeta(1, a, b), BetaFunction(a, b))),
    Variables(a, b),
    Assumptions(And(Element(a, SetMinus(CC, ZZLessEqual(0))), Element(b, SetMinus(CC, ZZLessEqual(0))))))

ff613a

I_{0}\!\left(a, b\right) = 0

Assumptions:

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; a + b \notin \{0, -1, \ldots\}

TeX:

I_{0}\!\left(a, b\right) = 0

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; a + b \notin \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`IncompleteBetaRegularized`	$I_{x}\!\left(a, b\right)$	Regularized incomplete beta function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("ff613a"),
    Formula(Equal(IncompleteBetaRegularized(0, a, b), 0)),
    Variables(x, a, b),
    Assumptions(And(Element(x, CC), Element(a, SetMinus(CC, ZZLessEqual(0))), Element(b, SetMinus(CC, ZZLessEqual(0))), NotElement(Add(a, b), ZZLessEqual(0)))))

6bcfa6

I_{1}\!\left(a, b\right) = 1

Assumptions:

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; a + b \notin \{0, -1, \ldots\}

TeX:

I_{1}\!\left(a, b\right) = 1

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; a + b \notin \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`IncompleteBetaRegularized`	$I_{x}\!\left(a, b\right)$	Regularized incomplete beta function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("6bcfa6"),
    Formula(Equal(IncompleteBetaRegularized(1, a, b), 1)),
    Variables(x, a, b),
    Assumptions(And(Element(x, CC), Element(a, SetMinus(CC, ZZLessEqual(0))), Element(b, SetMinus(CC, ZZLessEqual(0))), NotElement(Add(a, b), ZZLessEqual(0)))))

542cf7

\mathrm{B}\!\left(a, b\right) = \int_{0}^{1} {t}^{a - 1} {\left(1 - t\right)}^{b - 1} \, dt

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(b) > 0

TeX:

\mathrm{B}\!\left(a, b\right) = \int_{0}^{1} {t}^{a - 1} {\left(1 - t\right)}^{b - 1} \, dt

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(b) > 0

Definitions:

Fungrim symbol	Notation	Short description
`BetaFunction`	$\mathrm{B}\!\left(a, b\right)$	Beta function
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("542cf7"),
    Formula(Equal(BetaFunction(a, b), Integral(Mul(Pow(t, Sub(a, 1)), Pow(Sub(1, t), Sub(b, 1))), For(t, 0, 1)))),
    Variables(a, b),
    Assumptions(And(Element(a, CC), Element(b, CC), Greater(Re(a), 0), Greater(Re(b), 0))))

48910b

\mathrm{B}\!\left(a, b\right) = 2 \int_{0}^{\pi / 2} \sin^{2 a - 1}\!\left(t\right) \cos^{2 b - 1}\!\left(t\right) \, dt

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(b) > 0

TeX:

\mathrm{B}\!\left(a, b\right) = 2 \int_{0}^{\pi / 2} \sin^{2 a - 1}\!\left(t\right) \cos^{2 b - 1}\!\left(t\right) \, dt

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(b) > 0

Definitions:

Fungrim symbol	Notation	Short description
`BetaFunction`	$\mathrm{B}\!\left(a, b\right)$	Beta function
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Pow`	${a}^{b}$	Power
`Sin`	$\sin(z)$	Sine
`Cos`	$\cos(z)$	Cosine
`Pi`	$\pi$	The constant pi (3.14...)
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("48910b"),
    Formula(Equal(BetaFunction(a, b), Mul(2, Integral(Mul(Pow(Sin(t), Sub(Mul(2, a), 1)), Pow(Cos(t), Sub(Mul(2, b), 1))), For(t, 0, Div(Pi, 2)))))),
    Variables(a, b),
    Assumptions(And(Element(a, CC), Element(b, CC), Greater(Re(a), 0), Greater(Re(b), 0))))

3e08b6

\mathrm{B}_{x}\!\left(a, b\right) = \int_{0}^{x} {t}^{a - 1} {\left(1 - t\right)}^{b - 1} \, dt

Assumptions:

x \in \left[0, 1\right] \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(b) > 0

TeX:

\mathrm{B}_{x}\!\left(a, b\right) = \int_{0}^{x} {t}^{a - 1} {\left(1 - t\right)}^{b - 1} \, dt

x \in \left[0, 1\right] \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(b) > 0

Definitions:

Fungrim symbol	Notation	Short description
`IncompleteBeta`	$\mathrm{B}_{x}\!\left(a, b\right)$	Incomplete beta function
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Pow`	${a}^{b}$	Power
`ClosedInterval`	$\left[a, b\right]$	Closed interval
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("3e08b6"),
    Formula(Equal(IncompleteBeta(x, a, b), Integral(Mul(Pow(t, Sub(a, 1)), Pow(Sub(1, t), Sub(b, 1))), For(t, 0, x)))),
    Variables(x, a, b),
    Assumptions(And(Element(x, ClosedInterval(0, 1)), Element(a, CC), Element(b, CC), Greater(Re(a), 0), Greater(Re(b), 0))))

a1941b

I_{x}\!\left(a, b\right) = \frac{1}{\mathrm{B}\!\left(a, b\right)} \int_{0}^{x} {t}^{a - 1} {\left(1 - t\right)}^{b - 1} \, dt

Assumptions:

x \in \left[0, 1\right] \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(b) > 0

TeX:

I_{x}\!\left(a, b\right) = \frac{1}{\mathrm{B}\!\left(a, b\right)} \int_{0}^{x} {t}^{a - 1} {\left(1 - t\right)}^{b - 1} \, dt

x \in \left[0, 1\right] \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(b) > 0

Definitions:

Fungrim symbol	Notation	Short description
`IncompleteBetaRegularized`	$I_{x}\!\left(a, b\right)$	Regularized incomplete beta function
`BetaFunction`	$\mathrm{B}\!\left(a, b\right)$	Beta function
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Pow`	${a}^{b}$	Power
`ClosedInterval`	$\left[a, b\right]$	Closed interval
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("a1941b"),
    Formula(Equal(IncompleteBetaRegularized(x, a, b), Mul(Div(1, BetaFunction(a, b)), Integral(Mul(Pow(t, Sub(a, 1)), Pow(Sub(1, t), Sub(b, 1))), For(t, 0, x))))),
    Variables(x, a, b),
    Assumptions(And(Element(x, ClosedInterval(0, 1)), Element(a, CC), Element(b, CC), Greater(Re(a), 0), Greater(Re(b), 0))))

5ec9c0

\mathrm{B}_{x}\!\left(a, b\right) = \frac{{x}^{a}}{a} \,{}_2F_1\!\left(a, 1 - b, a + 1, x\right)

Assumptions:

a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \setminus \left\{1\right\} \;\mathbin{\operatorname{and}}\; \left(x \ne 0 \;\mathbin{\operatorname{or}}\; \operatorname{Re}(a) > 0\right)

TeX:

\mathrm{B}_{x}\!\left(a, b\right) = \frac{{x}^{a}}{a} \,{}_2F_1\!\left(a, 1 - b, a + 1, x\right)

a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \setminus \left\{1\right\} \;\mathbin{\operatorname{and}}\; \left(x \ne 0 \;\mathbin{\operatorname{or}}\; \operatorname{Re}(a) > 0\right)

Definitions:

Fungrim symbol	Notation	Short description
`IncompleteBeta`	$\mathrm{B}_{x}\!\left(a, b\right)$	Incomplete beta function
`Pow`	${a}^{b}$	Power
`Hypergeometric2F1`	$\,{}_2F_1\!\left(a, b, c, z\right)$	Gauss hypergeometric function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("5ec9c0"),
    Formula(Equal(IncompleteBeta(x, a, b), Mul(Div(Pow(x, a), a), Hypergeometric2F1(a, Sub(1, b), Add(a, 1), x)))),
    Variables(x, a, b),
    Assumptions(And(Element(a, SetMinus(CC, ZZLessEqual(0))), Element(b, CC), Element(x, SetMinus(CC, Set(1))), Or(NotEqual(x, 0), Greater(Re(a), 0)))))

cc2ebb

\mathrm{B}\!\left(a, b\right) = \mathrm{B}\!\left(b, a\right)

Assumptions:

a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \setminus \{0, -1, \ldots\}

TeX:

\mathrm{B}\!\left(a, b\right) = \mathrm{B}\!\left(b, a\right)

a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \setminus \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`BetaFunction`	$\mathrm{B}\!\left(a, b\right)$	Beta function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("cc2ebb"),
    Formula(Equal(BetaFunction(a, b), BetaFunction(b, a))),
    Variables(a, b),
    Assumptions(And(Element(a, SetMinus(CC, ZZLessEqual(0))), Element(b, SetMinus(CC, ZZLessEqual(0))))))

315b3d

I_{x}\!\left(a, b\right) = 1 - I_{1 - x}\!\left(b, a\right)

Assumptions:

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; a + b \notin \{0, -1, \ldots\}

TeX:

