Airy functions

Symbol: AiryAi —

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

— Airy function of the first kind

Definitions:

Fungrim symbol	Notation	Short description
`AiryAi`	$\operatorname{Ai}\!\left(z\right)$	Airy function of the first kind

Source code for this entry:

Entry(ID("9ac289"),
    SymbolDefinition(AiryAi, AiryAi(z), "Airy function of the first kind"))

5a9d3f

Symbol: AiryBi —

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

— Airy function of the second kind

Definitions:

Fungrim symbol	Notation	Short description
`AiryBi`	$\operatorname{Bi}\!\left(z\right)$	Airy function of the second kind

Source code for this entry:

Entry(ID("5a9d3f"),
    SymbolDefinition(AiryBi, AiryBi(z), "Airy function of the second kind"))

b4c968

Image: X-ray of

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

on

z \in \left[-6, 6\right] + \left[-6, 6\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
`AiryAi`	$\operatorname{Ai}\!\left(z\right)$	Airy function of the first kind
`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("b4c968"),
    Image(Description("X-ray of", AiryAi(z), "on", Element(z, Add(ClosedInterval(-6, 6), Mul(ClosedInterval(-6, 6), ConstI)))), ImageSource("xray_airy_ai")),
    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."))

fa65f3

Image: X-ray of

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

on

z \in \left[-6, 6\right] + \left[-6, 6\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
`AiryBi`	$\operatorname{Bi}\!\left(z\right)$	Airy function of the second kind
`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("fa65f3"),
    Image(Description("X-ray of", AiryBi(z), "on", Element(z, Add(ClosedInterval(-6, 6), Mul(ClosedInterval(-6, 6), ConstI)))), ImageSource("xray_airy_bi")),
    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."))

51b241

y''(z) - z y(z) = 0\; \text{ where } y(z) = C \operatorname{Ai}\!\left(z\right) + D \operatorname{Bi}\!\left(z\right)

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; C \in \mathbb{C} \;\mathbin{\operatorname{and}}\; D \in \mathbb{C}

TeX:

y''(z) - z y(z) = 0\; \text{ where } y(z) = C \operatorname{Ai}\!\left(z\right) + D \operatorname{Bi}\!\left(z\right)

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; C \in \mathbb{C} \;\mathbin{\operatorname{and}}\; D \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`AiryAi`	$\operatorname{Ai}\!\left(z\right)$	Airy function of the first kind
`AiryBi`	$\operatorname{Bi}\!\left(z\right)$	Airy function of the second kind
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("51b241"),
    Formula(Where(Equal(Sub(ComplexDerivative(y(z), For(z, z, 2)), Mul(z, y(z))), 0), Equal(y(z), Add(Mul(C, AiryAi(z)), Mul(D, AiryBi(z)))))),
    Variables(z, C, D),
    Assumptions(And(Element(z, CC), Element(C, CC), Element(D, CC))))

de9800

\operatorname{Ai}\!\left(z\right) \operatorname{Bi}'\!\left(z\right) - \operatorname{Ai}'\!\left(z\right) \operatorname{Bi}\!\left(z\right) = \frac{1}{\pi}

z \in \mathbb{C}

TeX:

\operatorname{Ai}\!\left(z\right) \operatorname{Bi}'\!\left(z\right) - \operatorname{Ai}'\!\left(z\right) \operatorname{Bi}\!\left(z\right) = \frac{1}{\pi}

Definitions:

Fungrim symbol	Notation	Short description
`AiryAi`	$\operatorname{Ai}\!\left(z\right)$	Airy function of the first kind
`AiryBi`	$\operatorname{Bi}\!\left(z\right)$	Airy function of the second kind
`Pi`	$\pi$	The constant pi (3.14...)
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("de9800"),
    Formula(Equal(Sub(Mul(AiryAi(z), AiryBi(z, 1)), Mul(AiryAi(z, 1), AiryBi(z))), Div(1, Pi))),
    Variables(z),
    Element(z, CC))

693cfe

\operatorname{Ai}\!\left(0\right) = \frac{1}{{3}^{2 / 3} \Gamma\!\left(\frac{2}{3}\right)} \in \left[0.355028053887817239260063186004 \pm 1.84 \cdot 10^{-31}\right]

TeX:

