Arithmetic-geometric mean

Symbol: AGM —

\operatorname{agm}\!\left(a, b\right)

— Arithmetic-geometric mean

This function can be called with one or two arguments, with

\operatorname{agm}(z) = \operatorname{agm}\!\left(1, z\right)

.

Definitions:

Fungrim symbol	Notation	Short description
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean

Source code for this entry:

Entry(ID("b0d256"),
    SymbolDefinition(AGM, AGM(a, b), "Arithmetic-geometric mean"),
    Description("This function can be called with one or two arguments, with", Equal(AGM(z), AGM(1, z)), "."))

eda1e3

Symbol: AGMSequence —

\operatorname{agm}_{n}\!\left(a, b\right)

— Convergents in AGM iteration

Represents the tuple

\left(a_{n}, b_{n}\right)

giving the

n

-th values in the arithmetic-geometric mean iteration with initial values

\left(a_{0}, b_{0}\right) = \left(a, b\right)

.

Definitions:

Fungrim symbol	Notation	Short description
`AGMSequence`	$\operatorname{agm}_{n}\!\left(a, b\right)$	Convergents in AGM iteration

Source code for this entry:

Entry(ID("eda1e3"),
    SymbolDefinition(AGMSequence, AGMSequence(n, a, b), "Convergents in AGM iteration"),
    Description("Represents the tuple", Tuple(a_(n), b_(n)), "giving the", n, "-th values in the arithmetic-geometric mean iteration with initial values", Equal(Tuple(a_(0), b_(0)), Tuple(a, b)), "."))

00d5df

Image: Plot of

\operatorname{agm}\!\left(1, x\right)

on

x \in \left[-2, 2\right]

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

Definitions:

Fungrim symbol	Notation	Short description
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`ClosedInterval`	$\left[a, b\right]$	Closed interval

Source code for this entry:

Entry(ID("00d5df"),
    Image(Description("Plot of", AGM(1, x), "on", Element(x, ClosedInterval(-2, 2))), ImageSource("plot_agm")))

c941c4

Image: X-ray of

\operatorname{agm}\!\left(1, 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(z)

. Thick black curves show where

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

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

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

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

\left|f(z)\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
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`ClosedInterval`	$\left[a, b\right]$	Closed interval
`ConstI`	$i$	Imaginary unit
`Im`	$\operatorname{Im}(z)$	Imaginary part
`Re`	$\operatorname{Re}(z)$	Real part
`Abs`	$\left\|z\right\|$	Absolute value

Source code for this entry:

Entry(ID("c941c4"),
    Image(Description("X-ray of", AGM(1, z), "on", Element(z, Add(ClosedInterval(-4, 4), Mul(ClosedInterval(-4, 4), ConstI)))), ImageSource("xray_agm")),
    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."))

21f412

\operatorname{agm}(z) = \operatorname{agm}\!\left(1, z\right) = \operatorname{agm}\!\left(z, 1\right)

Assumptions:

z \in \mathbb{C}

TeX:

\operatorname{agm}(z) = \operatorname{agm}\!\left(1, z\right) = \operatorname{agm}\!\left(z, 1\right)

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("21f412"),
    Formula(Equal(AGM(z), AGM(1, z), AGM(z, 1))),
    Variables(z),
    Assumptions(Element(z, CC)))

32b3b4

\left(a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}\right) \;\implies\; \operatorname{agm}\!\left(a, b\right) \in \mathbb{C}

TeX:

\left(a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}\right) \;\implies\; \operatorname{agm}\!\left(a, b\right) \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`CC`	$\mathbb{C}$	Complex numbers
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean

Source code for this entry:

Entry(ID("32b3b4"),
    Formula(Implies(And(Element(a, CC), Element(b, CC)), Element(AGM(a, b), CC))),
    Variables(a, b))

c38209

\left(a \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; b \in \left[0, \infty\right)\right) \;\implies\; \operatorname{agm}\!\left(a, b\right) \in \left[0, \infty\right)

TeX:

\left(a \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; b \in \left[0, \infty\right)\right) \;\implies\; \operatorname{agm}\!\left(a, b\right) \in \left[0, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`Infinity`	$\infty$	Positive infinity
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean

Source code for this entry:

Entry(ID("c38209"),
    Formula(Implies(And(Element(a, ClosedOpenInterval(0, Infinity)), Element(b, ClosedOpenInterval(0, Infinity))), Element(AGM(a, b), ClosedOpenInterval(0, Infinity)))),
    Variables(a, b))

f9caac

x \in \mathbb{C} \;\implies\; \operatorname{agm}(x) \in \mathbb{C}

TeX:

x \in \mathbb{C} \;\implies\; \operatorname{agm}(x) \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`CC`	$\mathbb{C}$	Complex numbers
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean

Source code for this entry:

Entry(ID("f9caac"),
    Formula(Implies(Element(x, CC), Element(AGM(x), CC))),
    Variables(x))

098757

x \in \left[0, \infty\right) \;\implies\; \operatorname{agm}(x) \in \left[0, \infty\right)

TeX:

x \in \left[0, \infty\right) \;\implies\; \operatorname{agm}(x) \in \left[0, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`Infinity`	$\infty$	Positive infinity
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean

Source code for this entry:

Entry(ID("098757"),
    Formula(Implies(Element(x, ClosedOpenInterval(0, Infinity)), Element(AGM(x), ClosedOpenInterval(0, Infinity)))),
    Variables(x))

08329d

\operatorname{agm}\!\left(0, b\right) = 0

Assumptions:

b \in \mathbb{C}

TeX:

\operatorname{agm}\!\left(0, b\right) = 0

b \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("08329d"),
    Formula(Equal(AGM(0, b), 0)),
    Variables(b),
    Assumptions(Element(b, CC)))

8f176c

\operatorname{agm}\!\left(a, 0\right) = 0

Assumptions:

a \in \mathbb{C}

TeX:

\operatorname{agm}\!\left(a, 0\right) = 0

a \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("8f176c"),
    Formula(Equal(AGM(a, 0), 0)),
    Variables(a),
    Assumptions(Element(a, CC)))

3e1398

\operatorname{agm}\!\left(a, -a\right) = 0

Assumptions:

a \in \mathbb{C}

TeX:

\operatorname{agm}\!\left(a, -a\right) = 0

a \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("3e1398"),
    Formula(Equal(AGM(a, Neg(a)), 0)),
    Variables(a),
    Assumptions(Element(a, CC)))

b41bdd

\operatorname{agm}\!\left(a, a\right) = a

Assumptions:

a \in \mathbb{C}

TeX:

\operatorname{agm}\!\left(a, a\right) = a

a \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("b41bdd"),
    Formula(Equal(AGM(a, a), a)),
    Variables(a),
    Assumptions(Element(a, CC)))

0d9352

\operatorname{agm}\!\left(1, \sqrt{2}\right) = \frac{2 \sqrt{2} {\pi}^{3 / 2}}{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}

TeX:

\operatorname{agm}\!\left(1, \sqrt{2}\right) = \frac{2 \sqrt{2} {\pi}^{3 / 2}}{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}

Definitions:

Fungrim symbol	Notation	Short description
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`Sqrt`	$\sqrt{z}$	Principal square root
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`Gamma`	$\Gamma(z)$	Gamma function

Source code for this entry:

Entry(ID("0d9352"),
    Formula(Equal(AGM(1, Sqrt(2)), Div(Mul(2, Sqrt(2), Pow(Pi, Div(3, 2))), Pow(Gamma(Div(1, 4)), 2)))))

e3896e

\operatorname{agm}\!\left(1, \frac{\sqrt{2}}{2}\right) = \frac{2 {\pi}^{3 / 2}}{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}

TeX:

\operatorname{agm}\!\left(1, \frac{\sqrt{2}}{2}\right) = \frac{2 {\pi}^{3 / 2}}{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}

Definitions:

Fungrim symbol	Notation	Short description
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`Sqrt`	$\sqrt{z}$	Principal square root
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`Gamma`	$\Gamma(z)$	Gamma function

Source code for this entry:

Entry(ID("e3896e"),
    Formula(Equal(AGM(1, Div(Sqrt(2), 2)), Div(Mul(2, Pow(Pi, Div(3, 2))), Pow(Gamma(Div(1, 4)), 2)))))

361801

\operatorname{agm}\!\left(1, 3 + 2 \sqrt{2}\right) = \frac{2 \left(2 + \sqrt{2}\right) {\pi}^{3 / 2}}{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}

TeX:

\operatorname{agm}\!\left(1, 3 + 2 \sqrt{2}\right) = \frac{2 \left(2 + \sqrt{2}\right) {\pi}^{3 / 2}}{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}

Definitions:

Fungrim symbol	Notation	Short description
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`Sqrt`	$\sqrt{z}$	Principal square root
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`Gamma`	$\Gamma(z)$	Gamma function

Source code for this entry:

Entry(ID("361801"),
    Formula(Equal(AGM(1, Add(3, Mul(2, Sqrt(2)))), Div(Mul(2, Add(2, Sqrt(2)), Pow(Pi, Div(3, 2))), Pow(Gamma(Div(1, 4)), 2)))))

f9190b

\operatorname{agm}\!\left(1, 3 - 2 \sqrt{2}\right) = \frac{2 \left(2 - \sqrt{2}\right) {\pi}^{3 / 2}}{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}

TeX:

\operatorname{agm}\!\left(1, 3 - 2 \sqrt{2}\right) = \frac{2 \left(2 - \sqrt{2}\right) {\pi}^{3 / 2}}{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}

Definitions:

Fungrim symbol	Notation	Short description
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`Sqrt`	$\sqrt{z}$	Principal square root
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`Gamma`	$\Gamma(z)$	Gamma function

Source code for this entry:

Entry(ID("f9190b"),
    Formula(Equal(AGM(1, Sub(3, Mul(2, Sqrt(2)))), Div(Mul(2, Sub(2, Sqrt(2)), Pow(Pi, Div(3, 2))), Pow(Gamma(Div(1, 4)), 2)))))

69d0a3

\operatorname{agm}\!\left(1, i\right) = \frac{\sqrt{2} \left(1 + i\right) {\pi}^{3 / 2}}{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}

TeX:

\operatorname{agm}\!\left(1, i\right) = \frac{\sqrt{2} \left(1 + i\right) {\pi}^{3 / 2}}{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}

Definitions:

Fungrim symbol	Notation	Short description
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`ConstI`	$i$	Imaginary unit
`Sqrt`	$\sqrt{z}$	Principal square root
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`Gamma`	$\Gamma(z)$	Gamma function

Source code for this entry:

Entry(ID("69d0a3"),
    Formula(Equal(AGM(1, ConstI), Div(Mul(Mul(Sqrt(2), Add(1, ConstI)), Pow(Pi, Div(3, 2))), Pow(Gamma(Div(1, 4)), 2)))))

5174ea

\operatorname{agm}\!\left(1, -i\right) = \frac{\sqrt{2} \left(1 - i\right) {\pi}^{3 / 2}}{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}

TeX:

\operatorname{agm}\!\left(1, -i\right) = \frac{\sqrt{2} \left(1 - i\right) {\pi}^{3 / 2}}{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{2}}

Definitions:

Fungrim symbol	Notation	Short description
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`ConstI`	$i$	Imaginary unit
`Sqrt`	$\sqrt{z}$	Principal square root
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`Gamma`	$\Gamma(z)$	Gamma function

Source code for this entry:

Entry(ID("5174ea"),
    Formula(Equal(AGM(1, Neg(ConstI)), Div(Mul(Mul(Sqrt(2), Sub(1, ConstI)), Pow(Pi, Div(3, 2))), Pow(Gamma(Div(1, 4)), 2)))))

7b362f

\operatorname{agm}\!\left(1, \sqrt{2}\right) = \frac{1}{\theta_{4}^{2}\!\left(0, i\right)}

TeX:

\operatorname{agm}\!\left(1, \sqrt{2}\right) = \frac{1}{\theta_{4}^{2}\!\left(0, i\right)}

Definitions:

Fungrim symbol	Notation	Short description
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`Sqrt`	$\sqrt{z}$	Principal square root
`Pow`	${a}^{b}$	Power
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("7b362f"),
    Formula(Equal(AGM(1, Sqrt(2)), Div(1, Pow(JacobiTheta(4, 0, ConstI), 2)))))

eb0661

\operatorname{agm}\!\left(1, 1\right) = 1

TeX:

\operatorname{agm}\!\left(1, 1\right) = 1

Definitions:

Fungrim symbol	Notation	Short description
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean

Source code for this entry:

Entry(ID("eb0661"),
    Formula(Equal(AGM(1, 1), 1)))

3da9b7

\left[ \frac{d}{d x}\, \operatorname{agm}\!\left(1, x\right) \right]_{x = 1} = \frac{1}{2}

TeX:

\left[ \frac{d}{d x}\, \operatorname{agm}\!\left(1, x\right) \right]_{x = 1} = \frac{1}{2}

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean

Source code for this entry:

Entry(ID("3da9b7"),
    Formula(Equal(ComplexDerivative(AGM(1, x), For(x, 1)), Div(1, 2))))

f178f2

\left[ \frac{d^{2}}{{d x}^{2}} \operatorname{agm}\!\left(1, x\right) \right]_{x = 1} = -\frac{1}{8}

TeX:

\left[ \frac{d^{2}}{{d x}^{2}} \operatorname{agm}\!\left(1, x\right) \right]_{x = 1} = -\frac{1}{8}

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean

Source code for this entry:

Entry(ID("f178f2"),
    Formula(Equal(ComplexDerivative(AGM(1, x), For(x, 1, 2)), Neg(Div(1, 8)))))

447541

\left[ \frac{d^{n}}{{d x}^{n}} \operatorname{agm}\!\left(1, x\right) \right]_{x = 1} = \frac{{\left(-1\right)}^{n} n !}{{8}^{n}} \text{A060691}\!\left(n\right)

Assumptions:

n \in \mathbb{Z}_{\ge 0}

References:

Sequence A060691 in Sloane's On-Line Encyclopedia of Integer Sequences (OEIS)

TeX:

\left[ \frac{d^{n}}{{d x}^{n}} \operatorname{agm}\!\left(1, x\right) \right]_{x = 1} = \frac{{\left(-1\right)}^{n} n !}{{8}^{n}} \text{A060691}\!\left(n\right)

n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`SloaneA`	$\text{A00000X}\!\left(n\right)$	Sequence X in Sloane's OEIS
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("447541"),
    Formula(Equal(ComplexDerivative(AGM(1, x), For(x, 1, n)), Mul(Div(Mul(Pow(-1, n), Factorial(n)), Pow(8, n)), SloaneA("060691", n)))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

95fb3e

\operatorname{agm}\!\left(a, b\right) = \lim_{n \to \infty} a_{n} = \lim_{n \to \infty} b_{n}\; \text{ where } \left(a_{n}, b_{n}\right) = \operatorname{agm}_{n}\!\left(a, b\right)

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}

TeX:

\operatorname{agm}\!\left(a, b\right) = \lim_{n \to \infty} a_{n} = \lim_{n \to \infty} b_{n}\; \text{ where } \left(a_{n}, b_{n}\right) = \operatorname{agm}_{n}\!\left(a, b\right)

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`SequenceLimit`	$\lim_{n \to a} f(n)$	Limiting value of sequence
`Infinity`	$\infty$	Positive infinity
`AGMSequence`	$\operatorname{agm}_{n}\!\left(a, b\right)$	Convergents in AGM iteration
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("95fb3e"),
    Formula(Where(Equal(AGM(a, b), SequenceLimit(a_(n), For(n, Infinity)), SequenceLimit(b_(n), For(n, Infinity))), Def(Tuple(a_(n), b_(n)), AGMSequence(n, a, b)))),
    Variables(a, b),
    Assumptions(And(Element(a, CC), Element(b, CC))))

84b888

\operatorname{agm}_{0}\!\left(a, b\right) = \left(a, b\right)

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}

TeX:

\operatorname{agm}_{0}\!\left(a, b\right) = \left(a, b\right)

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`AGMSequence`	$\operatorname{agm}_{n}\!\left(a, b\right)$	Convergents in AGM iteration
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("84b888"),
    Formula(Equal(AGMSequence(0, a, b), Tuple(a, b))),
    Variables(a, b),
    Assumptions(And(Element(a, CC), Element(b, CC))))

08b69d

\left(a_{n + 1}, b_{n + 1}\right) = \left(\frac{a_{n} + b_{n}}{2}, \sqrt{a_{n} b_{n}}\right)\; \text{ where } \left(a_{k}, b_{k}\right) = \operatorname{agm}_{k}\!\left(a, b\right)

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(a = 0 \;\mathbin{\operatorname{or}}\; b = 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Re}(a) > 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(b) > 0\right) \;\mathbin{\operatorname{or}}\; \left|\arg(a)\right| + \left|\arg(b)\right| < \pi\right)

TeX:

\left(a_{n + 1}, b_{n + 1}\right) = \left(\frac{a_{n} + b_{n}}{2}, \sqrt{a_{n} b_{n}}\right)\; \text{ where } \left(a_{k}, b_{k}\right) = \operatorname{agm}_{k}\!\left(a, b\right)

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(a = 0 \;\mathbin{\operatorname{or}}\; b = 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Re}(a) > 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(b) > 0\right) \;\mathbin{\operatorname{or}}\; \left|\arg(a)\right| + \left|\arg(b)\right| < \pi\right)

Definitions:

Fungrim symbol	Notation	Short description
`Sqrt`	$\sqrt{z}$	Principal square root
`AGMSequence`	$\operatorname{agm}_{n}\!\left(a, b\right)$	Convergents in AGM iteration
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part
`Abs`	$\left\|z\right\|$	Absolute value
`Arg`	$\arg(z)$	Complex argument
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("08b69d"),
    Formula(Where(Equal(Tuple(a_(Add(n, 1)), b_(Add(n, 1))), Tuple(Div(Add(a_(n), b_(n)), 2), Sqrt(Mul(a_(n), b_(n))))), Def(Tuple(a_(k), b_(k)), AGMSequence(k, a, b)))),
    Variables(n, a, b),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(a, CC), Element(b, CC), Or(Equal(a, 0), Equal(b, 0), And(Greater(Re(a), 0), Greater(Re(b), 0)), Less(Add(Abs(Arg(a)), Abs(Arg(b))), Pi)))))

41f67b

2 a_{n + 1} = a_{n} + b_{n} \;\mathbin{\operatorname{and}}\; b_{n + 1}^{2} = a_{n} b_{n}\; \text{ where } \left(a_{k}, b_{k}\right) = \operatorname{agm}_{k}\!\left(a, b\right)

Assumptions:

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

TeX:

2 a_{n + 1} = a_{n} + b_{n} \;\mathbin{\operatorname{and}}\; b_{n + 1}^{2} = a_{n} b_{n}\; \text{ where } \left(a_{k}, b_{k}\right) = \operatorname{agm}_{k}\!\left(a, b\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`AGMSequence`	$\operatorname{agm}_{n}\!\left(a, b\right)$	Convergents in AGM iteration
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("41f67b"),
    Formula(Where(And(Equal(Mul(2, a_(Add(n, 1))), Add(a_(n), b_(n))), Equal(Pow(b_(Add(n, 1)), 2), Mul(a_(n), b_(n)))), Def(Tuple(a_(k), b_(k)), AGMSequence(k, a, b)))),
    Variables(n, a, b),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(a, CC), Element(b, CC))))

a2b0f9

\left(a_{n + 1}, b_{n + 1}\right) = \left(x, s y\right)\; \text{ where } x = \frac{a_{n} + b_{n}}{2},\;y = \sqrt{a_{n} b_{n}},\;s = \begin{cases} +1, & y = 0 \;\mathbin{\operatorname{or}}\; \operatorname{Re}\!\left(\frac{x}{y}\right) \ge 0\\-1, & \text{otherwise}\\ \end{cases}\; \text{ where } \left(a_{k}, b_{k}\right) = \operatorname{agm}_{k}\!\left(a, b\right)

