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4 Elementary Functions Trigonometric Functions 4.18 Inequalities 4.20 Derivatives and Differential Equations

§4.19 Maclaurin Series and Laurent Series

ⓘ

Keywords:: Laurent series, Maclaurin series, trigonometric functions
Notes:: See Hobson (1928, pp. 288–293, 360–367).
Referenced by:: §24.2(i)
Permalink:: http://dlmf.nist.gov/4.19
See also:: Annotations for Ch.4

4.19.1		$\sin z=z-\frac{z^{3}}{3!}+\frac{z^{5}}{5!}-\frac{z^{7}}{7!}+\cdots,$
ⓘ Symbols: $!$ : factorial (as in $n!$ ), $\sin\NVar{z}$ : sine function and $z$ : complex variable A&S Ref: 4.3.65 Referenced by: §4.45(i) Permalink: http://dlmf.nist.gov/4.19.E1 Encodings: TeX, pMML, png See also: Annotations for §4.19 and Ch.4

4.19.2		$\cos z=1-\frac{z^{2}}{2!}+\frac{z^{4}}{4!}-\frac{z^{6}}{6!}+\cdots.$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $!$ : factorial (as in $n!$ ) and $z$ : complex variable A&S Ref: 4.3.66 Referenced by: §4.45(i) Permalink: http://dlmf.nist.gov/4.19.E2 Encodings: TeX, pMML, png See also: Annotations for §4.19 and Ch.4

In (4.19.3)–(4.19.9), $B_{n}$ are the Bernoulli numbers and $E_{n}$ are the Euler numbers (§§24.2(i)–24.2(ii)).

4.19.3		$\tan z=z+\frac{z^{3}}{3}+\frac{2}{15}z^{5}+\frac{17}{315}z^{7}+\cdots+\frac{(-% 1)^{n-1}2^{2n}(2^{2n}-1)B_{2n}}{(2n)!}z^{2n-1}+\cdots,$
$\|z\|<\frac{1}{2}\pi$ ,
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $\tan\NVar{z}$ : 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: TeX, pMML, png See also: Annotations for §4.19 and Ch.4

4.19.4		$\csc z=\frac{1}{z}+\frac{z}{6}+\frac{7}{360}z^{3}+\frac{31}{15120}z^{5}+\cdots% +\frac{(-1)^{n-1}2(2^{2n-1}-1)B_{2n}}{(2n)!}z^{2n-1}+\cdots,$
$0<\|z\|<\pi$ ,
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $\csc\NVar{z}$ : cosecant function, $!$ : factorial (as in $n!$ ), $n$ : integer and $z$ : complex variable A&S Ref: 4.3.68 Referenced by: §4.33 Permalink: http://dlmf.nist.gov/4.19.E4 Encodings: TeX, pMML, png See also: Annotations for §4.19 and Ch.4

4.19.5		$\sec z=1+\frac{z^{2}}{2}+\frac{5}{24}z^{4}+\frac{61}{720}z^{6}+\cdots+\frac{(-% 1)^{n}E_{2n}}{(2n)!}z^{2n}+\cdots,$
$\|z\|<\frac{1}{2}\pi$ ,
ⓘ Symbols: $E_{\NVar{n}}$ : Euler numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $\sec\NVar{z}$ : 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: TeX, pMML, png See also: Annotations for §4.19 and Ch.4

4.19.6		$\cot z=\frac{1}{z}-\frac{z}{3}-\frac{z^{3}}{45}-\frac{2}{945}z^{5}-\cdots-% \frac{(-1)^{n-1}2^{2n}B_{2n}}{(2n)!}z^{2n-1}-\cdots,$
$0<\|z\|<\pi$ ,
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cot\NVar{z}$ : cotangent function, $!$ : factorial (as in $n!$ ), $n$ : integer and $z$ : complex variable A&S Ref: 4.3.70 Permalink: http://dlmf.nist.gov/4.19.E6 Encodings: TeX, pMML, png See also: Annotations for §4.19 and Ch.4

4.19.7		$\ln\left(\frac{\sin z}{z}\right)=\sum_{n=1}^{\infty}\frac{(-1)^{n}2^{2n-1}B_{2% n}}{n(2n)!}z^{2n},$
$\|z\|<\pi$ ,
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function, $\sin\NVar{z}$ : sine function, $n$ : integer and $z$ : complex variable A&S Ref: 4.3.71 Permalink: http://dlmf.nist.gov/4.19.E7 Encodings: TeX, pMML, png See also: Annotations for §4.19 and Ch.4

4.19.8		$\ln\left(\cos z\right)=\sum_{n=1}^{\infty}\frac{(-1)^{n}2^{2n-1}(2^{2n}-1)B_{2% n}}{n(2n)!}z^{2n},$
$\|z\|<\frac{1}{2}\pi$ ,
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : integer and $z$ : complex variable A&S Ref: 4.3.72 Permalink: http://dlmf.nist.gov/4.19.E8 Encodings: TeX, pMML, png See also: Annotations for §4.19 and Ch.4

4.19.9		$\ln\left(\frac{\tan z}{z}\right)=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}2^{2n}(2^{% 2n-1}-1)B_{2n}}{n(2n)!}z^{2n},$
$\|z\|<\frac{1}{2}\pi$ .
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function, $\tan\NVar{z}$ : 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: TeX, pMML, png See also: Annotations for §4.19 and Ch.4

4.18 Inequalities 4.20 Derivatives and Differential Equations

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