principal branch (value)

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21: 14.28 Sums

… ►where the branches of the square roots have their principal values when $z_1,z_2\in (1,\mathrm{\infty })$ and are continuous when $z_1,z_2\in ℂ\setminus (0,1]$. …

22: 4.4 Special Values and Limits

… ►

4.4.19 $$\underset{n\to \mathrm{\infty }}{lim}\left(\left(\sum _{k=1}^n\frac{1}{k}\right)-\ln n\right)=\gamma =\mathrm{0.57721\; 56649\; 01532\; 86060}\mathrm{\dots },$$

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Symbols:

$\gamma$: Euler’s constant, $\ln z$: principal branch of logarithm function, $k$: integer and $n$: integer

A&S Ref:

4.1.32 (with 10D value)

Notes:

For more digits see OEIS Sequence A001620; see also Sloane (2003).

Permalink:

http://dlmf.nist.gov/4.4.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §4.4(iii), §4.4 and Ch.4

…

23: 5.2 Definitions

… ►

5.2.3 $$\gamma =\underset{n\to \mathrm{\infty }}{lim}\left(1+\frac{1}{2}+\frac{1}{3}+\mathrm{\cdots }+\frac{1}{n}-\ln n\right)=\mathrm{0.57721\hspace{0.33em}56649\hspace{0.33em}01532\hspace{0.33em}86060}\mathrm{\dots }.$$

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Defines:

$\gamma$: Euler’s constant

Symbols:

$\ln z$: principal branch of logarithm function and $n$: nonnegative integer

A&S Ref:

6.1.3 (where the 10D value is given, and Table 1.1 where the 24D value is given.)

Notes:

For more digits see OEIS Sequence A001620; see also Sloane (2003).

Referenced by:

§4.4(iii)

Permalink:

http://dlmf.nist.gov/5.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.2(ii), §5.2 and Ch.5

…

24: 31.9 Orthogonality

… ►The branches of the many-valued functions are continuous on the path, and assume their principal values at the beginning. …

25: 4.45 Methods of Computation

… ►For other values of $x$ set $x={10}^m\xi$, where $1/10\le \xi \le 10$ and $m\in \mathbb{Z}$. … ►Let $x$ have any real value. … ►Let $x$ have any real value. … ►For $x\in [-1/\mathrm{e},\mathrm{\infty })$ the principal branch $\mathrm{Wp}\left(x\right)$ can be computed by solving the defining equation $W{\mathrm{e}}^W=x$ numerically, for example, by Newton’s rule (§3.8(ii)). Initial approximations are obtainable, for example, from the power series (4.13.6) (with $t\ge 0$) when $x$ is close to $-1/\mathrm{e}$, from the asymptotic expansion (4.13.10) when $x$ is large, and by numerical integration of the differential equation (4.13.4) (§3.7) for other values of $x$. …

26: 1.10 Functions of a Complex Variable

… ►

1.10.3 $$\ln \left(1+z\right)=z-\frac{z^2}{2}+\frac{z^3}{3}-\mathrm{\cdots },$$ ,

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Symbols:

$\ln z$: principal branch of logarithm function, $z$: variable and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §1.10(i), §1.10(i), §1.10 and Ch.1

… ►One such branch is obtained by assigning ${(1-z)}^{\alpha }$ and ${(1+z)}^{\beta }$ their principal values (§4.2(iv)). … ►

1.10.20 $$\left|\ln \left(1+a_n(z)\right)\right|\le M_n,$$ $n\ge N$,

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Symbols:

$\ln z$: principal branch of logarithm function, $z$: variable, $n$: nonnegative integer and $\left|x\right|$: absolute value of $x$

Referenced by:

§1.10(ix)

Permalink:

http://dlmf.nist.gov/1.10.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §1.10(ix), §1.10(ix), §1.10 and Ch.1

…

27: 4.8 Identities

… ►This is interpreted that every value of $\mathrm{Ln}\left(z_1{z}_2\right)$ is one of the values of $\mathrm{Ln}{z}_1+\mathrm{Ln}{z}_2$, and vice versa. … ►

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

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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

►If $a\ne 0$ and $a^z$ has its general value, then …If $a\ne 0$ and $a^z$ has its principal value, then … ►

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

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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

…

28: 1.4 Calculus of One Variable

… ►

1.4.17

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, $n$: nonnegative integer, $C$: constant and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.4.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §1.4(iv), §1.4(iv), §1.4 and Ch.1

…

29: 13.21 Uniform Asymptotic Approximations for Large $\kappa$

… ►When $\kappa \to \mathrm{\infty }$ through positive real values with $\mu$ ($\ge 0$) fixed … ►Other types of approximations when $\kappa \to \mathrm{\infty }$ through positive real values with $\mu$ ($\ge 0$) fixed are as follows. … ►For extensions to complex values of $x$ see López (1999). … ►This reference also includes error bounds and extensions to asymptotic expansions and complex values of $x$. … ►This reference also includes error bounds and extensions to asymptotic expansions and complex values of $x$. …

30: 4.24 Inverse Trigonometric Functions: Further Properties

… ►The above equations are interpreted in the sense that every value of the left-hand side is a value of the right-hand side and vice versa. All square roots have either possible value.