I_{x}\!\left(a, b\right) = 1 - I_{1 - x}\!\left(b, a\right)

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; a + b \notin \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`IncompleteBetaRegularized`	$I_{x}\!\left(a, b\right)$	Regularized incomplete beta function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("315b3d"),
    Formula(Equal(IncompleteBetaRegularized(x, a, b), Sub(1, IncompleteBetaRegularized(Sub(1, x), b, a)))),
    Variables(a, b, x),
    Assumptions(And(Element(x, CC), Element(a, SetMinus(CC, ZZLessEqual(0))), Element(b, SetMinus(CC, ZZLessEqual(0))), NotElement(Add(a, b), ZZLessEqual(0)))))

fd0e48

\mathrm{B}\!\left(a, b\right) \mathrm{B}\!\left(a + b, c\right) = \mathrm{B}\!\left(b, c\right) \mathrm{B}\!\left(a, b + c\right)

Assumptions:

a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; c \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; a + b \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; b + c \notin \{0, -1, \ldots\}

TeX:

\mathrm{B}\!\left(a, b\right) \mathrm{B}\!\left(a + b, c\right) = \mathrm{B}\!\left(b, c\right) \mathrm{B}\!\left(a, b + c\right)

a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; c \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; a + b \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; b + c \notin \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`BetaFunction`	$\mathrm{B}\!\left(a, b\right)$	Beta function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("fd0e48"),
    Formula(Equal(Mul(BetaFunction(a, b), BetaFunction(Add(a, b), c)), Mul(BetaFunction(b, c), BetaFunction(a, Add(b, c))))),
    Variables(a, b, c),
    Assumptions(And(Element(a, SetMinus(CC, ZZLessEqual(0))), Element(b, SetMinus(CC, ZZLessEqual(0))), Element(c, SetMinus(CC, ZZLessEqual(0))), NotElement(Add(a, b), ZZLessEqual(0)), NotElement(Add(b, c), ZZLessEqual(0)))))

082a69

\mathrm{B}\!\left(m, n\right) = \frac{\left(m - 1\right)! \left(n - 1\right)!}{\left(m + n - 1\right)!}

Assumptions:

m \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1}

TeX:

\mathrm{B}\!\left(m, n\right) = \frac{\left(m - 1\right)! \left(n - 1\right)!}{\left(m + n - 1\right)!}

m \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`BetaFunction`	$\mathrm{B}\!\left(a, b\right)$	Beta function
`Factorial`	$n !$	Factorial
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("082a69"),
    Formula(Equal(BetaFunction(m, n), Div(Mul(Factorial(Sub(m, 1)), Factorial(Sub(n, 1))), Factorial(Sub(Add(m, n), 1))))),
    Variables(m, n),
    Assumptions(And(Element(m, ZZGreaterEqual(1)), Element(n, ZZGreaterEqual(1)))))

bb4f41

\mathrm{B}\!\left(m, n\right) = \frac{1}{m {m + n - 1 \choose m}}

Assumptions:

m \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1}

TeX:

\mathrm{B}\!\left(m, n\right) = \frac{1}{m {m + n - 1 \choose m}}

m \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`BetaFunction`	$\mathrm{B}\!\left(a, b\right)$	Beta function
`Binomial`	${n \choose k}$	Binomial coefficient
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("bb4f41"),
    Formula(Equal(BetaFunction(m, n), Div(1, Mul(m, Binomial(Sub(Add(m, n), 1), m))))),
    Variables(m, n),
    Assumptions(And(Element(m, ZZGreaterEqual(1)), Element(n, ZZGreaterEqual(1)))))

72db94

\mathrm{B}\!\left(n, b\right) = \begin{cases} {\tilde \infty}, & -b \in \{0, 1, \ldots, n - 1\}\\\frac{1}{n {n + b - 1 \choose n}}, & \text{otherwise}\\ \end{cases}

Assumptions:

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}

TeX:

\mathrm{B}\!\left(n, b\right) = \begin{cases} {\tilde \infty}, & -b \in \{0, 1, \ldots, n - 1\}\\\frac{1}{n {n + b - 1 \choose n}}, & \text{otherwise}\\ \end{cases}

