hyperbolic secant function

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21: 10.24 Functions of Imaginary Order

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$${\tilde{J}}_{\nu }\left(x\right)=\mathrm{sech}\left(\frac{1}{2}\pi \nu \right)\mathrm{\Re }\left(J_{\mathrm{i}\nu }\left(x\right)\right),$$

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$${\tilde{Y}}_{\nu }\left(x\right)=\mathrm{sech}\left(\frac{1}{2}\pi \nu \right)\mathrm{\Re }\left(Y_{\mathrm{i}\nu }\left(x\right)\right),$$

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22: 22.11 Fourier and Hyperbolic Series

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22.11.14 $$k^2{\mathrm{sn}}^2(z,k)=\frac{E^{\prime }}{K^{\prime }}-{\left(\frac{\pi }{2K^{\prime }}\right)}^2\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\left({\mathrm{sech}}^2\left(\frac{\pi }{2K^{\prime }}(z-2nK)\right)\right),$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $E^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the second kind, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{sech}z$: hyperbolic secant function, $z$: complex and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.11.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §22.11 and Ch.22

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23: 4.38 Inverse Hyperbolic Functions: Further Properties

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4.38.13 $$\frac{d}{dz}\mathrm{arcsech}z=-\frac{1}{z{(1-z^2)}^{1/2}}.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{arcsech}z$: inverse hyperbolic secant function and $z$: complex variable

A&S Ref:

4.6.41 (has an error.)

Permalink:

http://dlmf.nist.gov/4.38.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §4.38(ii), §4.38 and Ch.4

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24: 10.32 Integral Representations

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10.32.7 $$K_{\nu }\left(x\right)=\sec \left(\frac{1}{2}\nu \pi \right){\int }_0^{\mathrm{\infty }}\cos \left(x\sinh t\right)\cosh \left(\nu t\right)dt=\csc \left(\frac{1}{2}\nu \pi \right){\int }_0^{\mathrm{\infty }}\sin \left(x\sinh t\right)\sinh \left(\nu t\right)dt,$$ , $x>0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cos z$: cosine function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part, $\sec z$: secant function, $\sin z$: sine function, $x$: real variable and $\nu$: complex parameter

A&S Ref:

9.6.22

Referenced by:

§10.32(i)

Permalink:

http://dlmf.nist.gov/10.32.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.32(i), §10.32 and Ch.10

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25: 4.45 Methods of Computation

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Hyperbolic and Inverse Hyperbolic Functions

►The hyperbolic functions can be computed directly from the definitions (4.28.1)–(4.28.7). The inverses $\mathrm{arcsinh}$, $\mathrm{arccosh}$, and $\mathrm{arctanh}$ can be computed from the logarithmic forms given in §4.37(iv), with real arguments. For $\mathrm{arccsch}$, $\mathrm{arcsech}$, and $\mathrm{arccoth}$ we have (4.37.7)–(4.37.9). … ►Similarly for the hyperbolic and inverse hyperbolic functions; compare (4.28.1)–(4.28.7), §4.37(iv), and (4.37.7)–(4.37.9). …

26: 22.19 Physical Applications

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§22.19(i) Classical Dynamics: The Pendulum

… ►Both the $\mathrm{dn}$ and $\mathrm{cn}$ solutions approach $a\mathrm{sech}t$ as $a\to \sqrt{2/\beta }$ from the appropriate directions. ►

§22.19(iii) Nonlinear ODEs and PDEs

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§22.19(v) Other Applications

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27: 19.23 Integral Representations

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19.23.1 $$R_F(0,y,z)={\int }_0^{\pi /2}{(y{\cos }^2\theta +z{\sin }^2\theta )}^{-1/2}d\theta ,$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral and $\sin z$: sine function

Referenced by:

§19.23, §19.23

Permalink:

http://dlmf.nist.gov/19.23.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §19.23 and Ch.19

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19.23.4 $$R_F(0,y,z)=\frac{2}{\pi }{\int }_0^{\pi /2}{R}_C(y,z{\cos }^2\theta )d\theta =\frac{2}{\pi }{\int }_0^{\mathrm{\infty }}{R}_C(y{\cosh }^{2}t,z)dt.$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function and $\int$: integral

Referenced by:

§19.23

Permalink:

http://dlmf.nist.gov/19.23.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §19.23 and Ch.19

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19.23.5 $$R_F(x,y,z)=\frac{2}{\pi }{\int }_0^{\pi /2}{R}_C(x,y{\cos }^2\theta +z{\sin }^2\theta )d\theta ,$$ $\mathrm{\Re }y>0$, $\mathrm{\Re }z>0$,

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part and $\sin z$: sine function

Referenced by:

§19.2(iv), §19.23

Permalink:

http://dlmf.nist.gov/19.23.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.23 and Ch.19

… ►Also, in (19.23.8) and (19.23.10) $\mathrm{B}$ denotes the beta function (§5.12). … ►

19.23.8 $$R_{-a}(𝐛;𝐳)=\frac{2}{\mathrm{B}(b_1,b_2)}{\int }_0^{\pi /2}{(z_1{\cos }^2\theta +z_2{\sin }^2\theta )}^{-a}{(\cos \theta )}^{2b_1-1}{(\sin \theta )}^{2b_2-1}d\theta ,$$ $b_1,b_2>0$; $\mathrm{\Re }{z}_1,\mathrm{\Re }{z}_2>0$.

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Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, $\mathrm{B}(a,b)$: beta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part and $\sin z$: sine function

Referenced by:

§19.2(iv), §19.23, §19.23, §19.23

Permalink:

http://dlmf.nist.gov/19.23.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.23 and Ch.19

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28: 4.15 Graphics

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§4.15(i) Real Arguments

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§4.15(iii) Complex Arguments: Surfaces

►In the graphics shown in this subsection height corresponds to the absolute value of the function and color to the phase. … ►The corresponding surfaces for $\cos \left(x+\mathrm{i}y\right)$, $\cot \left(x+\mathrm{i}y\right)$, and $\sec \left(x+\mathrm{i}y\right)$ are similar. … ►The corresponding surfaces for $\arccos \left(x+\mathrm{i}y\right)$, $\mathrm{arccot}\left(x+\mathrm{i}y\right)$, $\mathrm{arcsec}\left(x+\mathrm{i}y\right)$ can be visualized from Figures 4.15.9, 4.15.11, 4.15.13 with the aid of equations (4.23.16)–(4.23.18).

29: 10.43 Integrals

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§10.43(i) Indefinite Integrals

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§10.43(iii) Fractional Integrals

… ► … ► … ►The Kontorovich–Lebedev transform of a function $g(x)$ is defined as …

30: 1.14 Integral Transforms

… ►The Fourier transform of a real- or complex-valued function $f(t)$ is defined by … ►Suppose $f(t)$ is a real- or complex-valued function and $s$ is a real or complex parameter. … ►

Periodic Functions

… ►The Mellin transform of a real- or complex-valued function $f(x)$ is defined by … ►The Stieltjes transform of a real-valued function $f(t)$ is defined by …