hyperbolic%20analog

SearchAdvancedHelp

(0.003 seconds)

1—10 of 269 matching pages

1: 4.37 Inverse Hyperbolic Functions

§4.37 Inverse Hyperbolic Functions

… ►The principal values of the inverse hyperbolic cosecant, hyperbolic secant, and hyperbolic tangent are given by … ►

Inverse Hyperbolic Sine

… ►

Inverse Hyperbolic Cosine

… ►

Inverse Hyperbolic Tangent

…

2: 36.2 Catastrophes and Canonical Integrals

… ►

Normal Forms for Umbilic Catastrophes with Codimension $K=3$

… ►(hyperbolic umbilic). ►

Canonical Integrals

… ►

36.2.27 $${\mathrm{\Psi }}^{(\mathrm{H})}\left(x,y,z\right)={\mathrm{\Psi }}^{(\mathrm{H})}\left(y,x,z\right).$$

ⓘ

Symbols:

${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $y$: real parameter, $z$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.2.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §36.2(iii), §36.2 and Ch.36

… ►

36.2.29 $${\mathrm{\Psi }}^{(\mathrm{H})}\left(0,0,z\right)=\overline{{\mathrm{\Psi }}^{(\mathrm{H})}\left(0,0,-z\right)}=\frac{2^{1/3}}{\sqrt{3}}\exp \left(\frac{1}{27}\mathrm{i}{z}^3\right){\mathrm{\Psi }}^{(\mathrm{E})}\left(0,0,-\frac{z}{2^{2/3}}\right),$$ .

ⓘ

Symbols:

$\overline{z}$: complex conjugate, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\exp z$: exponential function, ${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $\mathrm{i}$: imaginary unit and $z$: real parameter

Referenced by:

Erratum (V1.0.5) for Chapters 8, 20, 36

Permalink:

http://dlmf.nist.gov/36.2.E29

Encodings:

pMML, png, TeX

Addition (effective with 1.0.5):

This equation has been added. For the proof see Berry and Howls (2010).

See also:

Annotations for §36.2(iv), §36.2 and Ch.36

3: 4.35 Identities

§4.35 Identities

►

§4.35(i) Addition Formulas

… ►

§4.35(ii) Squares and Products

… ►

§4.35(iii) Multiples of the Argument

… ►

§4.35(iv) Real and Imaginary Parts; Moduli

…

4: 4.29 Graphics

… ►

§4.29(i) Real Arguments

… ►

►►

Figure 4.29.6: Principal values of $\mathrm{arccsch}x$ and $\mathrm{arcsech}x$. … Magnify

►

§4.29(ii) Complex Arguments

►The conformal mapping $w=\sinh z$ is obtainable from Figure 4.15.7 by rotating both the $w$-plane and the $z$-plane through an angle $\frac{1}{2}\pi$, compare (4.28.8). ►The surfaces for the complex hyperbolic and inverse hyperbolic functions are similar to the surfaces depicted in §4.15(iii) for the trigonometric and inverse trigonometric functions. …

5: 4.28 Definitions and Periodicity

§4.28 Definitions and Periodicity

… ►

Relations to Trigonometric Functions

… ►As a consequence, many properties of the hyperbolic functions follow immediately from the corresponding properties of the trigonometric functions. ►

Periodicity and Zeros

►The functions $\sinh z$ and $\cosh z$ have period $2\pi \mathrm{i}$, and $\tanh z$ has period $\pi \mathrm{i}$. …

6: 4.1 Special Notation

… ► ►►

$k,m,n$	integers.
…

… ►The main purpose of the present chapter is to extend these definitions and properties to complex arguments $z$. … ►; the hyperbolic trigonometric (or just hyperbolic) functions $\sinh z$, $\cosh z$, $\tanh z$, $\mathrm{csch}z$, $\mathrm{sech}z$, $\coth z$; the inverse hyperbolic functions $\mathrm{arcsinh}z$, $\mathrm{Arcsinh}z$, etc. …

7: 6.4 Analytic Continuation

… ►

6.4.4 $$\mathrm{Ci}\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)=\pm \pi \mathrm{i}+\mathrm{Ci}\left(z\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ci}\left(z\right)$: cosine integral, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

Referenced by:

§6.4

Permalink:

http://dlmf.nist.gov/6.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §6.4 and Ch.6

►

6.4.5 $$\mathrm{Chi}\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)=\pm \pi \mathrm{i}+\mathrm{Chi}\left(z\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{Chi}\left(z\right)$: hyperbolic cosine integral, $\mathrm{i}$: imaginary unit and $z$: complex variable

Referenced by:

§6.4

Permalink:

http://dlmf.nist.gov/6.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §6.4 and Ch.6

… ►Unless indicated otherwise, in the rest of this chapter and elsewhere in the DLMF the functions $E_1\left(z\right)$, $\mathrm{Ci}\left(z\right)$, $\mathrm{Chi}\left(z\right)$, $\mathrm{f}\left(z\right)$, and $\mathrm{g}\left(z\right)$ assume their principal values, that is, the branches that are real on the positive real axis and two-valued on the negative real axis. …

8: 6.2 Definitions and Interrelations

… ►This is also true of the functions $\mathrm{Ci}\left(z\right)$ and $\mathrm{Chi}\left(z\right)$ defined in §6.2(ii). … ►

Hyperbolic Analogs of the Sine and Cosine Integrals

►

6.2.15 $$\mathrm{Shi}\left(z\right)={\int }_0^z\frac{\sinh t}{t}dt,$$

ⓘ

Defines:

$\mathrm{Shi}\left(z\right)$: hyperbolic sine integral

Symbols:

$dx$: differential of $x$ , $\sinh z$: hyperbolic sine function, $\int$: integral and $z$: complex variable

A&S Ref:

5.2.3

Permalink:

http://dlmf.nist.gov/6.2.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §6.2(ii), §6.2(ii), §6.2 and Ch.6

►

6.2.16 $$\mathrm{Chi}\left(z\right)=\gamma +\ln z+{\int }_0^z\frac{\cosh t-1}{t}dt.$$

ⓘ

Defines:

$\mathrm{Chi}\left(z\right)$: hyperbolic cosine integral

Symbols:

$\gamma$: Euler’s constant, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\int$: integral, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

5.2.4

Referenced by:

§6.4

Permalink:

http://dlmf.nist.gov/6.2.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §6.2(ii), §6.2(ii), §6.2 and Ch.6

…

9: 4.32 Inequalities

§4.32 Inequalities

… ►

4.32.1 $$\cosh x\le {\left(\frac{\sinh x}{x}\right)}^3,$$

ⓘ

Symbols:

$\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.32.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.32 and Ch.4

►

4.32.2 $x>0$,

ⓘ

Symbols:

$\cos z$: cosine function, $\tanh z$: hyperbolic tangent function, $\sin z$: sine function and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.32.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.32 and Ch.4

►

4.32.3 $$|\cosh x-\cosh y|\ge |x-y|\sqrt{\sinh x\sinh y},$$ $x>0$, $y>0$,

ⓘ

Symbols:

$\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/4.32.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.32 and Ch.4

… ►For these and other inequalities involving hyperbolic functions see Mitrinović (1964, pp. 61, 76, 159) and Mitrinović (1970, p. 270).

10: 4.31 Special Values and Limits

§4.31 Special Values and Limits

►

Table 4.31.1: Hyperbolic functions: values at multiples of $\frac{1}{2}\pi \mathrm{i}$.

► ►►

$z$	0	$\frac{1}{2}\pi \mathrm{i}$	$\pi \mathrm{i}$	$\frac{3}{2}\pi \mathrm{i}$	$\mathrm{\infty }$
…

►

►

4.31.1 $$\underset{z\to 0}{lim}\frac{\sinh z}{z}=1,$$

ⓘ

Symbols:

$\sinh z$: hyperbolic sine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.31.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.31 and Ch.4

►

4.31.2 $$\underset{z\to 0}{lim}\frac{\tanh z}{z}=1,$$

ⓘ

Symbols:

$\tanh z$: hyperbolic tangent function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.31.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.31 and Ch.4

►

4.31.3 $$\underset{z\to 0}{lim}\frac{\cosh z-1}{z^2}=\frac{1}{2}.$$

ⓘ

Symbols:

$\cosh z$: hyperbolic cosine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.31.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.31 and Ch.4