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1: Wadim Zudilin

… ►Robinson Award of the Canadian Mathematical Society. …

2: Bibliography B

… ►

M. V. Berry (1989) Uniform asymptotic smoothing of Stokes’s discontinuities. Proc. Roy. Soc. London Ser. A 422, pp. 7–21.

ⓘ

External Links:: ISSN 0080-4630, FullText, MathReview (André Hautot), ZentralBlatt
Referenced by:: §2.11(iv).
Encodings:: BibTeX

… ►

J. M. Borwein and P. B. Borwein (1987) Pi and the AGM, A Study in Analytic Number Theory and Computational Complexity. Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons Inc., New York.

ⓘ

External Links:: ISBN 0-471-83138-7, MathReview (H. London), ZentralBlatt
Referenced by:: §19.35(i), §19.5, §19.8(i), §23.20(iv).
Encodings:: BibTeX

…

3: 1.4 Calculus of One Variable

… ► ►A simple discontinuity of

f(x)

x=c

occurs when

f(c+)

and

f(c-)

exist, but

f(c+)\not=f(c-)

. If

f(x)

is continuous on an interval

I

save for a finite number of simple discontinuities, then

f(x)

is piecewise (or sectionally) continuous on

I

. For an example, see Figure 1.4.1 … ►

Stieltjes Measure with $\alpha(x)$ Discontinuous

…

4: 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

(\tfrac{1}{2}z)^{\nu}

, is analytic in

\mathbb{C}\setminus(-\infty,0]

, and two-valued and discontinuous on the cut

\operatorname{ph}z=\pm\pi

. … ►The principal branch corresponds to the principal value of the square root in (10.25.3), is analytic in

\mathbb{C}\setminus(-\infty,0]

, and two-valued and discontinuous on the cut

\operatorname{ph}z=\pm\pi

. … ►

Symbol $\mathscr{Z}_{\nu}\left(z\right)$

►Corresponding to the symbol

\mathscr{C}_{\nu}

introduced in §10.2(ii), we sometimes use

\mathscr{Z}_{\nu}\left(z\right)

to denote

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

e^{\nu\pi i}K_{\nu}\left(z\right)

, or any nontrivial linear combination of these functions, the coefficients in which are independent of

z

and

\nu

. …

5: 10.2 Definitions

… ►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

\operatorname{ph}z=\pm\pi

; compare §4.2(i). … ►The principal branches of

{H^{(1)}_{\nu}}\left(z\right)

and

{H^{(2)}_{\nu}}\left(z\right)

are two-valued and discontinuous on the cut

\operatorname{ph}z=\pm\pi

. … ►The notation

\mathscr{C}_{\nu}\left(z\right)

denotes

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

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

{H^{(1)}_{\nu}}\left(z\right)

{H^{(2)}_{\nu}}\left(z\right)

, or any nontrivial linear combination of these functions, the coefficients in which are independent of

z

and

\nu

. …

6: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►Often circumstances allow rather stronger statements, such as uniform convergence, or pointwise convergence at points where

f(x)

is continuous, with convergence to

(f(x_{0}-)+f(x_{0}+))/2

x_{0}

is an isolated point of discontinuity. … ►Assume that

\mathcal{D}(T)

is dense in

V

, i. … ►In what follows

T

will be taken to be a self adjoint extension of

\mathcal{L}

following the discussion ending the prior sub-section. … ►This is the discontinuity across the branch cut in (1.18.52)

\boldsymbol{\sigma}_{c}\subset\mathbb{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,\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). …

7: Bernard Deconinck

… ►In addition, he has spent time at the University of Alberta, the Mathematical Sciences Research Institute in Berkeley, California, and Colorado State University. …

8: Bibliography W

… ►

R. Wong and Y.-Q. Zhao (1999a) Smoothing of Stokes’s discontinuity for the generalized Bessel function. II. Proc. Roy. Soc. London Ser. A 455, pp. 3065–3084.

ⓘ

External Links:: ISSN 1364-5021, Document, MathReview, ZentralBlatt
Referenced by:: §10.46.
Encodings:: BibTeX

►

R. Wong and Y.-Q. Zhao (1999b) Smoothing of Stokes’s discontinuity for the generalized Bessel function. Proc. Roy. Soc. London Ser. A 455, pp. 1381–1400.

ⓘ

External Links:: ISSN 1364-5021, Document, MathReview, ZentralBlatt
Referenced by:: §10.46.
Encodings:: BibTeX

…

9: 10.22 Integrals

… ►In this subsection

\mathscr{C}_{\nu}\left(z\right)

and

\mathscr{D}_{\mu}(z)

denote cylinder functions(§10.2(ii)) of orders

\nu

and

\mu

, respectively, not necessarily distinct. … ►

10.22.4

\int z\mathscr{C}_{\mu}\left(az\right)\mathscr{D}_{\mu}(bz)\,\mathrm{d}z=\frac% {z\left(a\mathscr{C}_{\mu+1}\left(az\right)\mathscr{D}_{\mu}(bz)-b\mathscr{C}_% {\mu}\left(az\right)\mathscr{D}_{\mu+1}(bz)\right)}{a^{2}-b^{2}},

a^{2}\neq b^{2}

ⓘ

Symbols:: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $z$ : complex variable and $\mathscr{D}_{\nu}(z)$ : cylinder function
Referenced by:: §10.22(ii)
Permalink:: http://dlmf.nist.gov/10.22.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §10.22(i), §10.22(i), §10.22 and Ch.10

►

10.22.5

\int z\mathscr{C}_{\mu}\left(az\right)\mathscr{D}_{\mu}(az)\,\mathrm{d}z=% \tfrac{1}{4}z^{2}\left(2\mathscr{C}_{\mu}\left(az\right)\mathscr{D}_{\mu}(az)-% \mathscr{C}_{\mu-1}\left(az\right)\mathscr{D}_{\mu+1}(az)-\mathscr{C}_{\mu+1}% \left(az\right)\mathscr{D}_{\mu-1}(az)\right),

ⓘ

Symbols:: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $z$ : complex variable and $\mathscr{D}_{\nu}(z)$ : cylinder function
Referenced by:: §10.22(ii)
Permalink:: http://dlmf.nist.gov/10.22.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §10.22(i), §10.22(i), §10.22 and Ch.10

►

10.22.6

\int\mathscr{C}_{\mu}\left(az\right)\mathscr{D}_{\nu}(az)\frac{\,\mathrm{d}z}{% z}=-\frac{az(\mathscr{C}_{\mu+1}\left(az\right)\mathscr{D}_{\nu}(az)-\mathscr{% C}_{\mu}\left(az\right)\mathscr{D}_{\nu+1}(az))}{\mu^{2}-\nu^{2}}+\frac{% \mathscr{C}_{\mu}\left(az\right)\mathscr{D}_{\nu}(az)}{\mu+\nu},

\mu^{2}\neq\nu^{2}

ⓘ

Symbols:: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $z$ : complex variable, $\nu$ : complex parameter and $\mathscr{D}_{\nu}(z)$ : cylinder function
Permalink:: http://dlmf.nist.gov/10.22.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §10.22(i), §10.22(i), §10.22 and Ch.10

… ►

Weber–Schafheitlin Discontinuous Integrals, including Special Cases

…

10: 2.11 Remainder Terms; Stokes Phenomenon

… ►That the change in their forms is discontinuous, even though the function being approximated is analytic, is an example of the Stokes phenomenon. … ►Optimum truncation occurs just prior to the numerically smallest term, that is, at

s_{4}

. …