\operatorname{Ai}\!\left(0\right) = \frac{1}{{3}^{2 / 3} \Gamma\!\left(\frac{2}{3}\right)} \in \left[0.355028053887817239260063186004 \pm 1.84 \cdot 10^{-31}\right]

Definitions:

Fungrim symbol	Notation	Short description
`AiryAi`	$\operatorname{Ai}\!\left(z\right)$	Airy function of the first kind
`Pow`	${a}^{b}$	Power
`Gamma`	$\Gamma(z)$	Gamma function

Source code for this entry:

Entry(ID("693cfe"),
    Formula(EqualAndElement(AiryAi(0), Div(1, Mul(Pow(3, Div(2, 3)), Gamma(Div(2, 3)))), RealBall(Decimal("0.355028053887817239260063186004"), Decimal("1.84e-31")))))

807917

\operatorname{Ai}'\!\left(0\right) = -\frac{1}{{3}^{1 / 3} \Gamma\!\left(\frac{1}{3}\right)} \in \left[-0.258819403792806798405183560189 \pm 2.04 \cdot 10^{-31}\right]

TeX:

\operatorname{Ai}'\!\left(0\right) = -\frac{1}{{3}^{1 / 3} \Gamma\!\left(\frac{1}{3}\right)} \in \left[-0.258819403792806798405183560189 \pm 2.04 \cdot 10^{-31}\right]

Definitions:

Fungrim symbol	Notation	Short description
`AiryAi`	$\operatorname{Ai}\!\left(z\right)$	Airy function of the first kind
`Pow`	${a}^{b}$	Power
`Gamma`	$\Gamma(z)$	Gamma function

Source code for this entry:

Entry(ID("807917"),
    Formula(EqualAndElement(AiryAi(0, 1), Neg(Div(1, Mul(Pow(3, Div(1, 3)), Gamma(Div(1, 3))))), RealBall(Decimal("-0.258819403792806798405183560189"), Decimal("2.04e-31")))))

9a8d4d

\operatorname{Bi}\!\left(0\right) = \frac{1}{{3}^{1 / 6} \Gamma\!\left(\frac{2}{3}\right)} \in \left[0.614926627446000735150922369094 \pm 3.87 \cdot 10^{-31}\right]

TeX:

\operatorname{Bi}\!\left(0\right) = \frac{1}{{3}^{1 / 6} \Gamma\!\left(\frac{2}{3}\right)} \in \left[0.614926627446000735150922369094 \pm 3.87 \cdot 10^{-31}\right]

Definitions:

Fungrim symbol	Notation	Short description
`AiryBi`	$\operatorname{Bi}\!\left(z\right)$	Airy function of the second kind
`Pow`	${a}^{b}$	Power
`Gamma`	$\Gamma(z)$	Gamma function

Source code for this entry:

Entry(ID("9a8d4d"),
    Formula(EqualAndElement(AiryBi(0), Div(1, Mul(Pow(3, Div(1, 6)), Gamma(Div(2, 3)))), RealBall(Decimal("0.614926627446000735150922369094"), Decimal("3.87e-31")))))

fba07c

\operatorname{Bi}'\!\left(0\right) = \frac{{3}^{1 / 6}}{\Gamma\!\left(\frac{1}{3}\right)} \in \left[0.448288357353826357914823710399 \pm 1.72 \cdot 10^{-31}\right]

TeX:

\operatorname{Bi}'\!\left(0\right) = \frac{{3}^{1 / 6}}{\Gamma\!\left(\frac{1}{3}\right)} \in \left[0.448288357353826357914823710399 \pm 1.72 \cdot 10^{-31}\right]

Definitions:

Fungrim symbol	Notation	Short description
`AiryBi`	$\operatorname{Bi}\!\left(z\right)$	Airy function of the second kind
`Pow`	${a}^{b}$	Power
`Gamma`	$\Gamma(z)$	Gamma function

Source code for this entry:

Entry(ID("fba07c"),
    Formula(EqualAndElement(AiryBi(0, 1), Div(Pow(3, Div(1, 6)), Gamma(Div(1, 3))), RealBall(Decimal("0.448288357353826357914823710399"), Decimal("1.72e-31")))))

b2e9d0

\operatorname{Ai}''\!\left(z\right) = z \operatorname{Ai}\!\left(z\right)

