4 Elementary Functions Logarithm, Exponential, Powers 4.5 Inequalities 4.7 Derivatives and Differential Equations

§4.6 Power Series

Permalink:: http://dlmf.nist.gov/4.6

§4.6(i) Logarithms
§4.6(ii) Powers

§4.6(i) Logarithms

Notes:: See Hardy (1952, pp. 471–473) for (4.6.1). The other equations are variations of this.
Keywords:: logarithm function, power series
Permalink:: http://dlmf.nist.gov/4.6.i

4.6.1 $\mathop{\ln\/}\nolimits\!\left(1+z\right)=z-\tfrac{1}{2}z^{2}+\tfrac{1}{3}z^{3% }-\cdots,$ $|z|\leq 1$ , $z\neq-1$ ,

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and : complex variable
A&S Ref:: 4.1.24
Referenced by:: ¶ ‣ §4.45(i), §4.6(i)
Permalink:: http://dlmf.nist.gov/4.6.E1
Encodings:: TeX, pMML, png

4.6.2 $\mathop{\ln\/}\nolimits z=\left(\frac{z-1}{z}\right)+\frac{1}{2}\left(\frac{z-% 1}{z}\right)^{2}+\frac{1}{3}\left(\frac{z-1}{z}\right)^{3}+\cdots,$ $\realpart{z}\geq\frac{1}{2}$ ,

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\realpart{}$ : real part and : complex variable
A&S Ref:: 4.1.25
Permalink:: http://dlmf.nist.gov/4.6.E2
Encodings:: TeX, pMML, png

4.6.3 $\mathop{\ln\/}\nolimits z=(z-1)-\tfrac{1}{2}(z-1)^{2}+\tfrac{1}{3}(z-1)^{3}-\cdots,$ $|z-1|\leq 1$ , $z\neq 0$ ,

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and : complex variable
A&S Ref:: 4.1.26
Permalink:: http://dlmf.nist.gov/4.6.E3
Encodings:: TeX, pMML, png

4.6.4 $\mathop{\ln\/}\nolimits z=2\left(\left(\frac{z-1}{z+1}\right)+\frac{1}{3}\left% (\frac{z-1}{z+1}\right)^{3}+\frac{1}{5}\left(\frac{z-1}{z+1}\right)^{5}+\cdots% \right),$ $\realpart{z}\geq 0$ , $z\neq 0$ ,

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\realpart{}$ : real part and : complex variable
A&S Ref:: 4.1.27
Permalink:: http://dlmf.nist.gov/4.6.E4
Encodings:: TeX, pMML, png

4.6.5 $\mathop{\ln\/}\nolimits\!\left(\frac{z+1}{z-1}\right)=2\left(\frac{1}{z}+\frac% {1}{3z^{3}}+\frac{1}{5z^{5}}+\cdots\right),$ $|z|\geq 1$ , $z\neq\pm 1$ ,

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and : complex variable
A&S Ref:: 4.1.28
Permalink:: http://dlmf.nist.gov/4.6.E5
Encodings:: TeX, pMML, png

4.6.6 $\mathop{\ln\/}\nolimits\!\left(z+a\right)=\mathop{\ln\/}\nolimits a+2\left(% \left(\frac{z}{2a+z}\right)+\frac{1}{3}\left(\frac{z}{2a+z}\right)^{3}+\frac{1% }{5}\left(\frac{z}{2a+z}\right)^{5}+\cdots\right),$