# ?????????,????app??,?www.22kk44.com,???????,??????,????????,??????????,????????,????????,??????,22kk44.com

(0.026 seconds)

## 1—10 of 480 matching pages

##### 1: 16.13 Appell Functions
###### §16.13 Appell Functions
16.13.1 ${F_{1}}\left(\alpha;\beta,\beta^{\prime};\gamma;x,y\right)=\sum_{m,n=0}^{% \infty}\frac{{\left(\alpha\right)_{m+n}}{\left(\beta\right)_{m}}{\left(\beta^{% \prime}\right)_{n}}}{{\left(\gamma\right)_{m+n}}m!n!}x^{m}y^{n},$ $\max\left(|x|,|y|\right)<1$,
16.13.3 ${F_{3}}\left(\alpha,\alpha^{\prime};\beta,\beta^{\prime};\gamma;x,y\right)=% \sum_{m,n=0}^{\infty}\frac{{\left(\alpha\right)_{m}}{\left(\alpha^{\prime}% \right)_{n}}{\left(\beta\right)_{m}}{\left(\beta^{\prime}\right)_{n}}}{{\left(% \gamma\right)_{m+n}}m!n!}x^{m}y^{n},$ $\max\left(|x|,|y|\right)<1$,
16.13.4 ${F_{4}}\left(\alpha,\beta;\gamma,\gamma^{\prime};x,y\right)=\sum_{m,n=0}^{% \infty}\frac{{\left(\alpha\right)_{m+n}}{\left(\beta\right)_{m+n}}}{{\left(% \gamma\right)_{m}}{\left(\gamma^{\prime}\right)_{n}}m!n!}x^{m}y^{n},$ $\sqrt{|x|}+\sqrt{|y|}<1$.
##### 2: 16.14 Partial Differential Equations
###### §16.14(i) Appell Functions
$x(1-x)\frac{{\partial}^{2}{F_{2}}}{{\partial x}^{2}}-xy\frac{\,{\partial}^{2}{% F_{2}}}{\,\partial x\,\partial y}+\left(\gamma-(\alpha+\beta+1)x\right)\frac{% \partial{F_{2}}}{\partial x}-\beta y\frac{\partial{F_{2}}}{\partial y}-\alpha% \beta{F_{2}}=0,$
$y(1-y)\frac{{\partial}^{2}{F_{2}}}{{\partial y}^{2}}-xy\frac{\,{\partial}^{2}{% F_{2}}}{\,\partial x\,\partial y}+\left(\gamma^{\prime}-(\alpha+\beta^{\prime}% +1)y\right)\frac{\partial{F_{2}}}{\partial y}-\beta^{\prime}x\frac{\partial{F_% {2}}}{\partial x}-\alpha\beta^{\prime}{F_{2}}=0,$
$x(1-x)\frac{{\partial}^{2}{F_{4}}}{{\partial x}^{2}}-2xy\frac{\,{\partial}^{2}% {F_{4}}}{\,\partial x\,\partial y}-y^{2}\frac{{\partial}^{2}{F_{4}}}{{\partial y% }^{2}}+\left(\gamma-(\alpha+\beta+1)x\right)\frac{\partial{F_{4}}}{\partial x}% -(\alpha+\beta+1)y\frac{\partial{F_{4}}}{\partial y}-\alpha\beta{F_{4}}=0,$
In addition to the four Appell functions there are $24$ other sums of double series that cannot be expressed as a product of two ${{}_{2}F_{1}}$ functions, and which satisfy pairs of linear partial differential equations of the second order. …
##### 3: 16.16 Transformations of Variables
###### §16.16(i) Reduction Formulas
16.16.4 ${F_{3}}\left(\alpha,\gamma-\alpha;\beta,\beta^{\prime};\gamma;x,y\right)=(1-y)% ^{-\beta^{\prime}}{F_{1}}\left(\alpha;\beta,\beta^{\prime};\gamma;x,\frac{y}{y% -1}\right),$
For quadratic transformations of Appell functions see Carlson (1976).
##### 4: 16.1 Special Notation
The main functions treated in this chapter are the generalized hypergeometric function ${{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$, the Appell (two-variable hypergeometric) functions ${F_{1}}\left(\alpha;\beta,\beta^{\prime};\gamma;x,y\right)$, ${F_{2}}\left(\alpha;\beta,\beta^{\prime};\gamma,\gamma^{\prime};x,y\right)$, ${F_{3}}\left(\alpha,\alpha^{\prime};\beta,\beta^{\prime};\gamma;x,y\right)$, ${F_{4}}\left(\alpha,\beta;\gamma,\gamma^{\prime};x,y\right)$, and the Meijer $G$-function ${G^{m,n}_{p,q}}\left(z;{a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}}\right)$. Alternative notations are ${{}_{p}F_{q}}\left({\mathbf{a}\atop\mathbf{b}};z\right)$, ${{}_{p}F_{q}}\left(a_{1},\dots,a_{p};b_{1},\dots,b_{q};z\right)$, and ${{}_{p}F_{q}}\left(\mathbf{a};\mathbf{b};z\right)$ for the generalized hypergeometric function, $F_{1}(\alpha,\beta,\beta^{\prime};\gamma;x,y)$, $F_{2}(\alpha,\beta,\beta^{\prime};\gamma,\gamma^{\prime};x,y)$, $F_{3}(\alpha,\alpha^{\prime},\beta,\beta^{\prime};\gamma;x,y)$, $F_{4}(\alpha,\beta;\gamma,\gamma^{\prime};x,y)$, for the Appell functions, and ${G^{m,n}_{p,q}}\left(z;\mathbf{a};\mathbf{b}\right)$ for the Meijer $G$-function.