Assumptions:

z \in \mathbb{C}

TeX:

\operatorname{Ai}''\!\left(z\right) = z \operatorname{Ai}\!\left(z\right)

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`AiryAi`	$\operatorname{Ai}\!\left(z\right)$	Airy function of the first kind
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("b2e9d0"),
    Formula(Equal(AiryAi(z, 2), Mul(z, AiryAi(z)))),
    Variables(z),
    Assumptions(Element(z, CC)))

70ec9f

\operatorname{Bi}''\!\left(z\right) = z \operatorname{Bi}\!\left(z\right)

Assumptions:

z \in \mathbb{C}

TeX:

\operatorname{Bi}''\!\left(z\right) = z \operatorname{Bi}\!\left(z\right)

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`AiryBi`	$\operatorname{Bi}\!\left(z\right)$	Airy function of the second kind
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("70ec9f"),
    Formula(Equal(AiryBi(z, 2), Mul(z, AiryBi(z)))),
    Variables(z),
    Assumptions(Element(z, CC)))

eadca2

{y}^{(n)}(z) = z {y}^{(n - 2)}(z) + \left(n - 2\right) {y}^{(n - 3)}(z)\; \text{ where } y(z) = C \operatorname{Ai}\!\left(z\right) + D \operatorname{Bi}\!\left(z\right)

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 3} \;\mathbin{\operatorname{and}}\; C \in \mathbb{C} \;\mathbin{\operatorname{and}}\; D \in \mathbb{C}

TeX:

{y}^{(n)}(z) = z {y}^{(n - 2)}(z) + \left(n - 2\right) {y}^{(n - 3)}(z)\; \text{ where } y(z) = C \operatorname{Ai}\!\left(z\right) + D \operatorname{Bi}\!\left(z\right)

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 3} \;\mathbin{\operatorname{and}}\; C \in \mathbb{C} \;\mathbin{\operatorname{and}}\; D \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`AiryAi`	$\operatorname{Ai}\!\left(z\right)$	Airy function of the first kind
`AiryBi`	$\operatorname{Bi}\!\left(z\right)$	Airy function of the second kind
`CC`	$\mathbb{C}$	Complex numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("eadca2"),
    Formula(Where(Equal(ComplexDerivative(y(z), For(z, z, n)), Add(Mul(z, ComplexDerivative(y(z), For(z, z, Sub(n, 2)))), Mul(Sub(n, 2), ComplexDerivative(y(z), For(z, z, Sub(n, 3)))))), Equal(y(z), Add(Mul(C, AiryAi(z)), Mul(D, AiryBi(z)))))),
    Variables(n, z, C, D),
    Assumptions(And(Element(z, CC), Element(n, ZZGreaterEqual(3)), Element(C, CC), Element(D, CC))))

01bbb6

\operatorname{Ai}\!\left(z\right) = \operatorname{Ai}\!\left(0\right) \,{}_0F_1\!\left(\frac{2}{3}, \frac{{z}^{3}}{9}\right) + z \operatorname{Ai}'\!\left(0\right) \,{}_0F_1\!\left(\frac{4}{3}, \frac{{z}^{3}}{9}\right)

Assumptions:

z \in \mathbb{C}

TeX:

\operatorname{Ai}\!\left(z\right) = \operatorname{Ai}\!\left(0\right) \,{}_0F_1\!\left(\frac{2}{3}, \frac{{z}^{3}}{9}\right) + z \operatorname{Ai}'\!\left(0\right) \,{}_0F_1\!\left(\frac{4}{3}, \frac{{z}^{3}}{9}\right)

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`AiryAi`	$\operatorname{Ai}\!\left(z\right)$	Airy function of the first kind
`Hypergeometric0F1`	$\,{}_0F_1\!\left(a, z\right)$	Confluent hypergeometric limit function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("01bbb6"),
    Formula(Equal(AiryAi(z), Add(Mul(AiryAi(0), Hypergeometric0F1(Div(2, 3), Div(Pow(z, 3), 9))), Mul(Mul(z, AiryAi(0, 1)), Hypergeometric0F1(Div(4, 3), Div(Pow(z, 3), 9)))))),
    Variables(z),
    Assumptions(Element(z, CC)))

bd319e

\operatorname{Bi}\!\left(z\right) = \operatorname{Bi}\!\left(0\right) \,{}_0F_1\!\left(\frac{2}{3}, \frac{{z}^{3}}{9}\right) + z \operatorname{Bi}'\!\left(0\right) \,{}_0F_1\!\left(\frac{4}{3}, \frac{{z}^{3}}{9}\right)