Assumptions:

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

TeX:

\left(a_{n + 1}, b_{n + 1}\right) = \left(x, s y\right)\; \text{ where } x = \frac{a_{n} + b_{n}}{2},\;y = \sqrt{a_{n} b_{n}},\;s = \begin{cases} +1, & y = 0 \;\mathbin{\operatorname{or}}\; \operatorname{Re}\!\left(\frac{x}{y}\right) \ge 0\\-1, & \text{otherwise}\\ \end{cases}\; \text{ where } \left(a_{k}, b_{k}\right) = \operatorname{agm}_{k}\!\left(a, b\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`Sqrt`	$\sqrt{z}$	Principal square root
`Re`	$\operatorname{Re}(z)$	Real part
`AGMSequence`	$\operatorname{agm}_{n}\!\left(a, b\right)$	Convergents in AGM iteration
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("a2b0f9"),
    Formula(Where(Equal(Tuple(a_(Add(n, 1)), b_(Add(n, 1))), Where(Tuple(x, Mul(s, y)), Def(x, Div(Add(a_(n), b_(n)), 2)), Def(y, Sqrt(Mul(a_(n), b_(n)))), Def(s, Cases(Tuple(Pos(1), Or(Equal(y, 0), GreaterEqual(Re(Div(x, y)), 0))), Tuple(Neg(1), Otherwise))))), Def(Tuple(a_(k), b_(k)), AGMSequence(k, a, b)))),
    Variables(n, a, b),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(a, CC), Element(b, CC))))

6d9ceb

\pi = \frac{4 {\left(\operatorname{agm}\!\left(1, \frac{1}{\sqrt{2}}\right)\right)}^{2}}{1 - \sum_{j=0}^{\infty} {2}^{j} c_{j}^{2}} = \lim_{n \to \infty} \frac{{\left(a_{n} + b_{n}\right)}^{2}}{1 - \sum_{j=0}^{n} {2}^{j} c_{j}^{2}}\; \text{ where } \left(a_{n}, b_{n}\right) = \operatorname{agm}_{n}\!\left(1, \frac{1}{\sqrt{2}}\right),\;c_{n} = a_{n} - b_{n}

TeX:

\pi = \frac{4 {\left(\operatorname{agm}\!\left(1, \frac{1}{\sqrt{2}}\right)\right)}^{2}}{1 - \sum_{j=0}^{\infty} {2}^{j} c_{j}^{2}} = \lim_{n \to \infty} \frac{{\left(a_{n} + b_{n}\right)}^{2}}{1 - \sum_{j=0}^{n} {2}^{j} c_{j}^{2}}\; \text{ where } \left(a_{n}, b_{n}\right) = \operatorname{agm}_{n}\!\left(1, \frac{1}{\sqrt{2}}\right),\;c_{n} = a_{n} - b_{n}

Definitions:

Fungrim symbol	Notation	Short description
`Pi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`Sqrt`	$\sqrt{z}$	Principal square root
`Sum`	$\sum_{n} f(n)$	Sum
`Infinity`	$\infty$	Positive infinity
`SequenceLimit`	$\lim_{n \to a} f(n)$	Limiting value of sequence
`AGMSequence`	$\operatorname{agm}_{n}\!\left(a, b\right)$	Convergents in AGM iteration

Source code for this entry:

Entry(ID("6d9ceb"),
    Formula(Where(Equal(Pi, Div(Mul(4, Pow(AGM(1, Div(1, Sqrt(2))), 2)), Sub(1, Sum(Mul(Pow(2, j), Pow(c_(j), 2)), For(j, 0, Infinity)))), SequenceLimit(Div(Pow(Add(a_(n), b_(n)), 2), Sub(1, Sum(Mul(Pow(2, j), Pow(c_(j), 2)), For(j, 0, n)))), For(n, Infinity))), Def(Tuple(a_(n), b_(n)), AGMSequence(n, 1, Div(1, Sqrt(2)))), Def(c_(n), Sub(a_(n), b_(n))))))

13c539

\left|\pi - \frac{{\left(a_{n} + b_{n}\right)}^{2}}{1 - \sum_{j=0}^{n} {2}^{j} c_{j}^{2}}\right| \le {2}^{n + 8} {e}^{-\pi {2}^{n + 1}}\; \text{ where } \left(a_{n}, b_{n}\right) = \operatorname{agm}_{n}\!\left(1, \frac{1}{\sqrt{2}}\right),\;c_{n} = a_{n} - b_{n}

References:

https://doi.org/10.2307/2005327

TeX:

\left|\pi - \frac{{\left(a_{n} + b_{n}\right)}^{2}}{1 - \sum_{j=0}^{n} {2}^{j} c_{j}^{2}}\right| \le {2}^{n + 8} {e}^{-\pi {2}^{n + 1}}\; \text{ where } \left(a_{n}, b_{n}\right) = \operatorname{agm}_{n}\!\left(1, \frac{1}{\sqrt{2}}\right),\;c_{n} = a_{n} - b_{n}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`Pi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`Sum`	$\sum_{n} f(n)$	Sum
`Exp`	${e}^{z}$	Exponential function
`AGMSequence`	$\operatorname{agm}_{n}\!\left(a, b\right)$	Convergents in AGM iteration
`Sqrt`	$\sqrt{z}$	Principal square root

Source code for this entry:

Entry(ID("13c539"),
    Formula(Where(LessEqual(Abs(Sub(Pi, Div(Pow(Add(a_(n), b_(n)), 2), Sub(1, Sum(Mul(Pow(2, j), Pow(c_(j), 2)), For(j, 0, n)))))), Mul(Pow(2, Add(n, 8)), Exp(Neg(Mul(Pi, Pow(2, Add(n, 1))))))), Def(Tuple(a_(n), b_(n)), AGMSequence(n, 1, Div(1, Sqrt(2)))), Def(c_(n), Sub(a_(n), b_(n))))),
    References("https://doi.org/10.2307/2005327"))

