§4.2 Definitions

§4.2(i) The Logarithm

The general logarithm function $\operatorname{Ln}z$ is defined by

 4.2.1 $\operatorname{Ln}z=\int_{1}^{z}\frac{\mathrm{d}t}{t},$ $z\neq 0$, ⓘ Defines: $\operatorname{Ln}\NVar{z}$: general logarithm function Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $z$: complex variable A&S Ref: 4.1.4 Referenced by: §4.2(i) Permalink: http://dlmf.nist.gov/4.2.E1 Encodings: TeX, pMML, png See also: Annotations for §4.2(i), §4.2 and Ch.4

where the integration path does not intersect the origin. This is a multivalued function of $z$ with branch point at $z=0$.

The principal value, or principal branch, is defined by

 4.2.2 $\ln z=\int_{1}^{z}\frac{\mathrm{d}t}{t},$ ⓘ Defines: $\ln\NVar{z}$: principal branch of logarithm function Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $z$: complex variable A&S Ref: 4.1.1 Permalink: http://dlmf.nist.gov/4.2.E2 Encodings: TeX, pMML, png See also: Annotations for §4.2(i), §4.2 and Ch.4

where the path does not intersect $(-\infty,0]$; see Figure 4.2.1. $\ln z$ is a single-valued analytic function on $\mathbb{C}\setminus(-\infty,0]$ and real-valued when $z$ ranges over the positive real numbers. Figure 4.2.1: z-plane: Branch cut for ln⁡z and zα. Magnify

The real and imaginary parts of $\ln z$ are given by

 4.2.3 $\ln z=\ln\left|z\right|+\mathrm{i}\operatorname{ph}z,$ $-\pi<\operatorname{ph}z<\pi$. ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$: principal branch of logarithm function, $\operatorname{ph}$: phase and $z$: complex variable A&S Ref: 4.1.2 Referenced by: §4.2(i) Permalink: http://dlmf.nist.gov/4.2.E3 Encodings: TeX, pMML, png See also: Annotations for §4.2(i), §4.2 and Ch.4

For $\operatorname{ph}z$ see §1.9(i).

The only zero of $\ln z$ is at $z=1$.

Most texts extend the definition of the principal value to include the branch cut

 4.2.4 $z=x,$ $-\infty, ⓘ Symbols: $x$: real variable and $z$: complex variable Permalink: http://dlmf.nist.gov/4.2.E4 Encodings: TeX, pMML, png See also: Annotations for §4.2(i), §4.2 and Ch.4

by replacing (4.2.3) with

 4.2.5 $\ln z=\ln\left|z\right|+\mathrm{i}\operatorname{ph}z,$ $-\pi<\operatorname{ph}z\leq\pi$. ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$: principal branch of logarithm function, $\operatorname{ph}$: phase and $z$: complex variable Referenced by: §4.2(i) Permalink: http://dlmf.nist.gov/4.2.E5 Encodings: TeX, pMML, png See also: Annotations for §4.2(i), §4.2 and Ch.4

With this definition the general logarithm is given by

 4.2.6 $\operatorname{Ln}z=\ln z+2k\pi\mathrm{i},$

where $k$ is the excess of the number of times the path in (4.2.1) crosses the negative real axis in the positive sense over the number of times in the negative sense.

In the DLMF we allow a further extension by regarding the cut as representing two sets of points, one set corresponding to the “upper side” and denoted by $z=x+\mathrm{i}0$, the other set corresponding to the “lower side” and denoted by $z=x-\mathrm{i}0$. Again see Figure 4.2.1. Then

 4.2.7 $\ln\left(x\pm\mathrm{i}0\right)=\ln|x|\pm i\pi,$ $-\infty, ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$: principal branch of logarithm function and $x$: real variable Permalink: http://dlmf.nist.gov/4.2.E7 Encodings: TeX, pMML, png See also: Annotations for §4.2(i), §4.2 and Ch.4

with either upper signs or lower signs taken throughout. Consequently $\ln z$ is two-valued on the cut, and discontinuous across the cut. We regard this as the closed definition of the principal value.

In contrast to (4.2.5) the closed definition is symmetric. As a consequence, it has the advantage of extending regions of validity of properties of principal values. For example, with the definition (4.2.5) the identity (4.8.7) is valid only when $\left|\operatorname{ph}z\right|<\pi$, but with the closed definition the identity (4.8.7) is valid when $\left|\operatorname{ph}z\right|\leq\pi$. For another example see (4.2.37).