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`BetaFunction`	$\mathrm{B}\!\left(a, b\right)$	Beta function
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity
`Range`	$\{a, a + 1, \ldots, b\}$	Integers between given endpoints
`Binomial`	${n \choose k}$	Binomial coefficient
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("72db94"),
    Formula(Equal(BetaFunction(n, b), Cases(Tuple(UnsignedInfinity, Element(Neg(b), Range(0, Sub(n, 1)))), Tuple(Div(1, Mul(n, Binomial(Sub(Add(n, b), 1), n))), Otherwise)))),
    Variables(n, b),
    Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(b, CC))))

a7dbf6

\mathrm{B}\!\left(-n, b\right) = \begin{cases} \frac{{\left(-1\right)}^{b}}{b {n \choose b}}, & b \in \{1, 2, \ldots, n\}\\{\tilde \infty}, & \text{otherwise}\\ \end{cases}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}

TeX:

\mathrm{B}\!\left(-n, b\right) = \begin{cases} \frac{{\left(-1\right)}^{b}}{b {n \choose b}}, & b \in \{1, 2, \ldots, n\}\\{\tilde \infty}, & \text{otherwise}\\ \end{cases}

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`BetaFunction`	$\mathrm{B}\!\left(a, b\right)$	Beta function
`Pow`	${a}^{b}$	Power
`Binomial`	${n \choose k}$	Binomial coefficient
`Range`	$\{a, a + 1, \ldots, b\}$	Integers between given endpoints
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("a7dbf6"),
    Formula(Equal(BetaFunction(Neg(n), b), Cases(Tuple(Div(Pow(-1, b), Mul(b, Binomial(n, b))), Element(b, Range(1, n))), Tuple(UnsignedInfinity, Otherwise)))),
    Variables(n, b),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(b, CC))))

1f72e9

\mathop{\operatorname{res}}\limits_{z=a} \mathrm{B}\!\left(z, b\right) = \begin{cases} {n - b \choose n}, & n \in \mathbb{Z}_{\ge 0}\\0, & \text{otherwise}\\ \end{cases}\; \text{ where } n = -a

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \setminus \{0, -1, \ldots\}

TeX:

\mathop{\operatorname{res}}\limits_{z=a} \mathrm{B}\!\left(z, b\right) = \begin{cases} {n - b \choose n}, & n \in \mathbb{Z}_{\ge 0}\\0, & \text{otherwise}\\ \end{cases}\; \text{ where } n = -a

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \setminus \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`Residue`	$\mathop{\operatorname{res}}\limits_{z=c} f(z)$	Complex residue
`BetaFunction`	$\mathrm{B}\!\left(a, b\right)$	Beta function
`Binomial`	${n \choose k}$	Binomial coefficient
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("1f72e9"),
    Formula(Equal(Residue(BetaFunction(z, b), For(z, a)), Where(Cases(Tuple(Binomial(Sub(n, b), n), Element(n, ZZGreaterEqual(0))), Tuple(0, Otherwise)), Equal(n, Neg(a))))),
    Variables(a, b),
    Assumptions(And(Element(a, CC), Element(b, SetMinus(CC, ZZLessEqual(0))))))

bdea17

\left(a + b\right) \mathrm{B}\!\left(a + 1, b\right) = a \mathrm{B}\!\left(a, b\right)

Assumptions:

a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \setminus \{0, -1, \ldots\}

TeX:

\left(a + b\right) \mathrm{B}\!\left(a + 1, b\right) = a \mathrm{B}\!\left(a, b\right)

a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \setminus \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`BetaFunction`	$\mathrm{B}\!\left(a, b\right)$	Beta function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("bdea17"),
    Formula(Equal(Mul(Add(a, b), BetaFunction(Add(a, 1), b)), Mul(a, BetaFunction(a, b)))),
    Variables(a, b),
    Assumptions(And(Element(a, SetMinus(CC, ZZLessEqual(0))), Element(b, SetMinus(CC, ZZLessEqual(0))))))

e9f966

\mathrm{B}\!\left(a, b\right) = \mathrm{B}\!\left(a + 1, b\right) + \mathrm{B}\!\left(a, b + 1\right)

Assumptions:

a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \setminus \{0, -1, \ldots\}

TeX:

\mathrm{B}\!\left(a, b\right) = \mathrm{B}\!\left(a + 1, b\right) + \mathrm{B}\!\left(a, b + 1\right)

a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \setminus \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`BetaFunction`	$\mathrm{B}\!\left(a, b\right)$	Beta function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("e9f966"),
    Formula(Equal(BetaFunction(a, b), Add(BetaFunction(Add(a, 1), b), BetaFunction(a, Add(b, 1))))),
    Variables(a, b),
    Assumptions(And(Element(a, SetMinus(CC, ZZLessEqual(0))), Element(b, SetMinus(CC, ZZLessEqual(0))))))

Definitions

Main formulas

Integral representations

Hypergeometric representations

Symmetry

Integer parameters

Recurrence relations