§4.8 Identities

Permalink:: http://dlmf.nist.gov/4.8

§4.8(i) Logarithms
§4.8(ii) Powers

§4.8(i) Logarithms

Notes:: See Levinson and Redheffer (1970, pp. 62–66).
Keywords:: logarithm function
Permalink:: http://dlmf.nist.gov/4.8.i

In (4.8.1)–(4.8.4) $z_{1}z_{2}\neq 0$ .

4.8.1 $\mathop{\mathrm{Ln}\/}\nolimits\!\left(z_{1}z_{2}\right)=\mathop{\mathrm{Ln}\/}\nolimits z_{1}+\mathop{\mathrm{Ln}\/}\nolimits z_{2}.$

Symbols:: $\mathop{\mathrm{Ln}\/}\nolimits z$ : general logarithm function and $z$ : complex variable
A&S Ref:: 4.1.6
Referenced by:: §4.8(i)
Permalink:: http://dlmf.nist.gov/4.8.E1
Encodings:: TeX, pMML, png

This is interpreted that every value of $\mathop{\mathrm{Ln}\/}\nolimits\!\left(z_{1}z_{2}\right)$ is one of the values of $\mathop{\mathrm{Ln}\/}\nolimits z_{1}+\mathop{\mathrm{Ln}\/}\nolimits z_{2}$ , and vice versa.

4.8.2 $\mathop{\ln\/}\nolimits\!\left(z_{1}z_{2}\right)=\mathop{\ln\/}\nolimits z_{1}+\mathop{\ln\/}\nolimits z_{2},$ $-\pi\leq\mathop{\mathrm{ph}\/}\nolimits z_{1}+\mathop{\mathrm{ph}\/}\nolimits z_{2}\leq\pi$ ,

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase and $z$ : complex variable
A&S Ref:: 4.1.7
Permalink:: http://dlmf.nist.gov/4.8.E2
Encodings:: TeX, pMML, png

4.8.3 $\mathop{\mathrm{Ln}\/}\nolimits\frac{z_{1}}{z_{2}}=\mathop{\mathrm{Ln}\/}\nolimits z_{1}-\mathop{\mathrm{Ln}\/}\nolimits z_{2},$

Symbols:: $\mathop{\mathrm{Ln}\/}\nolimits z$ : general logarithm function and $z$ : complex variable
A&S Ref:: 4.1.8
Permalink:: http://dlmf.nist.gov/4.8.E3
Encodings:: TeX, pMML, png

4.8.4 $\mathop{\ln\/}\nolimits\frac{z_{1}}{z_{2}}=\mathop{\ln\/}\nolimits z_{1}-\mathop{\ln\/}\nolimits z_{2},$ $-\pi\leq\mathop{\mathrm{ph}\/}\nolimits z_{1}-\mathop{\mathrm{ph}\/}\nolimits z_{2}\leq\pi$ .

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase and $z$ : complex variable
A&S Ref:: 4.1.9
Referenced by:: §4.8(i)
Permalink:: http://dlmf.nist.gov/4.8.E4
Encodings:: TeX, pMML, png

In (4.8.5)–(4.8.7) and (4.8.10) $z\neq 0$ .

4.8.5 $\mathop{\mathrm{Ln}\/}\nolimits\!\left(z^{n}\right)=n\mathop{\mathrm{Ln}\/}\nolimits z,$ $n\in\Integer$ ,

Symbols:: $\in$ : element of, $\Integer$ : set of all integers, $\mathop{\mathrm{Ln}\/}\nolimits z$ : general logarithm function, $n$ : integer and $z$ : complex variable
A&S Ref:: 4.1.10
Referenced by:: §4.8(i)
Permalink:: http://dlmf.nist.gov/4.8.E5
Encodings:: TeX, pMML, png

4.8.6 $\mathop{\ln\/}\nolimits\!\left(z^{n}\right)=n\mathop{\ln\/}\nolimits z,$ $n\in\Integer$ , $-\pi\leq n\mathop{\mathrm{ph}\/}\nolimits z\leq\pi$ ,

Symbols:: $\in$ : element of, $\Integer$ : set of all integers, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $n$ : integer and $z$ : complex variable
A&S Ref:: 4.1.11
Permalink:: http://dlmf.nist.gov/4.8.E6
Encodings:: TeX, pMML, png

4.8.7 $\mathop{\ln\/}\nolimits\frac{1}{z}=-\mathop{\ln\/}\nolimits z,$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi$ .