##### 5: 16.15 Integral Representations and Integrals
###### §16.15 Integral Representations and Integrals
16.15.3 ${F_{3}}\left(\alpha,\alpha^{\prime};\beta,\beta^{\prime};\gamma;x,y\right)=% \frac{\Gamma\left(\gamma\right)}{\Gamma\left(\beta\right)\Gamma\left(\beta^{% \prime}\right)\Gamma\left(\gamma-\beta-\beta^{\prime}\right)}\iint_{\Delta}% \frac{u^{\beta-1}v^{\beta^{\prime}-1}(1-u-v)^{\gamma-\beta-\beta^{\prime}-1}}{% (1-ux)^{\alpha}(1-vy)^{\alpha^{\prime}}}\,\mathrm{d}u\,\mathrm{d}v,$ $\Re\left(\gamma-\beta-\beta^{\prime}\right)>0$, $\Re\beta>0$, $\Re\beta^{\prime}>0$,
These representations can be used to derive analytic continuations of the Appell functions, including convergent series expansions for large $x$, large $y$, or both. For inverse Laplace transforms of Appell functions see Prudnikov et al. (1992b, §3.40).
##### 6: Bibliography L
• A. Laforgia (1979) On the Zeros of the Derivative of Bessel Functions of Second Kind. Pubblicazioni Serie III [Publication Series III], Vol. 179, Istituto per le Applicazioni del Calcolo “Mauro Picone” (IAC), Rome.
• S. Lang (1987) Elliptic Functions. 2nd edition, Graduate Texts in Mathematics, Vol. 112, Springer-Verlag, New York.
• D. J. Leeming (1989) The real zeros of the Bernoulli polynomials. J. Approx. Theory 58 (2), pp. 124–150.
• J. Lepowsky and S. Milne (1978) Lie algebraic approaches to classical partition identities. Adv. in Math. 29 (1), pp. 15–59.
• Y. L. Luke (1968) Approximations for elliptic integrals. Math. Comp. 22 (103), pp. 627–634.
• ##### 7: Bibliography W
• Z. Wang and R. Wong (2006) Uniform asymptotics of the Stieltjes-Wigert polynomials via the Riemann-Hilbert approach. J. Math. Pures Appl. (9) 85 (5), pp. 698–718.
• R. J. Wells (1999) Rapid approximation to the Voigt/Faddeeva function and its derivatives. J. Quant. Spect. and Rad. Transfer 62 (1), pp. 29–48.
• E. J. Weniger (2003) A rational approximant for the digamma function. Numer. Algorithms 33 (1-4), pp. 499–507.
• E. P. Wigner (1959) Group Theory and its Application to the Quantum Mechanics of Atomic Spectra. Pure and Applied Physics. Vol. 5, Academic Press, New York.
• R. Wong and Y. Zhao (2002a) Exponential asymptotics of the Mittag-Leffler function. Constr. Approx. 18 (3), pp. 355–385.
• ##### 8: Bibliography
• G. E. Andrews, I. P. Goulden, and D. M. Jackson (1986) Shanks’ convergence acceleration transform, Padé approximants and partitions. J. Combin. Theory Ser. A 43 (1), pp. 70–84.
• G. E. Andrews (1972) Summations and transformations for basic Appell series. J. London Math. Soc. (2) 4, pp. 618–622.
• G. E. Andrews (1996) Pfaff’s method II: Diverse applications. J. Comput. Appl. Math. 68 (1-2), pp. 15–23.
• H. Appel (1968) Numerical Tables for Angular Correlation Computations in $\alpha$-, $\beta$- and $\gamma$-Spectroscopy: $3j$-, $6j$-, $9j$-Symbols, F- and $\Gamma$-Coefficients. Landolt-Börnstein Numerical Data and Functional Relationships in Science and Technology, Springer-Verlag.
• P. Appell and J. Kampé de Fériet (1926) Fonctions hypergéométriques et hypersphériques. Polynomes d’Hermite. Gauthier-Villars, Paris.
• ##### 9: 5.23 Approximations
###### §5.23(i) Rational Approximations
For additional approximations see Hart et al. (1968, Appendix B), Luke (1975, pp. 22–23), and Weniger (2003).
##### 10: Christopher J. Howls
is Senior Lecturer in Applied Mathematics at the University of Southampton, U. … Howls has published numerous papers in the areas of asymptotics and its applications. …Institute of Mathematics and Its Applications in 2007. Since 2006 he has served as Chair of the Standing Committee of the British Applied Mathematics Colloquium. In 2008 he was appointed to the editorial board of the Proceedings of the Royal Society of London Series A. …