In the DLMF it is usually clear from the context which definition of principal value is being used. However, in the absence of any indication to the contrary it is assumed that the definition is the closed one. For other examples in this chapter see §§4.23, 4.24, 4.37, and 4.38.

§4.2(ii) Logarithms to a General Base $a$

With $a,b\neq 0$ or $1$,

 4.2.8 $\displaystyle\operatorname{log}_{a}z$ $\displaystyle=\ifrac{\ln z}{\ln a},$ ⓘ Symbols: $\operatorname{log}_{\NVar{a}}\NVar{z}$: logarithm to general base $a$, $\ln\NVar{z}$: principal branch of logarithm function, $a$: real or complex constant and $z$: complex variable A&S Ref: 4.1.18 Permalink: http://dlmf.nist.gov/4.2.E8 Encodings: TeX, pMML, png See also: Annotations for §4.2(ii), §4.2 and Ch.4 4.2.9 $\displaystyle\operatorname{log}_{a}z$ $\displaystyle=\frac{\operatorname{log}_{b}z}{\operatorname{log}_{b}a},$ ⓘ Symbols: $\operatorname{log}_{\NVar{a}}\NVar{z}$: logarithm to general base $a$, $a$: real or complex constant and $z$: complex variable A&S Ref: 4.1.19 Permalink: http://dlmf.nist.gov/4.2.E9 Encodings: TeX, pMML, png See also: Annotations for §4.2(ii), §4.2 and Ch.4 4.2.10 $\displaystyle\operatorname{log}_{a}b$ $\displaystyle=\frac{1}{\operatorname{log}_{b}a}.$ ⓘ Symbols: $\operatorname{log}_{\NVar{a}}\NVar{z}$: logarithm to general base $a$ and $a$: real or complex constant A&S Ref: 4.1.20 Permalink: http://dlmf.nist.gov/4.2.E10 Encodings: TeX, pMML, png See also: Annotations for §4.2(ii), §4.2 and Ch.4

Natural logarithms have as base the unique positive number

 4.2.11 $e=2.71828\ 18284\ 59045\ 23536\dots$ ⓘ Defines: $\mathrm{e}$: base of natural logarithm A&S Ref: 4.2.22 (with 10D value) Notes: For more digits see OEIS Sequence A001113; see also Sloane (2003). Permalink: http://dlmf.nist.gov/4.2.E11 Encodings: TeX, pMML, png See also: Annotations for §4.2(ii), §4.2 and Ch.4

such that

 4.2.12 $\ln e=1.$ ⓘ Symbols: $\mathrm{e}$: base of natural logarithm and $\ln\NVar{z}$: principal branch of logarithm function Permalink: http://dlmf.nist.gov/4.2.E12 Encodings: TeX, pMML, png See also: Annotations for §4.2(ii), §4.2 and Ch.4

Equivalently,

 4.2.13 $\int_{1}^{e}\frac{\mathrm{d}t}{t}=1.$ ⓘ Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\mathrm{e}$: base of natural logarithm and $\int$: integral Permalink: http://dlmf.nist.gov/4.2.E13 Encodings: TeX, pMML, png See also: Annotations for §4.2(ii), §4.2 and Ch.4

Thus

 4.2.14 $\operatorname{log}_{e}z=\ln z,$ ⓘ
 4.2.15 $\operatorname{log}_{10}z=\ifrac{(\ln z)}{(\ln 10)}=(\operatorname{log}_{10}e)% \ln z,$ ⓘ
 4.2.16 $\ln z=(\ln 10)\operatorname{log}_{10}z,$ ⓘ Symbols: $\operatorname{log}_{\NVar{a}}\NVar{z}$: logarithm to general base $a$, $\ln\NVar{z}$: principal branch of logarithm function and $z$: complex variable A&S Ref: 4.1.23 Permalink: http://dlmf.nist.gov/4.2.E16 Encodings: TeX, pMML, png See also: Annotations for §4.2(ii), §4.2 and Ch.4
 4.2.17 $\operatorname{log}_{10}e=0.43429\ 44819\ 03251\ 82765\dots,$ ⓘ Symbols: $\mathrm{e}$: base of natural logarithm and $\operatorname{log}_{\NVar{a}}\NVar{z}$: logarithm to general base $a$ A&S Ref: 4.1.22 (with 10D value) Notes: For more digits see OEIS Sequence A002285; see also Sloane (2003). Permalink: http://dlmf.nist.gov/4.2.E17 Encodings: TeX, pMML, png See also: Annotations for §4.2(ii), §4.2 and Ch.4
 4.2.18 $\ln 10=2.30258\ 50929\ 94045\ 68401\dots.$ ⓘ Symbols: $\ln\NVar{z}$: principal branch of logarithm function A&S Ref: 4.1.23 (with 10D value.) Notes: For more digits see OEIS Sequence A002392; see also Sloane (2003). Permalink: http://dlmf.nist.gov/4.2.E18 Encodings: TeX, pMML, png See also: Annotations for §4.2(ii), §4.2 and Ch.4

