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21: 1.10 Functions of a Complex Variable

… ►Note that (1.10.4) is a generalization of the binomial expansion (1.2.2) with the binomial coefficients defined in (1.2.6). … ►A cut neighborhood is formed by deleting a ray emanating from the center. (Or more generally, a simple contour that starts at the center and terminates on the boundary.) … ►It should be noted that different branches of

(w-w_{0})^{1/\mu}

used in forming

(w-w_{0})^{n/\mu}

in (1.10.16) give rise to different solutions of (1.10.12). … ►Let

F(x,z)

have a converging power series expansion of the form …

22: 36.5 Stokes Sets

… ►The Stokes set takes different forms for

z=0

z<0

, and

z>0

. … ►They generate a pair of cusp-edged sheets connected to the cusped sheets of the swallowtail bifurcation set (§36.4). … ►The first sheet corresponds to

x<0

and is generated as a solution of Equations (36.5.6)–(36.5.9). …For

\left|Y\right|>Y_{1}

the second sheet is generated by a second solution of (36.5.6)–(36.5.9), and for

\left|Y\right|<Y_{1}

it is generated by the roots of the polynomial equation … ►the intersection lines with the bifurcation set are generated by

|X|=X_{2}=0.45148

Y=Y_{2}=0.59693

. …

23: 25.16 Mathematical Applications

… ►Euler sums have the form … ►

H\left(s\right)

is the special case

H\left(s,1\right)

of the function …when both

H\left(s,z\right)

and

H\left(z,s\right)

are finite. ►For further properties of

H\left(s,z\right)

see Apostol and Vu (1984). … ►For further generalizations, see Flajolet and Salvy (1998).

24: 32.8 Rational Solutions

… ►

\mbox{P}_{\mbox{\scriptsize II}}

–

\mbox{P}_{\mbox{\scriptsize VI}}

possess hierarchies of rational solutions for special values of the parameters which are generated from “seed solutions” using the Bäcklund transformations and often can be expressed in the form of determinants. …

25: 21.5 Modular Transformations

… ►The modular transformations form a group under the composition of such transformations, the modular group, which is generated by simpler transformations, for which

\xi(\boldsymbol{{\Gamma}})

is determinate: …

26: 17.9 Further Transformations of ${{}_{r+1}\phi_{r}}$ Functions

… ►

17.9.3

{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=\frac{\left(abz/c;q\right)_{% \infty}}{\left(bz/c;q\right)_{\infty}}{{}_{3}\phi_{2}}\left({a,c/b,0\atop c,cq% /(bz)};q,q\right)+\frac{\left(a,bz,c/b;q\right)_{\infty}}{\left(c,z,c/(bz);q% \right)_{\infty}}{{}_{3}\phi_{2}}\left({z,abz/c,0\atop bz,bzq/c};q,q\right),

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Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base and $z$ : complex variable
Referenced by:: Erratum (V1.0.23) for Equation (17.9.3)
Permalink:: http://dlmf.nist.gov/17.9.E3
Encodings:: pMML, png, TeX
Correction (effective with 1.0.23):: The second term on the right-hand side was missing. The form of the equation where the second term is missing is correct if the ${{}_{2}\phi_{1}}$ is terminating. It is this form which appeared in the first edition of Gasper and Rahman (1990). The more general version which appears now is what is reproduced in Gasper and Rahman (2004, (III.5)).
Suggested 2019-04-26 by Roberto S. Costas-Santos
See also:: Annotations for §17.9(i), §17.9(i), §17.9 and Ch.17

…

27: 10.40 Asymptotic Expansions for Large Argument

… ►Corresponding expansions for

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

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

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

, and

K_{\nu}'\left(z\right)

for other ranges of

\operatorname{ph}z

are obtainable by combining (10.34.3), (10.34.4), (10.34.6), and their differentiated forms, with (10.40.2) and (10.40.4). In particular, use of (10.34.3) with

m=0

yields the following more general (and more accurate) version of (10.40.1): … ►The general terms in (10.40.6) and (10.40.7) can be written down by analogy with (10.18.17), (10.18.19), and (10.18.20). …

28: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►In general, operators

T

being formally self-adjoint second order differential operators of the form (1.18.28), with

X

unbounded, will have both a continuous and a point spectrum, and thus, correspondingly,

non-L^{2}\left(X\right)

eigenfunctions as in §1.18(vi) and

L^{2}\left(X\right)

eigenfunctions as in §1.18(v). …

29: 12.14 The Function $W\left(a,x\right)$

… ►In other cases the general theory of (12.2.2) is available.

W\left(a,x\right)

and

W\left(a,-x\right)

form a numerically satisfactory pair of solutions when

-\infty<x<\infty

. … ►These follow from the contour integrals of §12.5(ii), which are valid for general complex values of the argument

z

and parameter

a

. …

30: 13.6 Relations to Other Functions

… ►

13.6.6

U\left(a,a,z\right)=z^{1-a}U\left(1,2-a,z\right)=z^{1-a}e^{z}E_{a}\left(z% \right)=e^{z}\Gamma\left(1-a,z\right).

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Symbols:: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $E_{\NVar{p}}\left(\NVar{z}\right)$ : generalized exponential integral, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function and $z$ : complex variable
A&S Ref:: 13.6.28 (in different form)
Permalink:: http://dlmf.nist.gov/13.6.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §13.6(ii), §13.6 and Ch.13

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