埃因霍温理工大学市场开发文凭证书《做证微fuk7778》tAN

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21: 7.20 Mathematical Applications

… ►Furthermore, because $dy/dx=\tan \left(\frac{1}{2}\pi t^2\right)$, the angle between the $x$-axis and the tangent to the spiral at $P(t)$ is given by $\frac{1}{2}\pi t^2$. …

22: 31.10 Integral Equations and Representations

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31.10.8 $${\sin }^2\theta \left(\frac{{\partial }^2𝒦}{{\partial \theta }^2}+\left((1-2\gamma )\tan \theta +2(\delta +ϵ-\frac{1}{2})\cot \theta \right)\frac{\partial 𝒦}{\partial \theta }-4\alpha \beta 𝒦\right)+\frac{{\partial }^2𝒦}{{\partial \varphi }^2}+\left((1-2\delta )\cot \varphi -(1-2ϵ)\tan \varphi \right)\frac{\partial 𝒦}{\partial \varphi }=0.$$

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Symbols:

$\cot z$: cotangent function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function, $\tan z$: tangent function, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $\alpha$: real or complex parameter, $\beta$: real or complex parameter, $𝒦(z,t)$: kernel, $\theta$: angle and $\varphi$: angle

Referenced by:

§31.10(i)

Permalink:

http://dlmf.nist.gov/31.10.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §31.10(i), §31.10(i), §31.10 and Ch.31

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31.10.21 $$\frac{{\partial }^2𝒦}{{\partial r}^2}+\frac{2(\gamma +\delta +ϵ)-1}{r}\frac{\partial 𝒦}{\partial r}+\frac{1}{r^2}\frac{{\partial }^2𝒦}{{\partial \theta }^2}+\frac{(2(\delta +ϵ)-1)\cot \theta -(2\gamma -1)\tan \theta }{r^2}\frac{\partial 𝒦}{\partial \theta }+\frac{1}{r^2{\sin }^2\theta }\frac{{\partial }^2𝒦}{{\partial \varphi }^2}+\frac{(2\delta -1)\cot \varphi -(2ϵ-1)\tan \varphi }{r^2{\sin }^2\theta }\frac{\partial 𝒦}{\partial \varphi }=0.$$

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Symbols:

$\cot z$: cotangent function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function, $\tan z$: tangent function, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $𝒦(z;s,t)$: kernel, $r$: radius, $\theta$: angle and $\varphi$: angle

Permalink:

http://dlmf.nist.gov/31.10.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §31.10(ii), §31.10(ii), §31.10 and Ch.31

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23: 4.40 Integrals

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4.40.8 $${\int }_0^{\mathrm{\infty }}\frac{\sinh \left(ax\right)}{\sinh \left(\pi x\right)}dx=\frac{1}{2}\tan \left(\frac{1}{2}a\right),$$ ,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\sinh z$: hyperbolic sine function, $\int$: integral, $\tan z$: tangent function, $a$: real or complex constant and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.40.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(iii), §4.40 and Ch.4

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24: 24.15 Related Sequences of Numbers

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24.15.3 $$\tan t=\sum _{n=0}^{\mathrm{\infty }}{T}_n\frac{t^n}{n!},$$

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Symbols:

$!$: factorial (as in $n!$), $\tan z$: tangent function, $n$: integer, $t$: real or complex and $T_n$: tangent numbers

Referenced by:

§24.19(i)

Permalink:

http://dlmf.nist.gov/24.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §24.15(ii), §24.15 and Ch.24

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25: 3.8 Nonlinear Equations

… ► $f(x)=x-\tan x$. … ►

Table 3.8.1: Newton’s rule for $x-\tan x=0$.

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$n$	$x_n$
…

►

…

26: 19.6 Special Cases

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$\mathrm{\Pi }(\varphi ,1,0)=\tan \varphi .$

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$$\mathrm{\Pi }(\varphi ,1,k)=F(\varphi ,k)-\frac{1}{k_{}^{\prime }{}_{}{}^2}(E(\varphi ,k)-\mathrm{\Delta }\tan \varphi ).$$

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27: 18.21 Hahn Class: Interrelations

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18.21.10 $$\underset{t\to \mathrm{\infty }}{lim}{t}^{-n}{p}_n(x-t;\lambda +it,-t\tan \varphi ,\lambda -it,-t\tan \varphi )=\frac{{(-1)}^n}{{(\cos \varphi )}^n}{P}_n^{(\lambda )}(x;\varphi ).$$

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Symbols:

$P_n^{(\lambda )}(x;\varphi )$: Meixner–Pollaczek polynomial, $p_n(x;a,b,\overline{a},\overline{b})$: continuous Hahn polynomial, $\cos z$: cosine function, $\tan z$: tangent function, $t$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.21(ii), §18.22(ii), §18.22(iii)

Permalink:

http://dlmf.nist.gov/18.21.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

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28: 20.5 Infinite Products and Related Results

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20.5.11 $$\frac{{\theta }_2^{\prime }(z,q)}{{\theta }_2(z,q)}+\tan z=-4\sin \left(2z\right)\sum _{n=1}^{\mathrm{\infty }}\frac{q^{2n}}{1+2q^{2n}\cos \left(2z\right)+q^{4n}}=4\sum _{n=1}^{\mathrm{\infty }}{(-1)}^n\frac{q^{2n}}{1-q^{2n}}\sin \left(2nz\right).$$

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Symbols:

${\theta }_j(z,q)$: theta function, $\cos z$: cosine function, $\sin z$: sine function, $\tan z$: tangent function, $n$: integer, $z$: complex and $q$: nome

A&S Ref:

16.29.2

Referenced by:

§20.5(ii)

Permalink:

http://dlmf.nist.gov/20.5.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §20.5(ii), §20.5 and Ch.20

►The left-hand sides of (20.5.10) and (20.5.11) are replaced by their limiting values when $\cot z$ or $\tan z$ are undefined. …

29: 10.19 Asymptotic Expansions for Large Order

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10.19.5 $$\xi =\nu (\tan \beta -\beta )-\frac{1}{4}\pi ,$$

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Defines:

$\xi$ (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\tan z$: tangent function, $\nu$: complex parameter and $\beta$

Permalink:

http://dlmf.nist.gov/10.19.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.19(ii), §10.19 and Ch.10

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$$J_{\nu }\left(\nu \sec \beta \right)\sim {\left(\frac{2}{\pi \nu \tan \beta }\right)}^{\frac{1}{2}}\left(\cos \xi \sum _{k=0}^{\mathrm{\infty }}\frac{U_{2k}(\mathrm{i}\cot \beta )}{{\nu }^{2k}}-\mathrm{i}\sin \xi \sum _{k=0}^{\mathrm{\infty }}\frac{U_{2k+1}(\mathrm{i}\cot \beta )}{{\nu }^{2k+1}}\right),$$

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$$Y_{\nu }\left(\nu \sec \beta \right)\sim {\left(\frac{2}{\pi \nu \tan \beta }\right)}^{\frac{1}{2}}\left(\sin \xi \sum _{k=0}^{\mathrm{\infty }}\frac{U_{2k}(\mathrm{i}\cot \beta )}{{\nu }^{2k}}+\mathrm{i}\cos \xi \sum _{k=0}^{\mathrm{\infty }}\frac{U_{2k+1}(\mathrm{i}\cot \beta )}{{\nu }^{2k+1}}\right),$$

…

30: 4.23 Inverse Trigonometric Functions

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4.23.30 $$z=\tan w,$$

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Symbols:

$\tan z$: tangent function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.23.E30

Encodings:

pMML, png, TeX

See also:

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

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4.23.42 $${\mathrm{gd}}^{-1}\left(x\right)=\ln \tan \left(\frac{1}{2}x+\frac{1}{4}\pi \right)=\ln \left(\sec x+\tan x\right)=\mathrm{arcsinh}\left(\tan x\right)=\mathrm{arccsch}\left(\cot x\right)=\mathrm{arccosh}\left(\sec x\right)=\mathrm{arcsech}\left(\cos x\right)=\mathrm{arctanh}\left(\sin x\right)=\mathrm{arccoth}\left(\csc x\right).$$

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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, $\cot z$: cotangent function, $\mathrm{arccsch}z$: inverse hyperbolic cosecant function, $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\mathrm{arccoth}z$: inverse hyperbolic cotangent function, $\mathrm{arcsech}z$: inverse hyperbolic secant function, $\mathrm{arcsinh}z$: inverse hyperbolic sine function, $\mathrm{arctanh}z$: inverse hyperbolic tangent function, ${\mathrm{gd}}^{-1}\left(x\right)$: inverse Gudermannian function, $\ln z$: principal branch of logarithm function, $\sec z$: secant function, $\sin z$: sine function, $\tan z$: tangent function and $x$: real variable

Referenced by:

§19.9(ii), §4.23(viii), §4.26(ii)

Permalink:

http://dlmf.nist.gov/4.23.E42

Encodings:

pMML, png, TeX

See also:

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