$\operatorname{log}_{e}x=\ln x$ is also called the Napierian or hyperbolic logarithm. $\operatorname{log}_{10}x$ is the common or Briggs logarithm.

§4.2(iii) The Exponential Function

 4.2.19 $\exp z=1+\frac{z}{1!}+\frac{z^{2}}{2!}+\frac{z^{3}}{3!}+\cdots.$ ⓘ Defines: $\exp\NVar{z}$: exponential function Symbols: $!$: factorial (as in $n!$) and $z$: complex variable A&S Ref: 4.2.1 Referenced by: §4.45(i) Permalink: http://dlmf.nist.gov/4.2.E19 Encodings: TeX, pMML, png See also: Annotations for §4.2(iii), §4.2 and Ch.4

The function $\exp$ is an entire function of $z$, with no real or complex zeros. It has period $2\pi i$:

 4.2.20 $\exp\left(z+2\pi i\right)=\exp z.$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$: exponential function and $z$: complex variable Permalink: http://dlmf.nist.gov/4.2.E20 Encodings: TeX, pMML, png See also: Annotations for §4.2(iii), §4.2 and Ch.4

Also,

 4.2.21 $\exp\left(-z\right)=1/\exp\left(z\right).$ ⓘ Symbols: $\exp\NVar{z}$: exponential function and $z$: complex variable Permalink: http://dlmf.nist.gov/4.2.E21 Encodings: TeX, pMML, png See also: Annotations for §4.2(iii), §4.2 and Ch.4
 4.2.22 $|\exp z|=\exp\left(\Re z\right).$ ⓘ Symbols: $\exp\NVar{z}$: exponential function, $\Re$: real part and $z$: complex variable A&S Ref: 4.2.13 Permalink: http://dlmf.nist.gov/4.2.E22 Encodings: TeX, pMML, png See also: Annotations for §4.2(iii), §4.2 and Ch.4

The general value of the phase is given by

 4.2.23 $\operatorname{ph}\left(\exp z\right)=\Im z+2k\pi,$ $k\in\mathbb{Z}$.

If $z=x+iy$, then

 4.2.24 $\exp z=e^{x}\cos y+ie^{x}\sin y.$

If $\zeta\neq 0$ then

 4.2.25 $\exp z=\zeta\;\;\Longleftrightarrow\;\;z=\operatorname{Ln}\zeta.$ ⓘ Symbols: $\exp\NVar{z}$: exponential function, $\operatorname{Ln}\NVar{z}$: general logarithm function and $z$: complex variable A&S Ref: 4.2.4 Permalink: http://dlmf.nist.gov/4.2.E25 Encodings: TeX, pMML, png See also: Annotations for §4.2(iii), §4.2 and Ch.4

§4.2(iv) Powers

Powers with General Bases

The general $a^{\rm th}$ power of $z$ is defined by

 4.2.26 $z^{a}=\exp\left(a\operatorname{Ln}z\right),$ $z\neq 0$. ⓘ Symbols: $\exp\NVar{z}$: exponential function, $\operatorname{Ln}\NVar{z}$: general logarithm function, $a$: real or complex constant and $z$: complex variable Permalink: http://dlmf.nist.gov/4.2.E26 Encodings: TeX, pMML, png See also: Annotations for §4.2(iv), §4.2(iv), §4.2 and Ch.4

In particular, $z^{0}=1$, and if $a=n=1,2,3,\dots$, then

 4.2.27 $z^{a}=\underbrace{z\cdot z\cdots z}_{n\text{ times}}=1/z^{-a}.$ ⓘ Defines: $a=n$: integer (locally) Symbols: $n$: integer and $z$: complex variable Permalink: http://dlmf.nist.gov/4.2.E27 Encodings: TeX, pMML, png See also: Annotations for §4.2(iv), §4.2(iv), §4.2 and Ch.4

In all other cases, $z^{a}$ is a multivalued function with branch point at $z=0$. The principal value is