042551

{e}^{\pi} = 32 \prod_{n=0}^{\infty} {\left(\frac{a_{n + 1}}{a_{n}}\right)}^{{2}^{1 - n}}\; \text{ where } \left(a_{n}, b_{n}\right) = \operatorname{agm}_{n}\!\left(1, \frac{1}{\sqrt{2}}\right)

References:

https://doi.org/10.2307/2005327

TeX:

{e}^{\pi} = 32 \prod_{n=0}^{\infty} {\left(\frac{a_{n + 1}}{a_{n}}\right)}^{{2}^{1 - n}}\; \text{ where } \left(a_{n}, b_{n}\right) = \operatorname{agm}_{n}\!\left(1, \frac{1}{\sqrt{2}}\right)

Definitions:

Fungrim symbol	Notation	Short description
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`Product`	$\prod_{n} f(n)$	Product
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`AGMSequence`	$\operatorname{agm}_{n}\!\left(a, b\right)$	Convergents in AGM iteration
`Sqrt`	$\sqrt{z}$	Principal square root

Source code for this entry:

Entry(ID("042551"),
    Formula(Equal(Exp(Pi), Where(Mul(32, Product(Pow(Div(a_(Add(n, 1)), a_(n)), Pow(2, Sub(1, n))), For(n, 0, Infinity))), Def(Tuple(a_(n), b_(n)), AGMSequence(n, 1, Div(1, Sqrt(2))))))),
    References("https://doi.org/10.2307/2005327"))

59fab1

\operatorname{agm}\!\left(a, b\right) = \operatorname{agm}\!\left(b, a\right)

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}

TeX:

\operatorname{agm}\!\left(a, b\right) = \operatorname{agm}\!\left(b, a\right)

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("59fab1"),
    Formula(Equal(AGM(a, b), AGM(b, a))),
    Variables(a, b),
    Assumptions(And(Element(a, CC), Element(b, CC))))

c0dea0

\operatorname{agm}\!\left(\overline{a}, \overline{b}\right) = \overline{\operatorname{agm}\!\left(a, b\right)}

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(a = 0 \;\mathbin{\operatorname{or}}\; \frac{b}{a} \notin \left(-\infty, 0\right]\right)

TeX:

\operatorname{agm}\!\left(\overline{a}, \overline{b}\right) = \overline{\operatorname{agm}\!\left(a, b\right)}

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(a = 0 \;\mathbin{\operatorname{or}}\; \frac{b}{a} \notin \left(-\infty, 0\right]\right)

Definitions:

Fungrim symbol	Notation	Short description
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`Conjugate`	$\overline{z}$	Complex conjugate
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("c0dea0"),
    Formula(Equal(AGM(Conjugate(a), Conjugate(b)), Conjugate(AGM(a, b)))),
    Variables(a, b),
    Assumptions(And(Element(a, CC), Element(b, CC), Or(Equal(a, 0), NotElement(Div(b, a), OpenClosedInterval(Neg(Infinity), 0))))))

7189d6

\operatorname{agm}\!\left(-a, -b\right) = -\operatorname{agm}\!\left(a, b\right)

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(a = 0 \;\mathbin{\operatorname{or}}\; \frac{b}{a} \notin \left(-\infty, 0\right]\right)

TeX:

\operatorname{agm}\!\left(-a, -b\right) = -\operatorname{agm}\!\left(a, b\right)

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(a = 0 \;\mathbin{\operatorname{or}}\; \frac{b}{a} \notin \left(-\infty, 0\right]\right)

Definitions:

Fungrim symbol	Notation	Short description
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("7189d6"),
    Formula(Equal(AGM(Neg(a), Neg(b)), Neg(AGM(a, b)))),
    Variables(a, b),
    Assumptions(And(Element(a, CC), Element(b, CC), Or(Equal(a, 0), NotElement(Div(b, a), OpenClosedInterval(Neg(Infinity), 0))))))

ce2395

\operatorname{agm}\!\left(a, b\right) = a \operatorname{agm}\!\left(1, \frac{b}{a}\right)

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \ne 0 \;\mathbin{\operatorname{and}}\; \frac{b}{a} \notin \left(-\infty, 0\right]

TeX:

\operatorname{agm}\!\left(a, b\right) = a \operatorname{agm}\!\left(1, \frac{b}{a}\right)

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \ne 0 \;\mathbin{\operatorname{and}}\; \frac{b}{a} \notin \left(-\infty, 0\right]

Definitions:

Fungrim symbol	Notation	Short description
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("ce2395"),
    Formula(Equal(AGM(a, b), Mul(a, AGM(1, Div(b, a))))),
    Variables(a, b),
    Assumptions(And(Element(a, CC), Element(b, CC), NotEqual(a, 0), NotElement(Div(b, a), OpenClosedInterval(Neg(Infinity), 0)))))

ea1d58

\operatorname{agm}\!\left(a, b\right) = b \operatorname{agm}\!\left(1, \frac{a}{b}\right)

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \ne 0 \;\mathbin{\operatorname{and}}\; \frac{a}{b} \notin \left(-\infty, 0\right]

TeX:

\operatorname{agm}\!\left(a, b\right) = b \operatorname{agm}\!\left(1, \frac{a}{b}\right)

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \ne 0 \;\mathbin{\operatorname{and}}\; \frac{a}{b} \notin \left(-\infty, 0\right]

Definitions:

Fungrim symbol	Notation	Short description
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("ea1d58"),
    Formula(Equal(AGM(a, b), Mul(b, AGM(1, Div(a, b))))),
    Variables(a, b),
    Assumptions(And(Element(a, CC), Element(b, CC), NotEqual(b, 0), NotElement(Div(a, b), OpenClosedInterval(Neg(Infinity), 0)))))

d60119

\operatorname{agm}\!\left(c a, c b\right) = c \operatorname{agm}\!\left(a, b\right)

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; c \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(a = 0 \;\mathbin{\operatorname{or}}\; b = 0 \;\mathbin{\operatorname{or}}\; c = 0 \;\mathbin{\operatorname{or}}\; \frac{b}{a} \notin \left(-\infty, 0\right]\right)

TeX:

\operatorname{agm}\!\left(c a, c b\right) = c \operatorname{agm}\!\left(a, b\right)

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; c \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(a = 0 \;\mathbin{\operatorname{or}}\; b = 0 \;\mathbin{\operatorname{or}}\; c = 0 \;\mathbin{\operatorname{or}}\; \frac{b}{a} \notin \left(-\infty, 0\right]\right)

Definitions:

Fungrim symbol	Notation	Short description
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("d60119"),
    Formula(Equal(AGM(Mul(c, a), Mul(c, b)), Mul(c, AGM(a, b)))),
    Variables(a, b, c),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(c, CC), Or(Equal(a, 0), Equal(b, 0), Equal(c, 0), NotElement(Div(b, a), OpenClosedInterval(Neg(Infinity), 0))))))

c7f885

\operatorname{agm}\!\left(a, b\right) = \operatorname{agm}\!\left(\frac{a + b}{2}, \sqrt{a b}\right)

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(a = 0 \;\mathbin{\operatorname{or}}\; b = 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Re}(a) > 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(b) > 0\right) \;\mathbin{\operatorname{or}}\; \left|\arg(a)\right| + \left|\arg(b)\right| < \pi\right)

TeX:

\operatorname{agm}\!\left(a, b\right) = \operatorname{agm}\!\left(\frac{a + b}{2}, \sqrt{a b}\right)

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(a = 0 \;\mathbin{\operatorname{or}}\; b = 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Re}(a) > 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(b) > 0\right) \;\mathbin{\operatorname{or}}\; \left|\arg(a)\right| + \left|\arg(b)\right| < \pi\right)

Definitions:

Fungrim symbol	Notation	Short description
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`Sqrt`	$\sqrt{z}$	Principal square root
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part
`Abs`	$\left\|z\right\|$	Absolute value
`Arg`	$\arg(z)$	Complex argument
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("c7f885"),
    Formula(Equal(AGM(a, b), AGM(Div(Add(a, b), 2), Sqrt(Mul(a, b))))),
    Variables(a, b),
    Assumptions(And(Element(a, CC), Element(b, CC), Or(Equal(a, 0), Equal(b, 0), And(Greater(Re(a), 0), Greater(Re(b), 0)), Less(Add(Abs(Arg(a)), Abs(Arg(b))), Pi)))))

fa6ff7

\operatorname{agm}\!\left(a, b\right) = \operatorname{agm}\!\left(x, s y\right)\; \text{ where } x = \frac{a + b}{2},\;y = \sqrt{a b},\;s = \begin{cases} +1, & y = 0 \;\mathbin{\operatorname{or}}\; \operatorname{Re}\!\left(\frac{x}{y}\right) \ge 0\\-1, & \text{otherwise}\\ \end{cases}

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}

TeX:

\operatorname{agm}\!\left(a, b\right) = \operatorname{agm}\!\left(x, s y\right)\; \text{ where } x = \frac{a + b}{2},\;y = \sqrt{a b},\;s = \begin{cases} +1, & y = 0 \;\mathbin{\operatorname{or}}\; \operatorname{Re}\!\left(\frac{x}{y}\right) \ge 0\\-1, & \text{otherwise}\\ \end{cases}

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`Sqrt`	$\sqrt{z}$	Principal square root
`Re`	$\operatorname{Re}(z)$	Real part
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("fa6ff7"),
    Formula(Equal(AGM(a, b), Where(AGM(x, Mul(s, y)), Def(x, Div(Add(a, b), 2)), Def(y, Sqrt(Mul(a, b))), Def(s, Cases(Tuple(Pos(1), Or(Equal(y, 0), GreaterEqual(Re(Div(x, y)), 0))), Tuple(Neg(1), Otherwise)))))),
    Variables(a, b),
    Assumptions(And(Element(a, CC), Element(b, CC))))

8e80c6

\operatorname{agm}\!\left(1, b\right) = b \operatorname{agm}\!\left(1, \frac{1}{b}\right)

Assumptions:

b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \notin \left(-\infty, 0\right]

TeX:

\operatorname{agm}\!\left(1, b\right) = b \operatorname{agm}\!\left(1, \frac{1}{b}\right)

b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \notin \left(-\infty, 0\right]

Definitions:

Fungrim symbol	Notation	Short description
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("8e80c6"),
    Formula(Equal(AGM(1, b), Mul(b, AGM(1, Div(1, b))))),
    Variables(b),
    Assumptions(And(Element(b, CC), NotElement(b, OpenClosedInterval(Neg(Infinity), 0)))))

46c021

\operatorname{agm}\!\left(1, b\right) = \frac{b + 1}{2} \operatorname{agm}\!\left(1, \frac{2 \sqrt{b}}{b + 1}\right)

Assumptions:

b \in \mathbb{C}

TeX:

\operatorname{agm}\!\left(1, b\right) = \frac{b + 1}{2} \operatorname{agm}\!\left(1, \frac{2 \sqrt{b}}{b + 1}\right)

b \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`Sqrt`	$\sqrt{z}$	Principal square root
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("46c021"),
    Formula(Equal(AGM(1, b), Mul(Div(Add(b, 1), 2), AGM(1, Div(Mul(2, Sqrt(b)), Add(b, 1)))))),
    Variables(b),
    Assumptions(Element(b, CC)))

9d84d8

\operatorname{agm}\!\left(1 + b, 1 - b\right) = \operatorname{agm}\!\left(1, \sqrt{1 - {b}^{2}}\right)

Assumptions:

b \in \mathbb{C}

TeX:

\operatorname{agm}\!\left(1 + b, 1 - b\right) = \operatorname{agm}\!\left(1, \sqrt{1 - {b}^{2}}\right)

b \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`Sqrt`	$\sqrt{z}$	Principal square root
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("9d84d8"),
    Formula(Equal(AGM(Add(1, b), Sub(1, b)), AGM(1, Sqrt(Sub(1, Pow(b, 2)))))),
    Variables(b),
    Assumptions(Element(b, CC)))

d6d836

\operatorname{agm}\!\left(a, b\right) = \frac{a + b}{2 \,{}_2F_1\!\left(\frac{1}{2}, \frac{1}{2}, 1, {\left(\frac{a - b}{a + b}\right)}^{2}\right)}

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \ne 0 \;\mathbin{\operatorname{and}}\; \frac{a}{b} \notin \left(-\infty, 0\right]

TeX:

\operatorname{agm}\!\left(a, b\right) = \frac{a + b}{2 \,{}_2F_1\!\left(\frac{1}{2}, \frac{1}{2}, 1, {\left(\frac{a - b}{a + b}\right)}^{2}\right)}

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \ne 0 \;\mathbin{\operatorname{and}}\; \frac{a}{b} \notin \left(-\infty, 0\right]

Definitions:

