branch cuts

SearchAdvancedHelp

(0.001 seconds)

21—28 of 28 matching pages

21: 10.25 Definitions

… ►In particular, the principal branch of $I_{\nu }\left(z\right)$ is defined in a similar way: it corresponds to the principal value of ${(\frac{1}{2}z)}^{\nu }$, is analytic in $ℂ\setminus (-\mathrm{\infty },0]$, and two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi$. … ►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 $\mathrm{ph}z=\pm \pi$. …

22: 2.10 Sums and Sequences

… ►Here the branch of ${\left({\mathrm{e}}^{-\mathrm{i}\alpha }-z\right)}^{-1/2}$ is continuous in the $z$-plane cut along the outward-drawn ray through $z={\mathrm{e}}^{-\mathrm{i}\alpha }$ and equals ${\mathrm{e}}^{\mathrm{i}\alpha /2}$ at $z=0$. …

23: 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 $\mu$. ►The behavior of $P_{\nu }^{-\mu }\left(z\right)$ and ${𝑸}_{\nu }^{\mu }\left(z\right)$ as $z\to -1$ from the left on the upper or lower side of the cut from $-\mathrm{\infty }$ to $1$ can be deduced from (14.8.7)–(14.8.11), combined with (14.24.1) and (14.24.2) with $s=\pm 1$.

24: 31.11 Expansions in Series of Hypergeometric Functions

… ►Every Heun function (§31.4) can be represented by a series of Type I convergent in the whole plane cut along a line joining the two singularities of the Heun function. … ►The expansion (31.11.1) with (31.11.12) is convergent in the plane cut along the line joining the two singularities $z=0$ and $z=1$. … ►The expansion (31.11.1) for a Heun function that is associated with any branch of (31.11.2)—other than a multiple of the right-hand side of (31.11.12)—is convergent inside the ellipse $ℰ$. …

25: 22.18 Mathematical Applications

… ►The half-open rectangle $(-K,K)\times [-K^{\prime },K^{\prime }]$ maps onto $ℂ$ cut along the intervals $(-\mathrm{\infty },-1]$ and $[1,\mathrm{\infty })$. … ►This circumvents the cumbersome branch structure of the multivalued functions $x(y)$ or $y(x)$, and constitutes the process of uniformization; see Siegel (1988, Chapter II). …

26: 6.2 Definitions and Interrelations

… ►As in the case of the logarithm (§4.2(i)) there is a cut along the interval $(-\mathrm{\infty },0]$ and the principal value is two-valued on $(-\mathrm{\infty },0)$. … ►

6.2.4 $$E_1\left(z\right)=\mathrm{Ein}\left(z\right)-\ln z-\gamma .$$

ⓘ

Symbols:

$\gamma$: Euler’s constant, $\mathrm{Ein}\left(z\right)$: complementary exponential integral, $E_1\left(z\right)$: exponential integral, $\ln z$: principal branch of logarithm function and $z$: complex variable

Referenced by:

§6.10(ii), §6.4

Permalink:

http://dlmf.nist.gov/6.2.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §6.2(i), §6.2 and Ch.6

… ►

6.2.7 $$\mathrm{Ei}\left(\pm x\right)=-\mathrm{Ein}\left(\mp x\right)+\ln x+\gamma .$$

ⓘ

Symbols:

$\gamma$: Euler’s constant, $\mathrm{Ein}\left(z\right)$: complementary exponential integral, $\mathrm{Ei}\left(x\right)$: exponential integral, $\ln z$: principal branch of logarithm function and $x$: real variable

A&S Ref:

5.1.40 (in modified form)

Permalink:

http://dlmf.nist.gov/6.2.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §6.2(i), §6.2 and Ch.6

… ►

6.2.8 $$\mathrm{li}\left(x\right)={⨍}_0^x\frac{dt}{\ln t}=\mathrm{Ei}\left(\ln x\right),$$ $x>1$.

ⓘ

Defines:

$\mathrm{li}\left(x\right)$: logarithmic integral

Symbols:

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

A&S Ref:

5.1.3

Referenced by:

§27.12, §6.12(i)

Permalink:

http://dlmf.nist.gov/6.2.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §6.2(i), §6.2 and Ch.6

… ►

6.2.13 $$\mathrm{Ci}\left(z\right)=-\mathrm{Cin}\left(z\right)+\ln z+\gamma .$$

ⓘ

Symbols:

$\gamma$: Euler’s constant, $\mathrm{Ci}\left(z\right)$: cosine integral, $\mathrm{Cin}\left(z\right)$: cosine integral, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

5.2.2 (in modified form)

Referenced by:

§6.4

Permalink:

http://dlmf.nist.gov/6.2.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §6.2(ii), §6.2 and Ch.6

…

27: Mathematical Introduction

… ►These include, for example, multivalued functions of complex variables, for which new definitions of branch points and principal values are supplied (§§1.10(vi), 4.2(i)); the Dirac delta (or delta function), which is introduced in a more readily comprehensible way for mathematicians (§1.17); numerically satisfactory solutions of differential and difference equations (§§2.7(iv), 2.9(i)); and numerical analysis for complex variables (Chapter 3). … ►Other examples are: (a) the notation for the Ferrers functions—also known as associated Legendre functions on the cut—for which existing notations can easily be confused with those for other associated Legendre functions (§14.1); (b) the spherical Bessel functions for which existing notations are unsymmetric and inelegant (§§10.47(i) and 10.47(ii)); and (c) elliptic integrals for which both Legendre’s forms and the more recent symmetric forms are treated fully (Chapter 19). …

28: 1.14 Integral Transforms

… ►In this case, $𝒮f\left(s\right)$ represents an analytic function in the $s$-plane cut along the negative real axis, and …