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase and $z$ : complex variable
Referenced by:: §4.2(i), §4.8(i)
Permalink:: http://dlmf.nist.gov/4.8.E7
Encodings:: TeX, pMML, png

4.8.8 $\mathop{\mathrm{Ln}\/}\nolimits\!\left(\mathop{\exp\/}\nolimits z\right)=z+2k\pi i,$ $k\in\Integer$ ,

Symbols:: $\in$ : element of, $\mathop{\exp\/}\nolimits z$ : exponential function, $\Integer$ : set of all integers, $\mathop{\mathrm{Ln}\/}\nolimits z$ : general logarithm function, $k$ : integer and $z$ : complex variable
A&S Ref:: 4.2.2
Permalink:: http://dlmf.nist.gov/4.8.E8
Encodings:: TeX, pMML, png

4.8.9 $\mathop{\ln\/}\nolimits\!\left(\mathop{\exp\/}\nolimits z\right)=z,$ $-\pi\leq\imagpart{z}\leq\pi$ ,

Symbols:: $\mathop{\exp\/}\nolimits z$ : exponential function, $\imagpart{}$ : imaginary part, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $z$ : complex variable
A&S Ref:: 4.2.3
Permalink:: http://dlmf.nist.gov/4.8.E9
Encodings:: TeX, pMML, png

4.8.10 $\mathop{\exp\/}\nolimits\!\left(\mathop{\ln\/}\nolimits z\right)=\mathop{\exp\/}\nolimits\!\left(\mathop{\mathrm{Ln}\/}\nolimits z\right)=z.$

Symbols:: $\mathop{\exp\/}\nolimits z$ : exponential function, $\mathop{\mathrm{Ln}\/}\nolimits z$ : general logarithm function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $z$ : complex variable
A&S Ref:: 4.2.4
Referenced by:: §4.8(i)
Permalink:: http://dlmf.nist.gov/4.8.E10
Encodings:: TeX, pMML, png

If $a\neq 0$ and $a^{z}$ has its general value, then

4.8.11 $\mathop{\mathrm{Ln}\/}\nolimits\!\left(a^{z}\right)=z\mathop{\mathrm{Ln}\/}\nolimits a+2k\pi i,$ $k\in\Integer$ .

Symbols:: $\in$ : element of, $\Integer$ : set of all integers, $\mathop{\mathrm{Ln}\/}\nolimits z$ : general logarithm function, $k$ : integer, $a$ : real or complex constant and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/4.8.E11
Encodings:: TeX, pMML, png

If $a\neq 0$ and $a^{z}$ has its principal value, then

4.8.12 $\mathop{\ln\/}\nolimits\!\left(a^{z}\right)=z\mathop{\ln\/}\nolimits a+2k\pi i,$

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $k$ : integer, $a$ : real or complex constant and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/4.8.E12
Encodings:: TeX, pMML, png

where the integer $k$ is chosen so that $\realpart{(-iz\mathop{\ln\/}\nolimits a)}+2k\pi\in[-\pi,\pi]$ .

4.8.13 $\mathop{\ln\/}\nolimits\!\left(a^{x}\right)=x\mathop{\ln\/}\nolimits a,$ $a>0$ .

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $a$ : real or complex constant and $x$ : real variable
Permalink:: http://dlmf.nist.gov/4.8.E13
Encodings:: TeX, pMML, png

§4.8(ii) Powers

Notes:: See Hobson (1928, pp. 297–299).
Keywords:: exponential function, power function
Permalink:: http://dlmf.nist.gov/4.8.ii

4.8.14 $a^{{z_{1}}}a^{{z_{2}}}=a^{{z_{1}+z_{2}}},$

Symbols:: $a$ : real or complex constant and $z$ : complex variable
A&S Ref:: 4.2.15
Permalink:: http://dlmf.nist.gov/4.8.E14
Encodings:: TeX, pMML, png

4.8.15 $a^{z}b^{z}=(ab)^{z},$ $-\pi\leq\mathop{\mathrm{ph}\/}\nolimits a+\mathop{\mathrm{ph}\/}\nolimits b\leq\pi$ ,

Symbols:: $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $a$ : real or complex constant and $z$ : complex variable
A&S Ref:: 4.2.16
Permalink:: http://dlmf.nist.gov/4.8.E15
Encodings:: TeX, pMML, png

4.8.16 $e^{{z_{1}}}e^{{z_{2}}}=e^{{z_{1}+z_{2}}},$

Symbols:: $e$ : base of exponential function and $z$ : complex variable
A&S Ref:: 4.2.18
Permalink:: http://dlmf.nist.gov/4.8.E16
Encodings:: TeX, pMML, png

4.8.17 $(e^{{z_{1}}})^{{z_{2}}}=e^{{z_{1}z_{2}}},$ $-\pi\leq\imagpart{z_{1}}\leq\pi$ .

Symbols:: $e$ : base of exponential function, $\imagpart{}$ : imaginary part and $z$ : complex variable
A&S Ref:: 4.2.19
Permalink:: http://dlmf.nist.gov/4.8.E17
Encodings:: TeX, pMML, png

The restriction on $z_{1}$ can be removed when $z_{2}$ is an integer.

§4.8 Identities

Contents

§4.8(i) Logarithms

§4.8(ii) Powers