§4.19 Maclaurin Series and Laurent Series

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

4.19.1 $\mathop{\sin\/}\nolimits z=z-\frac{z^{3}}{3!}+\frac{z^{5}}{5!}-\frac{z^{7}}{7!}+\cdots,$

Symbols:: $!$ : $n!$ : factorial, $\mathop{\sin\/}\nolimits z$ : 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

4.19.2 $\mathop{\cos\/}\nolimits z=1-\frac{z^{2}}{2!}+\frac{z^{4}}{4!}-\frac{z^{6}}{6!}+\cdots.$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $!$ : $n!$ : factorial 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

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

4.19.3 $\mathop{\tan\/}\nolimits z=z+\frac{z^{3}}{3}+\frac{2}{15}z^{5}+\frac{17}{315}z^{7}+\cdots+\frac{(-1)^{{n-1}}2^{{2n}}(2^{{2n}}-1)\mathop{B_{{2n}}\/}\nolimits}{(2n)!}z^{{2n-1}}+\cdots,$ $|z|<\frac{1}{2}\pi$ ,

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $!$ : $n!$ : factorial, $\mathop{\tan\/}\nolimits z$ : 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

4.19.4 $\mathop{\csc\/}\nolimits z=\frac{1}{z}+\frac{z}{6}+\frac{7}{360}z^{3}+\frac{31}{15120}z^{5}+\cdots+\frac{(-1)^{{n-1}}2(2^{{2n-1}}-1)\mathop{B_{{2n}}\/}\nolimits}{(2n)!}z^{{2n-1}}+\cdots,$ $0<|z|<\pi$ ,

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{\csc\/}\nolimits z$ : cosecant function, $!$ : $n!$ : factorial, $n$ : 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

4.19.5 $\mathop{\sec\/}\nolimits z=1+\frac{z^{2}}{2}+\frac{5}{24}z^{4}+\frac{61}{720}z^{6}+\cdots+\frac{(-1)^{n}\mathop{E_{{2n}}\/}\nolimits}{(2n)!}z^{{2n}}+\cdots,$ $|z|<\frac{1}{2}\pi$ ,

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, $!$ : $n!$ : factorial, $\mathop{\sec\/}\nolimits z$ : 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

4.19.6 $\mathop{\cot\/}\nolimits z=\frac{1}{z}-\frac{z}{3}-\frac{z^{3}}{45}-\frac{2}{945}z^{5}-\cdots-\frac{(-1)^{{n-1}}2^{{2n}}\mathop{B_{{2n}}\/}\nolimits}{(2n)!}z^{{2n-1}}-\cdots,$ $0<|z|<\pi$ ,

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{\cot\/}\nolimits z$ : cotangent function, $!$ : $n!$ : factorial, $n$ : integer and $z$ : complex variable
A&S Ref:: 4.3.70
Permalink:: http://dlmf.nist.gov/4.19.E6
Encodings:: TeX, pMML, png

4.19.7 $\mathop{\ln\/}\nolimits\!\left(\frac{\mathop{\sin\/}\nolimits z}{z}\right)=\sum _{{n=1}}^{\infty}\frac{(-1)^{n}2^{{2n-1}}\mathop{B_{{2n}}\/}\nolimits}{n(2n)!}z^{{2n}},$ $|z|<\pi$ ,

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $!$ : $n!$ : factorial, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{\sin\/}\nolimits z$ : 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

4.19.8 $\mathop{\ln\/}\nolimits\!\left(\mathop{\cos\/}\nolimits z\right)=\sum _{{n=1}}^{\infty}\frac{(-1)^{n}2^{{2n-1}}(2^{{2n}}-1)\mathop{B_{{2n}}\/}\nolimits}{n(2n)!}z^{{2n}},$ $|z|<\frac{1}{2}\pi$ ,

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{\cos\/}\nolimits z$ : cosine function, $!$ : $n!$ : factorial, $\mathop{\ln\/}\nolimits z$ : 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

4.19.9 $\mathop{\ln\/}\nolimits\!\left(\frac{\mathop{\tan\/}\nolimits z}{z}\right)=\sum _{{n=1}}^{\infty}\frac{(-1)^{{n-1}}2^{{2n}}(2^{{2n-1}}-1)\mathop{B_{{2n}}\/}\nolimits}{n(2n)!}z^{{2n}},$ $|z|<\frac{1}{2}\pi$ .

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $!$ : $n!$ : factorial, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{\tan\/}\nolimits z$ : 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