 4.2.28 $z^{a}=\exp\left(a\ln z\right).$ ⓘ Symbols: $\exp\NVar{z}$: exponential function, $\ln\NVar{z}$: principal branch of logarithm function, $a$: real or complex constant and $z$: complex variable A&S Ref: 4.2.7 Permalink: http://dlmf.nist.gov/4.2.E28 Encodings: TeX, pMML, png See also: Annotations for §4.2(iv), §4.2(iv), §4.2 and Ch.4

This is an analytic function of $z$ on $\mathbb{C}\setminus(-\infty,0]$, and is two-valued and discontinuous on the cut shown in Figure 4.2.1, unless $a\in\mathbb{Z}$.

 4.2.29 $|z^{a}|=|z|^{\Re a}\exp\left(-(\Im a)\operatorname{ph}z\right),$ ⓘ Symbols: $\exp\NVar{z}$: exponential function, $\Im$: imaginary part, $\operatorname{ph}$: phase, $\Re$: real part, $a$: real or complex constant and $z$: complex variable A&S Ref: 4.2.9 Permalink: http://dlmf.nist.gov/4.2.E29 Encodings: TeX, pMML, png See also: Annotations for §4.2(iv), §4.2(iv), §4.2 and Ch.4
 4.2.30 $\operatorname{ph}\left(z^{a}\right)=(\Re a)\operatorname{ph}z+(\Im a)\ln|z|,$

where $\operatorname{ph}z\in[-\pi,\pi]$ for the principal value of $z^{a}$, and is unrestricted in the general case. When $a$ is real

 4.2.31 $\displaystyle|z^{a}|$ $\displaystyle=|z|^{a},$ $\displaystyle\operatorname{ph}\left(z^{a}\right)$ $\displaystyle=a\operatorname{ph}z.$ ⓘ Symbols: $\operatorname{ph}$: phase, $a$: real or complex constant and $z$: complex variable Permalink: http://dlmf.nist.gov/4.2.E31 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §4.2(iv), §4.2(iv), §4.2 and Ch.4

Unless indicated otherwise, it is assumed throughout the DLMF that a power assumes its principal value. With this convention,

 4.2.32 $e^{z}=\exp z,$ ⓘ Symbols: $\exp\NVar{z}$: exponential function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable Permalink: http://dlmf.nist.gov/4.2.E32 Encodings: TeX, pMML, png See also: Annotations for §4.2(iv), §4.2(iv), §4.2 and Ch.4

but the general value of $e^{z}$ is

 4.2.33 $e^{z}=(\exp z)\exp\left(2kz\pi\mathrm{i}\right),$ $k\in\mathbb{Z}$.

For $z=1$

 4.2.34 $e=1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\cdots.$ ⓘ Symbols: $\mathrm{e}$: base of natural logarithm and $!$: factorial (as in $n!$) Permalink: http://dlmf.nist.gov/4.2.E34 Encodings: TeX, pMML, png See also: Annotations for §4.2(iv), §4.2(iv), §4.2 and Ch.4

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

 4.2.35 $z^{a}=w\;\;\Longleftrightarrow\;\;z=\exp\left(\frac{1}{a}\operatorname{Ln}w% \right).$ ⓘ Defines: $a\neq 0$: complex constant (locally) Symbols: $\exp\NVar{z}$: exponential function, $\operatorname{Ln}\NVar{z}$: general logarithm function and $z$: complex variable Permalink: http://dlmf.nist.gov/4.2.E35 Encodings: TeX, pMML, png See also: Annotations for §4.2(iv), §4.2(iv), §4.2 and Ch.4

This result is also valid when $z^{a}$ has its principal value, provided that the branch of $\operatorname{Ln}w$ satisfies

 4.2.36 $-\pi\leq\Im\left(\frac{1}{a}\operatorname{Ln}w\right)\leq\pi.$

Another example of a principal value is provided by

 4.2.37 $\sqrt{z^{2}}=\begin{cases}z,&\Re z\geq 0,\\ -z,&\Re z\leq 0.\end{cases}$ ⓘ Symbols: $\Re$: real part and $z$: complex variable Referenced by: §4.2(i) Permalink: http://dlmf.nist.gov/4.2.E37 Encodings: TeX, pMML, png See also: Annotations for §4.2(iv), §4.2(iv), §4.2 and Ch.4

Again, without the closed definition the $\geq$ and $\leq$ signs would have to be replaced by $>$ and $<$, respectively.