§4.19 Maclaurin Series and Laurent Series

4.19.1		$$\sin z=z-\frac{z^3}{3!}+\frac{z^5}{5!}-\frac{z^7}{7!}+\mathrm{\cdots },$$
ⓘ Symbols: $!$: factorial (as in $n!$), $\sin z$: 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		$$\cos z=1-\frac{z^2}{2!}+\frac{z^4}{4!}-\frac{z^6}{6!}+\mathrm{\cdots }.$$
ⓘ Symbols: $\cos z$: 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		$$\tan z=z+\frac{z^3}{3}+\frac{2}{15}{z}^5+\frac{17}{315}{z}^7+\mathrm{\cdots }+\frac{{(-1)}^{n-1}{2}^{2n}(2^{2n}-1)B_{2n}}{(2n)!}{z}^{2n-1}+\mathrm{\cdots },$$
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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: TeX, pMML, png See also: Annotations for §4.19 and Ch.4

4.19.4		$$\csc z=\frac{1}{z}+\frac{z}{6}+\frac{7}{360}{z}^3+\frac{31}{15120}{z}^5+\mathrm{\cdots }+\frac{{(-1)}^{n-1}2(2^{2n-1}-1)B_{2n}}{(2n)!}{z}^{2n-1}+\mathrm{\cdots },$$
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ⓘ Symbols: $B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: 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		$$\sec z=1+\frac{z^2}{2}+\frac{5}{24}{z}^4+\frac{61}{720}{z}^6+\mathrm{\cdots }+\frac{{(-1)}^n{E}_{2n}}{(2n)!}{z}^{2n}+\mathrm{\cdots },$$
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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: TeX, pMML, png See also: Annotations for §4.19 and Ch.4

4.19.6		$$\cot z=\frac{1}{z}-\frac{z}{3}-\frac{z^3}{45}-\frac{2}{945}{z}^5-\mathrm{\cdots }-\frac{{(-1)}^{n-1}{2}^{2n}{B}_{2n}}{(2n)!}{z}^{2n-1}-\mathrm{\cdots },$$
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ⓘ Symbols: $B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: 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		$$\ln \left(\frac{\sin z}{z}\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n{2}^{2n-1}{B}_{2n}}{n(2n)!}{z}^{2n},$$
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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, $\sin z$: 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		$$\ln \left(\cos z\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n{2}^{2n-1}(2^{2n}-1)B_{2n}}{n(2n)!}{z}^{2n},$$
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ⓘ Symbols: $B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $!$: factorial (as in $n!$), $\ln z$: 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		$$\ln \left(\frac{\tan z}{z}\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^{n-1}{2}^{2n}(2^{2n-1}-1)B_{2n}}{n(2n)!}{z}^{2n},$$
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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: TeX, pMML, png See also: Annotations for §4.19 and Ch.4