principal branch (or value)

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1: 6.15 Sums

… ►

6.15.1 $$\sum _{n=1}^{\mathrm{\infty }}\mathrm{Ci}\left(\pi n\right)=\frac{1}{2}(\ln 2-\gamma ),$$

ⓘ

Symbols:

$\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ci}\left(z\right)$: cosine integral, $\ln z$: principal branch of logarithm function and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/6.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.15 and Ch.6

►

6.15.2 $$\sum _{n=1}^{\mathrm{\infty }}\frac{\mathrm{si}\left(\pi n\right)}{n}=\frac{1}{2}\pi (\ln \pi -1),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $\mathrm{si}\left(z\right)$: sine integral and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/6.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.15 and Ch.6

►

6.15.3 $$\sum _{n=1}^{\mathrm{\infty }}{(-1)}^n\mathrm{Ci}\left(2\pi n\right)=1-\ln 2-\frac{1}{2}\gamma ,$$

ⓘ

Symbols:

$\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ci}\left(z\right)$: cosine integral, $\ln z$: principal branch of logarithm function and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/6.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.15 and Ch.6

►

6.15.4 $$\sum _{n=1}^{\mathrm{\infty }}{(-1)}^n\frac{\mathrm{si}\left(2\pi n\right)}{n}=\pi (\frac{3}{2}\ln 2-1).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $\mathrm{si}\left(z\right)$: sine integral and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/6.15.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §6.15 and Ch.6

…

2: 4.10 Integrals

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4.10.1 $$\int \frac{dz}{z}=\ln z,$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.1.48

Referenced by:

§4.10(i)

Permalink:

http://dlmf.nist.gov/4.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.10(i), §4.10 and Ch.4

►

4.10.2 $$\int \ln zdz=z\ln z-z,$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.1.49

Permalink:

http://dlmf.nist.gov/4.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.10(i), §4.10 and Ch.4

… ►

4.10.4 $$\int \frac{dz}{z\ln z}=\ln \left(\ln z\right),$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.1.52

Referenced by:

§4.10(i)

Permalink:

http://dlmf.nist.gov/4.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.10(i), §4.10 and Ch.4

►

4.10.5 $${\int }_0^1\frac{\ln t}{1-t}dt=-\frac{{\pi }^2}{6},$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral and $\ln z$: principal branch of logarithm function

A&S Ref:

4.1.55

Referenced by:

§4.10(i)

Permalink:

http://dlmf.nist.gov/4.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.10(i), §4.10 and Ch.4

… ►

4.10.7 $${⨍}_0^x\frac{dt}{\ln t}=\mathrm{li}\left(x\right),$$ $x>1$.

ⓘ

Symbols:

$dx$: differential of $x$ , $\mathrm{li}\left(x\right)$: logarithmic integral, $\ln z$: principal branch of logarithm function, ${⨍}_a^b$: Cauchy principal value and $x$: real variable

A&S Ref:

4.1.57

Referenced by:

§4.10(i)

Permalink:

http://dlmf.nist.gov/4.10.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §4.10(i), §4.10 and Ch.4

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3: 4.1 Special Notation

… ► ►►

$k,m,n$	integers.
…

…

4: 4.6 Power Series

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4.6.1 $$\ln \left(1+z\right)=z-\frac{1}{2}{z}^2+\frac{1}{3}{z}^3-\mathrm{\cdots },$$ $|z|\le 1$, $z\ne -1$,

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.1.24

Referenced by:

§4.45(i), §4.6(i), (8.15.2)

Permalink:

http://dlmf.nist.gov/4.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.6(i), §4.6 and Ch.4

►

4.6.2 $$\ln z=\left(\frac{z-1}{z}\right)+\frac{1}{2}{\left(\frac{z-1}{z}\right)}^2+\frac{1}{3}{\left(\frac{z-1}{z}\right)}^3+\mathrm{\cdots },$$ $\mathrm{\Re }z\ge \frac{1}{2}$,

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

4.1.25

Permalink:

http://dlmf.nist.gov/4.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.6(i), §4.6 and Ch.4

►

4.6.3 $$\ln z=(z-1)-\frac{1}{2}{(z-1)}^2+\frac{1}{3}{(z-1)}^3-\mathrm{\cdots },$$ $|z-1|\le 1$, $z\ne 0$,

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.1.26

Permalink:

http://dlmf.nist.gov/4.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.6(i), §4.6 and Ch.4

►

4.6.4 $$\ln z=2\left(\left(\frac{z-1}{z+1}\right)+\frac{1}{3}{\left(\frac{z-1}{z+1}\right)}^3+\frac{1}{5}{\left(\frac{z-1}{z+1}\right)}^5+\mathrm{\cdots }\right),$$ $\mathrm{\Re }z\ge 0$, $z\ne 0$,

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

4.1.27

Permalink:

http://dlmf.nist.gov/4.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.6(i), §4.6 and Ch.4

►

4.6.5 $$\ln \left(\frac{z+1}{z-1}\right)=2\left(\frac{1}{z}+\frac{1}{3z^3}+\frac{1}{5z^5}+\mathrm{\cdots }\right),$$ $|z|\ge 1$, $z\ne \pm 1$,

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.1.28

Permalink:

http://dlmf.nist.gov/4.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.6(i), §4.6 and Ch.4

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5: 6.6 Power Series

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6.6.1 $$\mathrm{Ei}\left(x\right)=\gamma +\ln x+\sum _{n=1}^{\mathrm{\infty }}\frac{x^n}{n!n},$$ $x>0$.