Fungrim symbol	Notation	Short description
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`Hypergeometric2F1`	$\,{}_2F_1\!\left(a, b, c, z\right)$	Gauss hypergeometric function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("d6d836"),
    Formula(Equal(AGM(a, b), Div(Add(a, b), Mul(2, Hypergeometric2F1(Div(1, 2), Div(1, 2), 1, Pow(Div(Sub(a, b), Add(a, b)), 2)))))),
    Variables(a, b),
    Assumptions(And(Element(a, CC), Element(b, CC), NotEqual(b, 0), NotElement(Div(a, b), OpenClosedInterval(Neg(Infinity), 0)))))

71a0ff

\operatorname{agm}\!\left(a, b\right) = \frac{\pi}{4} \frac{a + b}{K\!\left({\left(\frac{a - b}{a + b}\right)}^{2}\right)}

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \ne 0 \;\mathbin{\operatorname{and}}\; \frac{a}{b} \notin \left(-\infty, 0\right]

TeX:

\operatorname{agm}\!\left(a, b\right) = \frac{\pi}{4} \frac{a + b}{K\!\left({\left(\frac{a - b}{a + b}\right)}^{2}\right)}

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \ne 0 \;\mathbin{\operatorname{and}}\; \frac{a}{b} \notin \left(-\infty, 0\right]

Definitions:

Fungrim symbol	Notation	Short description
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`Pi`	$\pi$	The constant pi (3.14...)
`EllipticK`	$K(m)$	Legendre complete elliptic integral of the first kind
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("71a0ff"),
    Formula(Equal(AGM(a, b), Mul(Div(Pi, 4), Div(Add(a, b), EllipticK(Pow(Div(Sub(a, b), Add(a, b)), 2)))))),
    Variables(a, b),
    Assumptions(And(Element(a, CC), Element(b, CC), NotEqual(b, 0), NotElement(Div(a, b), OpenClosedInterval(Neg(Infinity), 0)))))

e15f43

K(m) = \frac{\pi}{2 \operatorname{agm}\!\left(1, \sqrt{1 - m}\right)}

Assumptions:

m \in \mathbb{C}

TeX:

K(m) = \frac{\pi}{2 \operatorname{agm}\!\left(1, \sqrt{1 - m}\right)}

m \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`EllipticK`	$K(m)$	Legendre complete elliptic integral of the first kind
`Pi`	$\pi$	The constant pi (3.14...)
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`Sqrt`	$\sqrt{z}$	Principal square root
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("e15f43"),
    Formula(Equal(EllipticK(m), Div(Pi, Mul(2, AGM(1, Sqrt(Sub(1, m))))))),
    Variables(m),
    Assumptions(Element(m, CC)))

26fd1b

\log\!\left(\frac{1}{q}\right) = \frac{\pi}{\operatorname{agm}\!\left(\theta_{2}^{2}\!\left(0, q\right), \theta_{3}^{2}\!\left(0, q\right)\right)}

Assumptions:

q \in \left(0, 1\right)

TeX:

\log\!\left(\frac{1}{q}\right) = \frac{\pi}{\operatorname{agm}\!\left(\theta_{2}^{2}\!\left(0, q\right), \theta_{3}^{2}\!\left(0, q\right)\right)}

q \in \left(0, 1\right)

Definitions:

Fungrim symbol	Notation	Short description
`Log`	$\log(z)$	Natural logarithm
`Pi`	$\pi$	The constant pi (3.14...)
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`Pow`	${a}^{b}$	Power
`OpenInterval`	$\left(a, b\right)$	Open interval

Source code for this entry:

Entry(ID("26fd1b"),
    Formula(Equal(Log(Div(1, q)), Div(Pi, AGM(Pow(JacobiThetaQ(2, 0, q), 2), Pow(JacobiThetaQ(3, 0, q), 2))))),
    Variables(q),
    Assumptions(Element(q, OpenInterval(0, 1))))

20828c

\frac{d}{d a}\, \operatorname{agm}\!\left(a, b\right) = \frac{\operatorname{agm}\!\left(a, b\right)}{\pi a \left(a - b\right)} \left(\pi a - 2 \operatorname{agm}\!\left(a, b\right) E\!\left({\left(\frac{a - b}{a + b}\right)}^{2}\right)\right)

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \ne 0 \;\mathbin{\operatorname{and}}\; a \ne b \;\mathbin{\operatorname{and}}\; \frac{a}{b} \notin \left(-\infty, 0\right]

References:

http://functions.wolfram.com/09.54.20.0001.01

TeX:

\frac{d}{d a}\, \operatorname{agm}\!\left(a, b\right) = \frac{\operatorname{agm}\!\left(a, b\right)}{\pi a \left(a - b\right)} \left(\pi a - 2 \operatorname{agm}\!\left(a, b\right) E\!\left({\left(\frac{a - b}{a + b}\right)}^{2}\right)\right)

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \ne 0 \;\mathbin{\operatorname{and}}\; a \ne b \;\mathbin{\operatorname{and}}\; \frac{a}{b} \notin \left(-\infty, 0\right]

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`Pi`	$\pi$	The constant pi (3.14...)
`EllipticE`	$E(m)$	Legendre complete elliptic integral of the second kind
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("20828c"),
    Formula(Equal(ComplexDerivative(AGM(a, b), For(a, a)), Mul(Div(AGM(a, b), Mul(Mul(Pi, a), Sub(a, b))), Sub(Mul(Pi, a), Mul(Mul(2, AGM(a, b)), EllipticE(Pow(Div(Sub(a, b), Add(a, b)), 2))))))),
    Variables(a, b),
    Assumptions(And(Element(a, CC), Element(b, CC), NotEqual(b, 0), NotEqual(a, b), NotElement(Div(a, b), OpenClosedInterval(Neg(Infinity), 0)))),
    References("http://functions.wolfram.com/09.54.20.0001.01"))

a4cc5a

2 a \left({b}^{2} - {a}^{2}\right) {\left(f'(a)\right)}^{2} - a {\left(f(a)\right)}^{2} + \left(\left(3 {a}^{2} - {b}^{2}\right) f'(a) + a \left({a}^{2} - {b}^{2}\right) f''(a)\right) f(a) = 0\; \text{ where } f(a) = \operatorname{agm}\!\left(a, b\right)

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \ne 0 \;\mathbin{\operatorname{and}}\; \frac{a}{b} \notin \left(-\infty, 0\right]

References:

http://functions.wolfram.com/09.54.13.0001.01

TeX:

2 a \left({b}^{2} - {a}^{2}\right) {\left(f'(a)\right)}^{2} - a {\left(f(a)\right)}^{2} + \left(\left(3 {a}^{2} - {b}^{2}\right) f'(a) + a \left({a}^{2} - {b}^{2}\right) f''(a)\right) f(a) = 0\; \text{ where } f(a) = \operatorname{agm}\!\left(a, b\right)

