analytical properties

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21: 9.2 Differential Equation

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9.2.1 $$\frac{d^{2}w}{{dz}^2}=zw.$$

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Defines:

$w$: ODE solution (locally)

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ and $z$: complex variable

Source:

Olver (1997b, (1.01), p. 392)

A&S Ref:

10.4.1 (in slightly different form)

Referenced by:

§36.8, (9.10.10), (9.10.20), (9.10.21), (9.10.8), (9.10.9), §9.10(iii), (9.11.2), §9.11(i), §9.11(iv), §9.12(i), §9.17(ii), §9.17(ii), (9.2.16), §9.2(iii), §9.2(vi), (9.8.14), (9.8.16), (9.8.18), (9.8.19)

Permalink:

http://dlmf.nist.gov/9.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §9.2(i), §9.2 and Ch.9

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22: 22.2 Definitions

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23: 23.2 Definitions and Periodic Properties

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§23.2(ii) Weierstrass Elliptic Functions

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24: 4.37 Inverse Hyperbolic Functions

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4.37.6 $$\mathrm{Arccoth}z=\mathrm{Arctanh}\left(1/z\right).$$

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Defines:

$\mathrm{Arccoth}z$: general inverse hyperbolic cotangent function

Symbols:

$\mathrm{Arctanh}z$: general inverse hyperbolic tangent function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.37.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(i), §4.37 and Ch.4

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25: 10.21 Zeros

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§10.21(ii) Analytic Properties

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26: 8.21 Generalized Sine and Cosine Integrals

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8.21.2 $$\mathrm{Ci}(a,z)\pm \mathrm{i}\mathrm{Si}(a,z)={\mathrm{e}}^{\pm \frac{1}{2}\pi \mathrm{i}a}\gamma (a,z{\mathrm{e}}^{\mp \frac{1}{2}\pi \mathrm{i}}).$$

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Defines:

$\mathrm{Ci}(a,z)$: generalized cosine integral and $\mathrm{Si}(a,z)$: generalized sine integral

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\gamma (a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Referenced by:

§8.21(ii)

Permalink:

http://dlmf.nist.gov/8.21.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(i), §8.21 and Ch.8

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27: 10.47 Definitions and Basic Properties

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28: 28.2 Definitions and Basic Properties

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§28.2(ii) Basic Solutions $w_{\text{I}}$, $w_{\text{II}}$

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29: 4.23 Inverse Trigonometric Functions

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4.23.6 $$\mathrm{Arccot}z=\mathrm{Arctan}\left(1/z\right).$$

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Defines:

$\mathrm{Arccot}z$: general arccotangent function

Symbols:

$\mathrm{Arctan}z$: general arctangent function and $z$: complex variable

A&S Ref:

4.4.8 (modified)

Permalink:

http://dlmf.nist.gov/4.23.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.23(i), §4.23 and Ch.4

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30: 9.12 Scorer Functions

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9.12.5 $$\mathrm{Hi}\left(z\right)=\mathrm{Bi}\left(z\right){\int }_{-\mathrm{\infty }}^z\mathrm{Ai}\left(t\right)dt-\mathrm{Ai}\left(z\right){\int }_{-\mathrm{\infty }}^z\mathrm{Bi}\left(t\right)dt.$$

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Defines:

$\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function)

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $dx$: differential of $x$ , $\int$: integral and $z$: complex variable

Source:

Olver (1997b, (12.09), p. 430)

Referenced by:

(9.10.3)

Permalink:

http://dlmf.nist.gov/9.12.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(i), §9.12 and Ch.9

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