branch

SearchAdvancedHelp

(0.001 seconds)

11—20 of 201 matching pages

11: 14.24 Analytic Continuation

… ►Let $s$ be an arbitrary integer, and $P_{\nu }^{-\mu }\left(z{\mathrm{e}}^{s\pi \mathrm{i}}\right)$ and ${𝑸}_{\nu }^{\mu }\left(z{\mathrm{e}}^{s\pi \mathrm{i}}\right)$ denote the branches obtained from the principal branches by making $\frac{1}{2}s$ circuits, in the positive sense, of the ellipse having $\pm 1$ as foci and passing through $z$. … ►Next, let $P_{\nu ,s}^{-\mu }\left(z\right)$ and ${𝑸}_{\nu ,s}^{\mu }\left(z\right)$ denote the branches obtained from the principal branches by encircling the branch point $1$ (but not the branch point $-1$) $s$ times in the positive sense. … ►For fixed $z$, other than $\pm 1$ or $\mathrm{\infty }$, each branch of $P_{\nu }^{-\mu }\left(z\right)$ and ${𝑸}_{\nu }^{\mu }\left(z\right)$ is an entire function of each parameter $\nu$ and $\mu$. …

12: 22.14 Integrals

… ►

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

… ►The branches of the inverse trigonometric functions are chosen so that they are continuous. … ►

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

… ►Again, the branches of the inverse trigonometric functions must be continuous. … ►

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

…

13: 1.10 Functions of a Complex Variable

… ►

§1.10(vi) Multivalued Functions

… ► ►Branches can be constructed in two ways: … ► ►

Example

…

14: 10.25 Definitions

… ►

§10.25(ii) Standard Solutions

… ►It has a branch point at $z=0$ for all $\nu \in ℂ$. The principal branch corresponds to the principal value of the square root in (10.25.3), is analytic in $ℂ\setminus (-\mathrm{\infty },0]$, and two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi$. … ►For fixed $z$ $(\ne 0)$ each branch of $I_{\nu }\left(z\right)$ and $K_{\nu }\left(z\right)$ is entire in $\nu$. ►

Branch Conventions

…

15: 15.2 Definitions and Analytical Properties

… ► … ►again with analytic continuation for other values of $z$, and with the principal branch defined in a similar way. … ►The difference between the principal branches on the two sides of the branch cut (§4.2(i)) is given by … ►

§15.2(ii) Analytic Properties

… ►The same is true of other branches, provided that $z=0$, $1$, and $\mathrm{\infty }$ are excluded. …

16: 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

…

17: 4.1 Special Notation

… ► ►►

$k,m,n$	integers.
…

…

18: 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

…

19: 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

…

20: 6.14 Integrals

… ►

6.14.1 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}{E}_1\left(t\right)dt=\frac{1}{a}\ln \left(1+a\right),$$ $\mathrm{\Re }a>-1$,

ⓘ

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\int$: integral, $\ln z$: principal branch of logarithm function and $\mathrm{\Re }$: real part

A&S Ref:

5.1.34 (special case of)

Permalink:

http://dlmf.nist.gov/6.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.14(i), §6.14 and Ch.6

►

6.14.2 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}\mathrm{Ci}\left(t\right)dt=-\frac{1}{2a}\ln \left(1+a^2\right),$$ $\mathrm{\Re }a>0$,

ⓘ

Symbols:

$\mathrm{Ci}\left(z\right)$: cosine integral, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\ln z$: principal branch of logarithm function and $\mathrm{\Re }$: real part

A&S Ref:

5.2.28

Referenced by:

§6.14(i)

Permalink:

http://dlmf.nist.gov/6.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.14(i), §6.14 and Ch.6

… ►

6.14.4 $${\int }_0^{\mathrm{\infty }}{E}_1^2\left(t\right)dt=2\ln 2,$$

ⓘ

Symbols:

$dx$: differential of $x$ , $E_1\left(z\right)$: exponential integral, $\int$: integral and $\ln z$: principal branch of logarithm function

A&S Ref:

5.1.33

Permalink:

http://dlmf.nist.gov/6.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §6.14(ii), §6.14 and Ch.6

… ►

6.14.7 $${\int }_0^{\mathrm{\infty }}\mathrm{Ci}\left(t\right)\mathrm{si}\left(t\right)dt=\ln 2.$$

ⓘ

Symbols:

$\mathrm{Ci}\left(z\right)$: cosine integral, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function and $\mathrm{si}\left(z\right)$: sine integral

A&S Ref:

5.2.32

Referenced by:

§6.14(ii)

Permalink:

http://dlmf.nist.gov/6.14.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §6.14(ii), §6.14 and Ch.6

…