branch cuts

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11: Bibliography K

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Branch Cuts for Complex Elementary Functions or Much Ado About Nothing’s Sign Bit. In The State of the Art in Numerical Analysis (Birmingham, 1986), A. Iserles and M. J. D. Powell (Eds.), Inst. Math. Appl. Conf. Ser. New Ser., Vol. 9, pp. 165–211.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§4.23(iv), §4.37(iv).

Encodings:

BibTeX

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12: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►This is the discontinuity across the branch cut in (1.18.52) ${𝝈}_c\subset ℝ$, from $z$ below to above the cut, divided by $2\pi \mathrm{i}$. … ►This representation has poles with residues ${\left|\widehat{f}({\lambda }_n)\right|}^2$ at the discrete eigenvalues and a branch cut along $[0,\mathrm{\infty })$ with discontinuity, from below to above the cut, $2\pi \mathrm{i}{\left|\widehat{f}(\lambda )\right|}^2$, as in (1.18.53), see Newton (2002, §7.1.1). … ►For this latter see Simon (1973), and Reinhardt (1982); wherein advantage is taken of the fact that although branch points are actual singularities of an analytic function, the location of the branch cuts are often at our disposal, as they are not singularities of the function, but simply arbitrary lines to keep a function single valued, and thus only singularities of a specific representation of that analytic function. This is accomplished by the variable change $x\to x{\mathrm{e}}^{\mathrm{i}\theta }$, in $ℒ$, which rotates the continuous spectrum ${𝝈}_c\to {𝝈}_c{\mathrm{e}}^{-2\mathrm{i}\theta }$ and the branch cut of (1.18.66) into the lower half complex plain by the angle $-2\theta$, with respect to the unmoved branch point at $\lambda =0$; thus, providing access to resonances on the higher Riemann sheet should $\theta$ be large enough to expose them. …

13: 1.10 Functions of a Complex Variable

… ►Suppose $F(z)$ is multivalued and $a$ is a point such that there exists a branch of $F(z)$ in a cut neighborhood of $a$, but there does not exist a branch of $F(z)$ in any punctured neighborhood of $a$. … ►(a) By introducing appropriate cuts from the branch points and restricting $F(z)$ to be single-valued in the cut plane (or domain). … ►Branches of $F(z)$ can be defined, for example, in the cut plane $D$ obtained from $ℂ$ by removing the real axis from $1$ to $\mathrm{\infty }$ and from $-1$ to $-\mathrm{\infty }$; see Figure 1.10.1. …

14: 19.2 Definitions

… ►The principal branch of $K\left(k\right)$ and $E\left(k\right)$ is $|\mathrm{ph}\left(1-k^2\right)|\le \pi$, that is, the branch-cuts are $(-\mathrm{\infty },-1]\cup [1,+\mathrm{\infty })$. …

15: 25.12 Polylogarithms

… ►The principal branch has a cut along the interval $[1,\mathrm{\infty })$ and agrees with (25.12.1) when $|z|\le 1$; see also §4.2(i). …

16: 16.2 Definition and Analytic Properties

… ►The branch obtained by introducing a cut from $1$ to $+\mathrm{\infty }$ on the real axis, that is, the branch in the sector $|\mathrm{ph}\left(1-z\right)|\le \pi$, is the principal branch (or principal value) of ${}_{q+1}{}^{}F_q^{}(𝐚;𝐛;z)$; compare §4.2(i). …

17: 10.2 Definitions

… ►The principal branch of $J_{\nu }\left(z\right)$ corresponds to the principal value of ${(\frac{1}{2}z)}^{\nu }$ (§4.2(iv)) and is analytic in the $z$-plane cut along the interval $(-\mathrm{\infty },0]$. … ►The principal branch corresponds to the principal branches of $J_{\pm \nu }\left(z\right)$ in (10.2.3) and (10.2.4), with a cut in the $z$-plane along the interval $(-\mathrm{\infty },0]$. ►Except in the case of $J_{\pm n}\left(z\right)$, the principal branches of $J_{\nu }\left(z\right)$ and $Y_{\nu }\left(z\right)$ are two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi$; compare §4.2(i). … ►The principal branches correspond to principal values of the square roots in (10.2.5) and (10.2.6), again with a cut in the $z$-plane along the interval $(-\mathrm{\infty },0]$. ►The principal branches of $H_{\nu }^{(1)}\left(z\right)$ and $H_{\nu }^{(2)}\left(z\right)$ are two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi$. …

18: 28.7 Analytic Continuation of Eigenvalues

… ►The only singularities are algebraic branch points, with $a_n\left(q\right)$ and $b_n\left(q\right)$ finite at these points. The number of branch points is infinite, but countable, and there are no finite limit points. In consequence, the functions can be defined uniquely by introducing suitable cuts in the $q$-plane. …The branch points are called the exceptional values, and the other points normal values. … ► …

19: 14.21 Definitions and Basic Properties

… ►When $z$ is complex $P_{\nu }^{\pm \mu }\left(z\right)$, $Q_{\nu }^{\mu }\left(z\right)$, and ${𝑸}_{\nu }^{\mu }\left(z\right)$ are defined by (14.3.6)–(14.3.10) with $x$ replaced by $z$: the principal branches are obtained by taking the principal values of all the multivalued functions appearing in these representations when $z\in (1,\mathrm{\infty })$, and by continuity elsewhere in the $z$-plane with a cut along the interval $(-\mathrm{\infty },1]$; compare §4.2(i). …

20: 13.2 Definitions and Basic Properties

… ►The principal branch corresponds to the principal value of $z^{-a}$ in (13.2.6), and has a cut in the $z$-plane along the interval $(-\mathrm{\infty },0]$; compare §4.2(i). …