ⓘ

Symbols:

$\gamma$: Euler’s constant, $\mathrm{Ei}\left(x\right)$: exponential integral, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $x$: real variable and $n$: nonnegative integer

A&S Ref:

5.1.10

Referenced by:

§6.5

Permalink:

http://dlmf.nist.gov/6.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.6 and Ch.6

►

6.6.2 $$E_1\left(z\right)=-\gamma -\ln z-\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n{z}^n}{n!n}.$$

ⓘ

Symbols:

$\gamma$: Euler’s constant, $E_1\left(z\right)$: exponential integral, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

5.1.11

Referenced by:

§6.4, §6.5

Permalink:

http://dlmf.nist.gov/6.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.6 and Ch.6

►

6.6.3 $$E_1\left(z\right)=-\ln z+{\mathrm{e}}^{-z}\sum _{n=0}^{\mathrm{\infty }}\frac{z^n}{n!}\psi \left(n+1\right),$$

ⓘ

Symbols:

$\psi \left(z\right)$: psi (or digamma) function, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§2.5(iii), §6.6, §8.19(iv)

Permalink:

http://dlmf.nist.gov/6.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.6 and Ch.6

… ►

6.6.6 $$\mathrm{Ci}\left(z\right)=\gamma +\ln z+\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n{z}^{2n}}{(2n)!(2n)}.$$

ⓘ

Symbols:

$\gamma$: Euler’s constant, $\mathrm{Ci}\left(z\right)$: cosine integral, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

5.2.16

Referenced by:

§6.5

Permalink:

http://dlmf.nist.gov/6.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §6.6 and Ch.6

…

6: 4.5 Inequalities

… ►

4.5.1 $x>-1$, $x\ne 0$,

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function and $x$: real variable

A&S Ref:

4.1.33

Referenced by:

§4.5(i), §4.5(ii)

Permalink:

http://dlmf.nist.gov/4.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.5(i), §4.5 and Ch.4

►

4.5.2 , $x\ne 0$,

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function and $x$: real variable

A&S Ref:

4.1.34

Referenced by:

§4.5(i), §4.5(ii)

Permalink:

http://dlmf.nist.gov/4.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.5(i), §4.5 and Ch.4

►

4.5.3 ,

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function and $x$: real variable

A&S Ref:

4.1.35

Referenced by:

§4.5(i)

Permalink:

http://dlmf.nist.gov/4.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.5(i), §4.5 and Ch.4

►

4.5.4 $$\ln x\le x-1,$$ $x>0$,

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function and $x$: real variable

A&S Ref:

4.1.36

Referenced by:

§4.5(i)

Permalink:

http://dlmf.nist.gov/4.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.5(i), §4.5 and Ch.4

►

4.5.5 $$\ln x\le a(x^{1/a}-1),$$ $a$, $x>0$,

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function, $a$: real or complex constant and $x$: real variable

A&S Ref:

4.1.37

Referenced by:

§4.5(i)

Permalink:

http://dlmf.nist.gov/4.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.5(i), §4.5 and Ch.4

…

7: 4.2 Definitions

… ►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 $$\ln z={\int }_1^z\frac{dt}{t},$$

ⓘ

Defines:

$\ln z$: principal branch of logarithm function

Symbols:

$dx$: differential of $x$ , $\int$: integral and $z$: complex variable

A&S Ref:

4.1.1

Permalink:

http://dlmf.nist.gov/4.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.2(i), §4.2 and Ch.4

… ►

4.2.12 $$\ln \mathrm{e}=1.$$

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm and $\ln z$: principal branch of logarithm function

Permalink:

http://dlmf.nist.gov/4.2.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §4.2(ii), §4.2 and Ch.4

… ►

4.2.16 $$\ln z=(\ln 10){\log }_{10}z,$$

ⓘ

Symbols:

${\log }_{a}z$: logarithm to general base, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.1.23

Permalink:

http://dlmf.nist.gov/4.2.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §4.2(ii), §4.2 and Ch.4

…

8: 22.14 Integrals

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22.14.1 $$\int \mathrm{sn}(x,k)dx=k^{-1}\ln \left(\mathrm{dn}(x,k)-k\mathrm{cn}(x,k)\right),$$

ⓘ

Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, $x$: real and $k$: modulus

A&S Ref:

16.24.1

Permalink:

http://dlmf.nist.gov/22.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §22.14(i), §22.14 and Ch.22

