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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11: 4.19 Maclaurin Series and Laurent Series

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4.19.3 $$\tan z=z+\frac{z^3}{3}+\frac{2}{15}{z}^5+\frac{17}{315}{z}^7+\mathrm{\cdots }+\frac{{(-1)}^{n-1}{2}^{2n}(2^{2n}-1)B_{2n}}{(2n)!}{z}^{2n-1}+\mathrm{\cdots },$$ ,

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Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\tan z$: tangent function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.67

Referenced by:

§4.19

Permalink:

http://dlmf.nist.gov/4.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.19 and Ch.4

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4.19.9 $$\ln \left(\frac{\tan z}{z}\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^{n-1}{2}^{2n}(2^{2n-1}-1)B_{2n}}{n(2n)!}{z}^{2n},$$ .

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Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $\tan z$: tangent function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.73

Referenced by:

§4.19, §4.33

Permalink:

http://dlmf.nist.gov/4.19.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §4.19 and Ch.4

12: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series

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22.12.4 $$2\mathrm{i}K\mathrm{dn}(2Kt,k)=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\frac{\pi }{\tan \left(\pi (t-(n+\frac{1}{2})\tau )\right)}=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\left(\underset{M\to \mathrm{\infty }}{lim}\sum _{m=-M}^M\frac{1}{t-m-(n+\frac{1}{2})\tau }\right).$$

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Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $\tan z$: tangent function, $k$: modulus and $\tau$: ratio

Referenced by:

§22.12, §22.12

Permalink:

http://dlmf.nist.gov/22.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §22.12 and Ch.22

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22.12.7 $$2\mathrm{i}K{k}^{\prime }\mathrm{nd}(2Kt,k)=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\frac{\pi }{\tan \left(\pi (t+\frac{1}{2}-(n+\frac{1}{2})\tau )\right)}=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\underset{M\to \mathrm{\infty }}{lim}\left(\sum _{m=-M}^M\frac{1}{t+\frac{1}{2}-m-(n+\frac{1}{2})\tau }\right),$$

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Symbols:

$\mathrm{nd}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $\tan z$: tangent function, $k$: modulus, $k^{\prime }$: complementary modulus and $\tau$: ratio

Permalink:

http://dlmf.nist.gov/22.12.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §22.12 and Ch.22

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22.12.10 $$-2K{k}^{\prime }\mathrm{sc}(2Kt,k)=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\frac{\pi }{\tan \left(\pi (t+\frac{1}{2}-n\tau )\right)}=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\left(\underset{M\to \mathrm{\infty }}{lim}\sum _{m=-M}^M\frac{1}{t+\frac{1}{2}-m-n\tau }\right),$$

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Symbols:

$\mathrm{sc}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\tan z$: tangent function, $k$: modulus, $k^{\prime }$: complementary modulus and $\tau$: ratio

Permalink:

http://dlmf.nist.gov/22.12.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §22.12 and Ch.22

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22.12.13 $$2K\mathrm{cs}(2Kt,k)=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\frac{\pi }{\tan \left(\pi (t-n\tau )\right)}=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\left(\underset{M\to \mathrm{\infty }}{lim}\sum _{m=-M}^M\frac{1}{t-m-n\tau }\right).$$

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Symbols:

$\mathrm{cs}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\tan z$: tangent function, $k$: modulus and $\tau$: ratio

Referenced by:

§22.12

Permalink:

http://dlmf.nist.gov/22.12.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §22.12 and Ch.22

13: 4.1 Special Notation

… ►The main functions treated in this chapter are the logarithm $\ln z$, $\mathrm{Ln}z$; the exponential $\exp z$, ${\mathrm{e}}^z$; the circular trigonometric (or just trigonometric) functions $\sin z$, $\cos z$, $\tan z$, $\csc z$, $\sec z$, $\cot z$; the inverse trigonometric functions $\arcsin z$, $\mathrm{Arcsin}z$, etc. …

14: 4.42 Solution of Triangles

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4.42.3 $$\tan A=\frac{a}{b}=\frac{1}{\cot A}.$$

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Symbols:

$\cot z$: cotangent function, $\tan z$: tangent function, $A$: angle, $a$: height and $b$: base

A&S Ref:

4.3.147

Permalink:

http://dlmf.nist.gov/4.42.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.42(i), §4.42 and Ch.4

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15: 4.28 Definitions and Periodicity

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4.28.10 $$\tan \left(\mathrm{i}z\right)=\mathrm{i}\tanh z,$$

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Symbols:

$\tanh z$: hyperbolic tangent function, $\mathrm{i}$: imaginary unit, $\tan z$: tangent function and $z$: complex variable

A&S Ref:

4.5.9

Referenced by:

§22.10(ii)

Permalink:

http://dlmf.nist.gov/4.28.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §4.28, §4.28 and Ch.4

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16: 12.13 Sums

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12.13.5 $$U(a,x\cos t+y\sin t)={\mathrm{e}}^{\frac{1}{4}{(x\sin t-y\cos t)}^2}\sum _{m=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0.0pt}{}{-a-\frac{1}{2}}{m}\right){(\tan t)}^{m}U(m+a,x)U(-m-\frac{1}{2},y),$$ $\mathrm{\Re }a\le -\frac{1}{2},0\le t\le \frac{1}{4}\pi$.

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, $\mathrm{\Re }$: real part, $\sin z$: sine function, $\tan z$: tangent function, $x$: real variable, $y$: real variable and $a$: real or complex parameter

Referenced by:

§12.13(i)

Permalink:

http://dlmf.nist.gov/12.13.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §12.13(i), §12.13 and Ch.12

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17: Guide to Searching the DLMF

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Table 1: Query Examples

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Query	Matching records contain
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trigonometric	the word ”trigonometric” or any of the various trigonometric functions such as $\sin$, $\cos$, $\tan$, and $\cot$.
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Table 3: A sample of recognized symbols

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Symbols	Comments
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All elementary functions	Such as sin, cos, tan, Ln, log, exp
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18: 4.18 Inequalities

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4.18.2 $$x\le \tan x,$$ ,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\tan z$: tangent function and $x$: real variable

A&S Ref:

4.3.80

Referenced by:

§4.18

Permalink:

http://dlmf.nist.gov/4.18.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.18, §4.18 and Ch.4

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19: 5.5 Functional Relations

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5.5.4 $$\psi \left(z\right)-\psi \left(1-z\right)=-\pi /\tan \left(\pi z\right),$$ $z\ne 0,\pm 1,\mathrm{\dots }$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $\tan z$: tangent function and $z$: complex variable

A&S Ref:

6.3.7 (without the condition on $z$.)

Permalink:

http://dlmf.nist.gov/5.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §5.5(ii), §5.5 and Ch.5

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20: 4.26 Integrals

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4.26.3 $$\int \tan xdx=-\ln \left(\cos x\right),$$ .

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, $\tan z$: tangent function and $x$: real variable

A&S Ref:

4.3.115

Permalink:

http://dlmf.nist.gov/4.26.E3

Encodings:

pMML, png, TeX

See also:

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

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4.26.4 $$\int \csc xdx=\ln \left(\tan \frac{1}{2}x\right),$$ .

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, $\tan z$: tangent function and $x$: real variable

A&S Ref:

4.3.116

Permalink:

http://dlmf.nist.gov/4.26.E4

Encodings:

pMML, png, TeX

See also:

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

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