Assumptions:

z \in \mathbb{C}

TeX:

\operatorname{Bi}\!\left(z\right) = \operatorname{Bi}\!\left(0\right) \,{}_0F_1\!\left(\frac{2}{3}, \frac{{z}^{3}}{9}\right) + z \operatorname{Bi}'\!\left(0\right) \,{}_0F_1\!\left(\frac{4}{3}, \frac{{z}^{3}}{9}\right)

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`AiryBi`	$\operatorname{Bi}\!\left(z\right)$	Airy function of the second kind
`Hypergeometric0F1`	$\,{}_0F_1\!\left(a, z\right)$	Confluent hypergeometric limit function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("bd319e"),
    Formula(Equal(AiryBi(z), Add(Mul(AiryBi(0), Hypergeometric0F1(Div(2, 3), Div(Pow(z, 3), 9))), Mul(Mul(z, AiryBi(0, 1)), Hypergeometric0F1(Div(4, 3), Div(Pow(z, 3), 9)))))),
    Variables(z),
    Assumptions(Element(z, CC)))

20e530

\operatorname{Ai}'\!\left(z\right) = \operatorname{Ai}'\!\left(0\right) \,{}_0F_1\!\left(\frac{1}{3}, \frac{{z}^{3}}{9}\right) + \frac{{z}^{2}}{2} \operatorname{Ai}\!\left(0\right) \,{}_0F_1\!\left(\frac{5}{3}, \frac{{z}^{3}}{9}\right)

Assumptions:

z \in \mathbb{C}

TeX:

\operatorname{Ai}'\!\left(z\right) = \operatorname{Ai}'\!\left(0\right) \,{}_0F_1\!\left(\frac{1}{3}, \frac{{z}^{3}}{9}\right) + \frac{{z}^{2}}{2} \operatorname{Ai}\!\left(0\right) \,{}_0F_1\!\left(\frac{5}{3}, \frac{{z}^{3}}{9}\right)

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`AiryAi`	$\operatorname{Ai}\!\left(z\right)$	Airy function of the first kind
`Hypergeometric0F1`	$\,{}_0F_1\!\left(a, z\right)$	Confluent hypergeometric limit function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("20e530"),
    Formula(Equal(AiryAi(z, 1), Add(Mul(AiryAi(0, 1), Hypergeometric0F1(Div(1, 3), Div(Pow(z, 3), 9))), Mul(Mul(Div(Pow(z, 2), 2), AiryAi(0)), Hypergeometric0F1(Div(5, 3), Div(Pow(z, 3), 9)))))),
    Variables(z),
    Assumptions(Element(z, CC)))

4d65e5

\operatorname{Bi}'\!\left(z\right) = \operatorname{Bi}'\!\left(0\right) \,{}_0F_1\!\left(\frac{1}{3}, \frac{{z}^{3}}{9}\right) + \frac{{z}^{2}}{2} \operatorname{Bi}\!\left(0\right) \,{}_0F_1\!\left(\frac{5}{3}, \frac{{z}^{3}}{9}\right)

Assumptions:

z \in \mathbb{C}

TeX:

\operatorname{Bi}'\!\left(z\right) = \operatorname{Bi}'\!\left(0\right) \,{}_0F_1\!\left(\frac{1}{3}, \frac{{z}^{3}}{9}\right) + \frac{{z}^{2}}{2} \operatorname{Bi}\!\left(0\right) \,{}_0F_1\!\left(\frac{5}{3}, \frac{{z}^{3}}{9}\right)

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`AiryBi`	$\operatorname{Bi}\!\left(z\right)$	Airy function of the second kind
`Hypergeometric0F1`	$\,{}_0F_1\!\left(a, z\right)$	Confluent hypergeometric limit function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("4d65e5"),
    Formula(Equal(AiryBi(z, 1), Add(Mul(AiryBi(0, 1), Hypergeometric0F1(Div(1, 3), Div(Pow(z, 3), 9))), Mul(Mul(Div(Pow(z, 2), 2), AiryBi(0)), Hypergeometric0F1(Div(5, 3), Div(Pow(z, 3), 9)))))),
    Variables(z),
    Assumptions(Element(z, CC)))

def37e

C \operatorname{Ai}\!\left(z\right) + D \operatorname{Bi}\!\left(z\right) \text{ is holomorphic on } z \in \mathbb{C}