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \ne 0 \;\mathbin{\operatorname{and}}\; \frac{a}{b} \notin \left(-\infty, 0\right]

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("a4cc5a"),
    Formula(Where(Equal(Add(Sub(Mul(Mul(Mul(2, a), Sub(Pow(b, 2), Pow(a, 2))), Pow(ComplexDerivative(f(a), For(a, a)), 2)), Mul(a, Pow(f(a), 2))), Mul(Add(Mul(Sub(Mul(3, Pow(a, 2)), Pow(b, 2)), ComplexDerivative(f(a), For(a, a))), Mul(Mul(a, Sub(Pow(a, 2), Pow(b, 2))), ComplexDerivative(f(a), For(a, a, 2)))), f(a))), 0), Def(f(a), AGM(a, b)))),
    Variables(a, b),
    Assumptions(And(Element(a, CC), Element(b, CC), NotEqual(b, 0), NotElement(Div(a, b), OpenClosedInterval(Neg(Infinity), 0)))),
    References("http://functions.wolfram.com/09.54.13.0001.01"))

cfefa9

\operatorname{agm}\!\left(1, 1 + x\right) = \sum_{n=0}^{\infty} \frac{\text{A060691}\!\left(n\right)}{{8}^{n}} {\left(-x\right)}^{n}

Assumptions:

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|x\right| < 1

References:

Sequence A060691 in Sloane's On-Line Encyclopedia of Integer Sequences (OEIS)

TeX:

\operatorname{agm}\!\left(1, 1 + x\right) = \sum_{n=0}^{\infty} \frac{\text{A060691}\!\left(n\right)}{{8}^{n}} {\left(-x\right)}^{n}

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|x\right| < 1

Definitions:

Fungrim symbol	Notation	Short description
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`Sum`	$\sum_{n} f(n)$	Sum
`SloaneA`	$\text{A00000X}\!\left(n\right)$	Sequence X in Sloane's OEIS
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`Abs`	$\left\|z\right\|$	Absolute value

Source code for this entry:

Entry(ID("cfefa9"),
    Formula(Equal(AGM(1, Add(1, x)), Sum(Mul(Div(SloaneA("060691", n), Pow(8, n)), Pow(Neg(x), n)), For(n, 0, Infinity)))),
    Variables(x),
    Assumptions(And(Element(x, CC), Less(Abs(x), 1))))

417619

\operatorname{agm}\!\left(a, b\right) = \frac{\pi}{2 I}\; \text{ where } I = \int_{0}^{\pi / 2} \frac{1}{\sqrt{{a}^{2} \cos^{2}\!\left(x\right) + {b}^{2} \sin^{2}\!\left(x\right)}} \, dx

Assumptions:

a \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; b \in \left(0, \infty\right)

TeX:

\operatorname{agm}\!\left(a, b\right) = \frac{\pi}{2 I}\; \text{ where } I = \int_{0}^{\pi / 2} \frac{1}{\sqrt{{a}^{2} \cos^{2}\!\left(x\right) + {b}^{2} \sin^{2}\!\left(x\right)}} \, dx

a \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; b \in \left(0, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`Pi`	$\pi$	The constant pi (3.14...)
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Sqrt`	$\sqrt{z}$	Principal square root
`Pow`	${a}^{b}$	Power
`Cos`	$\cos(z)$	Cosine
`Sin`	$\sin(z)$	Sine
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("417619"),
    Formula(Equal(AGM(a, b), Where(Div(Pi, Mul(2, I)), Def(I, Integral(Div(1, Sqrt(Add(Mul(Pow(a, 2), Pow(Cos(x), 2)), Mul(Pow(b, 2), Pow(Sin(x), 2))))), For(x, 0, Div(Pi, 2))))))),
    Variables(a, b),
    Assumptions(And(Element(a, OpenInterval(0, Infinity)), Element(b, OpenInterval(0, Infinity)))))

162ecf

\sqrt{a b} \le \operatorname{agm}\!\left(a, b\right) \le \frac{a + b}{2}

Assumptions:

a \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; b \in \left[0, \infty\right)

TeX:

\sqrt{a b} \le \operatorname{agm}\!\left(a, b\right) \le \frac{a + b}{2}

a \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; b \in \left[0, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`Sqrt`	$\sqrt{z}$	Principal square root
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("162ecf"),
    Formula(LessEqual(Sqrt(Mul(a, b)), AGM(a, b), Div(Add(a, b), 2))),
    Variables(a),
    Assumptions(And(Element(a, ClosedOpenInterval(0, Infinity)), Element(b, ClosedOpenInterval(0, Infinity)))))

23ee29

\left|\operatorname{agm}\!\left(a, b\right)\right| \le \left|\operatorname{agm}\!\left(\left|a\right|, \left|b\right|\right)\right|

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}

TeX:

\left|\operatorname{agm}\!\left(a, b\right)\right| \le \left|\operatorname{agm}\!\left(\left|a\right|, \left|b\right|\right)\right|

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("23ee29"),
    Formula(LessEqual(Abs(AGM(a, b)), Abs(AGM(Abs(a), Abs(b))))),
    Variables(a, b),
    Assumptions(And(Element(a, CC), Element(b, CC))))

75e692

\left|\operatorname{agm}\!\left(1, z\right) - a_{n}\right| \le \left|a_{n} - b_{n}\right|\; \text{ where } \left(a_{n}, b_{n}\right) = \operatorname{agm}_{n}\!\left(1, z\right)

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) \ge 0

TeX:

\left|\operatorname{agm}\!\left(1, z\right) - a_{n}\right| \le \left|a_{n} - b_{n}\right|\; \text{ where } \left(a_{n}, b_{n}\right) = \operatorname{agm}_{n}\!\left(1, z\right)

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) \ge 0

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`AGMSequence`	$\operatorname{agm}_{n}\!\left(a, b\right)$	Convergents in AGM iteration
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("75e692"),
    Formula(Where(LessEqual(Abs(Sub(AGM(1, z), a_(n))), Abs(Sub(a_(n), b_(n)))), Def(Tuple(a_(n), b_(n)), AGMSequence(n, 1, z)))),
    Variables(z),
    Assumptions(And(Element(z, CC), GreaterEqual(Re(z), 0))))

Definitions

Illustrations

Single parameter

Domain

Specific values

AGM iteration

Recurrence and limit

Correct square root for complex variables

Brent-Salamin algorithm for pi

Functional equations

Representation by other functions

Representation of other functions

Derivatives and differential equations

Series expansions

Integral representations

Bounds and inequalities