# large variable and/or large parameter

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## 11—20 of 118 matching pages

##### 11: 12.9 Asymptotic Expansions for Large Variable
###### §12.9 Asymptotic Expansions for LargeVariable
12.9.1 $U\left(a,z\right)\sim e^{-\frac{1}{4}z^{2}}z^{-a-\frac{1}{2}}\sum_{s=0}^{% \infty}(-1)^{s}\frac{{\left(\frac{1}{2}+a\right)_{2s}}}{s!(2z^{2})^{s}},$ $|\operatorname{ph}z|\leq\tfrac{3}{4}\pi-\delta(<\tfrac{3}{4}\pi)$ ,
12.9.2 $V\left(a,z\right)\sim\sqrt{\frac{2}{\pi}}e^{\frac{1}{4}z^{2}}z^{a-\frac{1}{2}}% \sum_{s=0}^{\infty}\frac{{\left(\frac{1}{2}-a\right)_{2s}}}{s!(2z^{2})^{s}},$ $|\operatorname{ph}z|\leq\tfrac{1}{4}\pi-\delta(<\tfrac{1}{4}\pi)$ .
12.9.4 $V\left(a,z\right)\sim\sqrt{\frac{2}{\pi}}e^{\frac{1}{4}z^{2}}z^{a-\frac{1}{2}}% \sum_{s=0}^{\infty}\frac{{\left(\tfrac{1}{2}-a\right)_{2s}}}{s!(2z^{2})^{s}}% \pm\frac{i}{\Gamma\left(\tfrac{1}{2}-a\right)}e^{-\frac{1}{4}z^{2}}z^{-a-\frac% {1}{2}}\sum_{s=0}^{\infty}(-1)^{s}\frac{{\left(\tfrac{1}{2}+a\right)_{2s}}}{s!% (2z^{2})^{s}},$ $-\tfrac{1}{4}\pi+\delta\leq\pm\operatorname{ph}z\leq\tfrac{3}{4}\pi-\delta$.
##### 12: 16.22 Asymptotic Expansions
For asymptotic expansions of Meijer $G$-functions with large parameters see Fields (1973, 1983).
##### 14: 28.8 Asymptotic Expansions for Large $q$
###### Barrett’s Expansions
The approximations apply when the parameters $a$ and $q$ are real and large, and are uniform with respect to various regions in the $z$-plane. …
##### 17: Bibliography U
• F. Ursell (1972) Integrals with a large parameter. Several nearly coincident saddle-points. Proc. Cambridge Philos. Soc. 72, pp. 49–65.
• F. Ursell (1980) Integrals with a large parameter: A double complex integral with four nearly coincident saddle-points. Math. Proc. Cambridge Philos. Soc. 87 (2), pp. 249–273.
• F. Ursell (1984) Integrals with a large parameter: Legendre functions of large degree and fixed order. Math. Proc. Cambridge Philos. Soc. 95 (2), pp. 367–380.
##### 20: Bibliography L
• R. E. Langer (1934) The solutions of the Mathieu equation with a complex variable and at least one parameter large. Trans. Amer. Math. Soc. 36 (3), pp. 637–695.
• J. L. López, P. Pagola, and E. Pérez Sinusía (2013a) Asymptotics of the first Appell function $F_{1}$ with large parameters II. Integral Transforms Spec. Funct. 24 (12), pp. 982–999.
• J. L. López, P. Pagola, and E. Pérez Sinusía (2013b) Asymptotics of the first Appell function $F_{1}$ with large parameters. Integral Transforms Spec. Funct. 24 (9), pp. 715–733.
• J. L. López (1999) Asymptotic expansions of the Whittaker functions for large order parameter. Methods Appl. Anal. 6 (2), pp. 249–256.