Assumptions:

C \in \mathbb{C} \;\mathbin{\operatorname{and}}\; D \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{not} \left(C = 0 \;\mathbin{\operatorname{and}}\; D = 0\right)

TeX:

C \operatorname{Ai}\!\left(z\right) + D \operatorname{Bi}\!\left(z\right) \text{ is holomorphic on } z \in \mathbb{C}

C \in \mathbb{C} \;\mathbin{\operatorname{and}}\; D \in \mathbb{C} \;\mathbin{\operatorname{and}}\;  \operatorname{not} \left(C = 0 \;\mathbin{\operatorname{and}}\; D = 0\right)

Definitions:

Fungrim symbol	Notation	Short description
`IsHolomorphic`	$f(z) \text{ is holomorphic at } z = c$	Holomorphic predicate
`AiryAi`	$\operatorname{Ai}\!\left(z\right)$	Airy function of the first kind
`AiryBi`	$\operatorname{Bi}\!\left(z\right)$	Airy function of the second kind
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("def37e"),
    Formula(IsHolomorphic(Add(Mul(C, AiryAi(z)), Mul(D, AiryBi(z))), ForElement(z, CC))),
    Variables(C, D),
    Assumptions(And(Element(C, CC), Element(D, CC), Not(And(Equal(C, 0), Equal(D, 0))))))

1f0577

\mathop{\operatorname{poles}\,}\limits_{z \in \mathbb{C} \cup \left\{{\tilde \infty}\right\}} \left[C \operatorname{Ai}\!\left(z\right) + D \operatorname{Bi}\!\left(z\right)\right] = \left\{\right\}

Assumptions:

C \in \mathbb{C} \;\mathbin{\operatorname{and}}\; D \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{not} \left(C = 0 \;\mathbin{\operatorname{and}}\; D = 0\right)

TeX:

\mathop{\operatorname{poles}\,}\limits_{z \in \mathbb{C} \cup \left\{{\tilde \infty}\right\}} \left[C \operatorname{Ai}\!\left(z\right) + D \operatorname{Bi}\!\left(z\right)\right] = \left\{\right\}

C \in \mathbb{C} \;\mathbin{\operatorname{and}}\; D \in \mathbb{C} \;\mathbin{\operatorname{and}}\;  \operatorname{not} \left(C = 0 \;\mathbin{\operatorname{and}}\; D = 0\right)

Definitions:

Fungrim symbol	Notation	Short description
`AiryAi`	$\operatorname{Ai}\!\left(z\right)$	Airy function of the first kind
`AiryBi`	$\operatorname{Bi}\!\left(z\right)$	Airy function of the second kind
`CC`	$\mathbb{C}$	Complex numbers
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity

Source code for this entry:

Entry(ID("1f0577"),
    Formula(Equal(Poles(Add(Mul(C, AiryAi(z)), Mul(D, AiryBi(z))), ForElement(z, Union(CC, Set(UnsignedInfinity)))), Set())),
    Variables(C, D),
    Assumptions(And(Element(C, CC), Element(D, CC), Not(And(Equal(C, 0), Equal(D, 0))))))

90f31e

\operatorname{EssentialSingularities}\!\left(C \operatorname{Ai}\!\left(z\right) + D \operatorname{Bi}\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{{\tilde \infty}\right\}

Assumptions:

C \in \mathbb{C} \;\mathbin{\operatorname{and}}\; D \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{not} \left(C = 0 \;\mathbin{\operatorname{and}}\; D = 0\right)

TeX:

\operatorname{EssentialSingularities}\!\left(C \operatorname{Ai}\!\left(z\right) + D \operatorname{Bi}\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{{\tilde \infty}\right\}

C \in \mathbb{C} \;\mathbin{\operatorname{and}}\; D \in \mathbb{C} \;\mathbin{\operatorname{and}}\;  \operatorname{not} \left(C = 0 \;\mathbin{\operatorname{and}}\; D = 0\right)

Definitions:

Fungrim symbol	Notation	Short description
`AiryAi`	$\operatorname{Ai}\!\left(z\right)$	Airy function of the first kind
`AiryBi`	$\operatorname{Bi}\!\left(z\right)$	Airy function of the second kind
`CC`	$\mathbb{C}$	Complex numbers
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity

Source code for this entry:

Entry(ID("90f31e"),
    Formula(Equal(EssentialSingularities(Add(Mul(C, AiryAi(z)), Mul(D, AiryBi(z))), z, Union(CC, Set(UnsignedInfinity))), Set(UnsignedInfinity))),
    Variables(C, D),
    Assumptions(And(Element(C, CC), Element(D, CC), Not(And(Equal(C, 0), Equal(D, 0))))))

b88f65

\operatorname{BranchPoints}\!\left(C \operatorname{Ai}\!\left(z\right) + D \operatorname{Bi}\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}

Assumptions:

C \in \mathbb{C} \;\mathbin{\operatorname{and}}\; D \in \mathbb{C}

TeX:

\operatorname{BranchPoints}\!\left(C \operatorname{Ai}\!\left(z\right) + D \operatorname{Bi}\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}

C \in \mathbb{C} \;\mathbin{\operatorname{and}}\; D \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`AiryAi`	$\operatorname{Ai}\!\left(z\right)$	Airy function of the first kind
`AiryBi`	$\operatorname{Bi}\!\left(z\right)$	Airy function of the second kind
`CC`	$\mathbb{C}$	Complex numbers
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity

Source code for this entry:

Entry(ID("b88f65"),
    Formula(Equal(BranchPoints(Add(Mul(C, AiryAi(z)), Mul(D, AiryBi(z))), z, Union(CC, Set(UnsignedInfinity))), Set())),
    Variables(C, D),
    Assumptions(And(Element(C, CC), Element(D, CC))))

7194d4

\operatorname{BranchCuts}\!\left(C \operatorname{Ai}\!\left(z\right) + D \operatorname{Bi}\!\left(z\right), z, \mathbb{C}\right) = \left\{\right\}

Assumptions:

C \in \mathbb{C} \;\mathbin{\operatorname{and}}\; D \in \mathbb{C}

TeX:

\operatorname{BranchCuts}\!\left(C \operatorname{Ai}\!\left(z\right) + D \operatorname{Bi}\!\left(z\right), z, \mathbb{C}\right) = \left\{\right\}

C \in \mathbb{C} \;\mathbin{\operatorname{and}}\; D \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`AiryAi`	$\operatorname{Ai}\!\left(z\right)$	Airy function of the first kind
`AiryBi`	$\operatorname{Bi}\!\left(z\right)$	Airy function of the second kind
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("7194d4"),
    Formula(Equal(BranchCuts(Add(Mul(C, AiryAi(z)), Mul(D, AiryBi(z))), z, CC), Set())),
    Variables(C, D),
    Assumptions(And(Element(C, CC), Element(D, CC))))

d1f9d0

\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \operatorname{Ai}\!\left(z\right) \subset \mathbb{R}

TeX:

\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \operatorname{Ai}\!\left(z\right) \subset \mathbb{R}

Definitions:

Fungrim symbol	Notation	Short description
`Zeros`	$\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)$	Zeros (roots) of function
`AiryAi`	$\operatorname{Ai}\!\left(z\right)$	Airy function of the first kind
`CC`	$\mathbb{C}$	Complex numbers
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("d1f9d0"),
    Formula(Subset(Zeros(AiryAi(z), ForElement(z, CC)), RR)))

a2df77

\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \operatorname{Ai}'\!\left(z\right) \subset \mathbb{R}

TeX:

\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \operatorname{Ai}'\!\left(z\right) \subset \mathbb{R}

Definitions:

Fungrim symbol	Notation	Short description
`Zeros`	$\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)$	Zeros (roots) of function
`AiryAi`	$\operatorname{Ai}\!\left(z\right)$	Airy function of the first kind
`CC`	$\mathbb{C}$	Complex numbers
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("a2df77"),
    Formula(Subset(Zeros(AiryAi(z, 1), ForElement(z, CC)), RR)))

Definitions

Illustrations

Differential equation

Special values

Higher derivatives

Hypergeometric representations

Analytic properties