… ►

22.14.4 $$\int \mathrm{cd}(x,k)dx=k^{-1}\ln \left(\mathrm{nd}(x,k)+k\mathrm{sd}(x,k)\right),$$

ⓘ

Symbols:

$\mathrm{cd}(z,k)$: Jacobian elliptic function, $\mathrm{nd}(z,k)$: Jacobian elliptic function, $\mathrm{sd}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, $x$: real and $k$: modulus

A&S Ref:

16.24.4

Permalink:

http://dlmf.nist.gov/22.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §22.14(i), §22.14 and Ch.22

… ►

22.14.7 $$\int \mathrm{dc}(x,k)dx=\ln \left(\mathrm{nc}(x,k)+\mathrm{sc}(x,k)\right),$$

ⓘ

Symbols:

$\mathrm{dc}(z,k)$: Jacobian elliptic function, $\mathrm{nc}(z,k)$: Jacobian elliptic function, $\mathrm{sc}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, $x$: real and $k$: modulus

A&S Ref:

16.24.7

Permalink:

http://dlmf.nist.gov/22.14.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §22.14(i), §22.14 and Ch.22

►

22.14.8 $$\int \mathrm{nc}(x,k)dx=k_{}^{\prime }{}_{}{}^{-1}\ln \left(\mathrm{dc}(x,k)+k^{\prime }\mathrm{sc}(x,k)\right),$$

ⓘ

Symbols:

$\mathrm{dc}(z,k)$: Jacobian elliptic function, $\mathrm{nc}(z,k)$: Jacobian elliptic function, $\mathrm{sc}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, $x$: real, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

16.24.8

Permalink:

http://dlmf.nist.gov/22.14.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §22.14(i), §22.14 and Ch.22

►

22.14.9 $$\int \mathrm{sc}(x,k)dx=k_{}^{\prime }{}_{}{}^{-1}\ln \left(\mathrm{dc}(x,k)+k^{\prime }\mathrm{nc}(x,k)\right).$$

ⓘ

Symbols:

$\mathrm{dc}(z,k)$: Jacobian elliptic function, $\mathrm{nc}(z,k)$: Jacobian elliptic function, $\mathrm{sc}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, $x$: real, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

16.24.9

Permalink:

http://dlmf.nist.gov/22.14.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §22.14(i), §22.14 and Ch.22

…

9: 4.8 Identities

… ►

4.8.7 $$\ln \frac{1}{z}=-\ln z,$$ $|\mathrm{ph}z|\le \pi$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase and $z$: complex variable

Referenced by:

§4.2(i), §4.8(i)

Permalink:

http://dlmf.nist.gov/4.8.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §4.8(i), §4.8 and Ch.4

… ►

4.8.9 $$\ln \left(\exp z\right)=z,$$ $-\pi \le \mathrm{\Im }z\le \pi$,

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\mathrm{\Im }$: imaginary part, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.2.3

Permalink:

http://dlmf.nist.gov/4.8.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §4.8(i), §4.8 and Ch.4

►

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

ⓘ

Symbols:

$\exp z$: exponential function, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.2.4

Referenced by:

(25.10.2), §4.8(i)

Permalink:

http://dlmf.nist.gov/4.8.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §4.8(i), §4.8 and Ch.4

… ►

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

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $\ln z$: 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:

pMML, png, TeX

See also:

Annotations for §4.8(i), §4.8 and Ch.4

… ►

4.8.13 $$\ln \left(a^x\right)=x\ln a,$$ $a>0$.

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function, $a$: real or complex constant and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.8.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §4.8(i), §4.8 and Ch.4

…

10: 5.10 Continued Fractions

… ►

5.10.1 $$\mathrm{Ln}\mathrm{\Gamma }\left(z\right)+z-\left(z-\frac{1}{2}\right)\ln z-\frac{1}{2}\ln \left(2\pi \right)=\frac{a_0}{z+}\frac{a_1}{z+}\frac{a_2}{z+}\frac{a_3}{z+}\frac{a_4}{z+}\frac{a_5}{z+}\mathrm{\cdots },$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function, $z$: complex variable and $a_k$: coefficient

Referenced by:

Erratum (V1.0.10) for Equations (5.9.10), (5.9.11), (5.10.1), (5.11.1), (5.11.8)

Permalink:

http://dlmf.nist.gov/5.10.E1

Encodings:

pMML, png, TeX

Addition (effective with 1.0.10):

To increase the region of validity of this equation, the logarithm of the gamma function that appears on its left-hand side has been changed to $\mathrm{Ln}\mathrm{\Gamma }\left(z\right)$, where $\mathrm{Ln}$ is the general logarithm. Originally $\ln \mathrm{\Gamma }\left(z\right)$ was used, where $\ln$ is the principal branch of the logarithm.

Suggested 2015-02-13 by Philippe Spindel

See also:

Annotations for §5.10 and Ch.5

…