secant function

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21: 28.20 Definitions and Basic Properties

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28.20.9 $${\mathrm{M}}_{\nu }^{(3)}(z,h)=H_{\nu }^{(1)}\left(2h\cosh z\right)\left(1+O\left(\mathrm{sech}z\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.20.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §28.20(iii), §28.20 and Ch.28

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28.20.10 $${\mathrm{M}}_{\nu }^{(4)}(z,h)=H_{\nu }^{(2)}\left(2h\cosh z\right)\left(1+O\left(\mathrm{sech}z\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.20.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §28.20(iii), §28.20 and Ch.28

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28.20.11 $${\mathrm{M}}_{\nu }^{(1)}(z,h)=J_{\nu }\left(2h\cosh z\right)+{\mathrm{e}}^{|\mathrm{\Im }\left(2h\cosh z\right)|}O\left({\left(\mathrm{sech}z\right)}^{3/2}\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $O\left(x\right)$: order not exceeding, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, $\mathrm{\Im }$: imaginary part, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§28.33(ii)

Permalink:

http://dlmf.nist.gov/28.20.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §28.20(iii), §28.20 and Ch.28

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28.20.12 $${\mathrm{M}}_{\nu }^{(2)}(z,h)=Y_{\nu }\left(2h\cosh z\right)+{\mathrm{e}}^{|\mathrm{\Im }\left(2h\cosh z\right)|}O\left({(\mathrm{sech}z)}^{3/2}\right),$$

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Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $O\left(x\right)$: order not exceeding, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, $\mathrm{\Im }$: imaginary part, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§28.33(ii)

Permalink:

http://dlmf.nist.gov/28.20.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §28.20(iii), §28.20 and Ch.28

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22: 4.19 Maclaurin Series and Laurent Series

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4.19.5 $$\sec z=1+\frac{z^2}{2}+\frac{5}{24}{z}^4+\frac{61}{720}{z}^6+\mathrm{\cdots }+\frac{{(-1)}^n{E}_{2n}}{(2n)!}{z}^{2n}+\mathrm{\cdots },$$ ,

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Symbols:

$E_n$: Euler numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\sec z$: secant function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.69

Referenced by:

§24.1, §24.19(i), §24.2(ii)

Permalink:

http://dlmf.nist.gov/4.19.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.19 and Ch.4

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23: 5.6 Inequalities

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5.6.7 $$|\mathrm{\Gamma }\left(x+\mathrm{i}y\right)|\ge {(\mathrm{sech}\left(\pi y\right))}^{1/2}\mathrm{\Gamma }\left(x\right),$$ $x\ge \frac{1}{2}$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{sech}z$: hyperbolic secant function, $\mathrm{i}$: imaginary unit, $x$: real variable and $y$: real variable

Referenced by:

§5.6(ii)

Permalink:

http://dlmf.nist.gov/5.6.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §5.6(ii), §5.6 and Ch.5

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24: 4.26 Integrals

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4.26.5 $$\int \sec xdx={\mathrm{gd}}^{-1}\left(x\right),$$ .

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, ${\mathrm{gd}}^{-1}\left(x\right)$: inverse Gudermannian function, $\sec z$: secant function and $x$: real variable

A&S Ref:

4.3.117

Permalink:

http://dlmf.nist.gov/4.26.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.26(ii), §4.26 and Ch.4

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4.26.18 $$\int \mathrm{arcsec}xdx=x\mathrm{arcsec}x-\ln \left(x+{(x^2-1)}^{1/2}\right),$$ ,

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $\mathrm{arcsec}z$: arcsecant function, $\ln z$: principal branch of logarithm function and $x$: real variable

A&S Ref:

4.4.62 (is stated differently.)

Permalink:

http://dlmf.nist.gov/4.26.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §4.26(iv), §4.26 and Ch.4

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25: 20.10 Integrals

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20.10.4 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_1\left(\frac{\beta \pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_2\left(\frac{(1+\beta )\pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt=-\frac{\mathrm{\ell }}{\sqrt{s}}\sinh \left(\beta \sqrt{s}\right)\mathrm{sech}\left(\mathrm{\ell }\sqrt{s}\right),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{sech}z$: hyperbolic secant function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $s$: parameter, $\beta$: parameter and $\mathrm{\ell }$: parameter

Permalink:

http://dlmf.nist.gov/20.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §20.10(ii), §20.10 and Ch.20

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26: 22.16 Related Functions

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22.16.8 $$\mathrm{am}(x,k)=\mathrm{gd}x-\frac{1}{4}k_{}^{\prime }{}_{}{}^2(x-\sinh x\cosh x)\mathrm{sech}x+O\left(k_{}^{\prime }{}_{}{}^4\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{gd}x$: Gudermannian function, $\mathrm{am}(x,k)$: Jacobi’s amplitude function, $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, $\sinh z$: hyperbolic sine function, $x$: real, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

16.15.4

Permalink:

http://dlmf.nist.gov/22.16.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22

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27: 14.5 Special Values

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14.5.26 $${𝑸}_{\frac{1}{2}}\left(\cosh \xi \right)=2{\pi }^{-1/2}\cosh \xi \mathrm{sech}\left(\frac{1}{2}\xi \right)K\left(\mathrm{sech}\left(\frac{1}{2}\xi \right)\right)-4{\pi }^{-1/2}\cosh \left(\frac{1}{2}\xi \right)E\left(\mathrm{sech}\left(\frac{1}{2}\xi \right)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, ${𝑸}_{\nu }\left(z\right)={𝑸}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function and $\xi >0$: variable

A&S Ref:

8.13.7 (in slightly different form)

Permalink:

http://dlmf.nist.gov/14.5.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §14.5(v), §14.5 and Ch.14

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28: 4.21 Identities

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4.21.13 $${\sec }^{2}z=1+{\tan }^{2}z,$$

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Symbols:

$\sec z$: secant function, $\tan z$: tangent function and $z$: complex variable

A&S Ref:

4.3.11

Permalink:

http://dlmf.nist.gov/4.21.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §4.21(ii), §4.21 and Ch.4

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29: 19.30 Lengths of Plane Curves

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19.30.11 $$s=2a^2{\int }_0^r\frac{dt}{\sqrt{4a^4-t^4}}=\sqrt{2a^2}{R}_F(q-1,q,q+1),$$ $q=2a^2/r^2=\sec \left(2\theta \right)$,

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $dx$: differential of $x$ , $\int$: integral, $\sec z$: secant function, $r$: length, $q$ and $s$: arclength

Permalink:

http://dlmf.nist.gov/19.30.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §19.30(iii), §19.30 and Ch.19

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30: 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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