branch points

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1: 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. …The branch points are called the exceptional values, and the other points normal values. … ►For a visualization of the first branch point of

a_{0}\left(\mathrm{i}\hat{q}\right)

and

a_{2}\left(\mathrm{i}\hat{q}\right)

see Figure 28.7.1. …

2: Mark J. Ablowitz

… ►ODEs which do not have moveable branch point singularities. …

3: 1.10 Functions of a Complex Variable

… ►Then

a

is a branch point of

F(z)

. For example,

z=0

is a branch point of

\sqrt{z}

. … ►(a) By introducing appropriate cuts from the branch points and restricting

F(z)

to be single-valued in the cut plane (or domain). ►(b) By specifying the value of

F(z)

at a point

z_{0}

(not a branch point), and requiring

F(z)

to be continuous on any path that begins at

z_{0}

and does not pass through any branch points or other singularities of

F(z)

. …

4: 14.24 Analytic Continuation

… ►Next, let

P^{-\mu}_{\nu,s}\left(z\right)

and

\boldsymbol{Q}^{\mu}_{\nu,s}\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. …

5: 19.14 Reduction of General Elliptic Integrals

… ►A similar remark applies to the transformations given in Erdélyi et al. (1953b, §13.5) and to the choice among explicit reductions in the extensive table of Byrd and Friedman (1971), in which one limit of integration is assumed to be a branch point of the integrand at which the integral converges. If no such branch point is accessible from the interval of integration (for example, if the integrand is

(t(3-t)(4-t))^{-3/2}

and the interval is [1,2]), then no method using this assumption succeeds. …

6: 21.7 Riemann Surfaces

… ►The zeros

\lambda_{j}

j=1,2,\dots,2g+1

Q(\lambda)

specify the finite branch points

P_{j}

, that is, points at which

\mu_{j}=0

, on the Riemann surface. Denote the set of all branch points by

B=\{P_{1},P_{2},\dots,P_{2g+1},P_{\infty}\}

. … ►

21.7.17

\sum_{P_{j}\in U}\prod_{k=1}^{4}\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{% \alpha}}_{k}+\boldsymbol{{\eta}}^{1}(P_{j})}{\boldsymbol{{\beta}}_{k}+% \boldsymbol{{\eta}}^{2}(P_{j})}\left(\mathbf{z}_{k}\middle|\boldsymbol{{\Omega% }}\right)=\sum_{P_{j}\in U^{c}}\prod_{k=1}^{4}\theta\genfrac{[}{]}{0.0pt}{}{% \boldsymbol{{\alpha}}_{k}+\boldsymbol{{\eta}}^{1}(P_{j})}{\boldsymbol{{\beta}}% _{k}+\boldsymbol{{\eta}}^{2}(P_{j})}\left(\mathbf{z}_{k}\middle|\boldsymbol{{% \Omega}}\right).

ⓘ

Symbols:: $\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function with characteristics, $\in$ : element of, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : $g$ -dimensional vector, $\boldsymbol{{\beta}}$ : $g$ -dimensional vector, $P_{j}$ : branch points and $U$ : subset of $B$
Permalink:: http://dlmf.nist.gov/21.7.E17
Encodings:: pMML, png, TeX
See also:: Annotations for §21.7(iii), §21.7 and Ch.21

7: 4.37 Inverse Hyperbolic Functions

… ►

4.37.6

\operatorname{Arccoth}z=\operatorname{Arctanh}\left(1/z\right).

ⓘ

Defines:: $\operatorname{Arccoth}\NVar{z}$ : general inverse hyperbolic cotangent function
Symbols:: $\operatorname{Arctanh}\NVar{z}$ : general inverse hyperbolic tangent function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/4.37.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §4.37(i), §4.37 and Ch.4

… ►

\operatorname{Arcsinh}z

and

\operatorname{Arccsch}z

have branch points at

z=\pm i

; the other four functions have branch points at

z=\pm 1

. …

8: 6.4 Analytic Continuation

… ►Analytic continuation of the principal value of

E_{1}\left(z\right)

yields a multi-valued function with branch points at

z=0

and

z=\infty

. …

9: 15.2 Definitions and Analytical Properties

… ►

§15.2(ii) Analytic Properties

… ►As a multivalued function of

z

\mathbf{F}\left(a,b;c;z\right)

is analytic everywhere except for possible branch points at

z=0

1

, and

\infty

. …

10: 10.2 Definitions

… ►This solution of (10.2.1) is an analytic function of

z\in\mathbb{C}

, except for a branch point at

z=0

when

\nu

is not an integer. … ►Whether or not

\nu

is an integer

Y_{\nu}\left(z\right)

has a branch point at

z=0

. … ►Each solution has a branch point at

z=0

for all

\nu\in\mathbb{C}

. …