# §15.12 Asymptotic Approximations

## §15.12(i) Large Variable

For the asymptotic behavior of as with , , fixed, combine (15.2.2) with (15.8.2) or (15.8.8).

## §15.12(ii) Large

Let denote an arbitrary small positive constant. Also let be real or complex and fixed, and at least one of the following conditions be satisfied:

Then for fixed ,

Similar results for other sectors are given in Wagner (1988). For the more general case in which and see Wagner (1990).

## §15.12(iii) Other Large Parameters

Again, throughout this subsection denotes an arbitrary small positive constant, and are real or complex and fixed.

If , then as with ,

where

For see §10.25(ii). For this result and an extension to an asymptotic expansion with error bounds see Jones (2001).

See also Dunster (1999) where the asymptotics of Jacobi polynomials is described; compare (15.9.1).

If , then as with ,

where

with the branch chosen to be continuous and when . For see §12.2, and for an extension to an asymptotic expansion see Olde Daalhuis (2003a).

If , then as with ,

where

with the branch chosen to be continuous and when . Also,

15.12.12

where

For see §9.2, and for further information and an extension to an asymptotic expansion see Olde Daalhuis (2003b). (Two errors in this reference are corrected in (15.12.9).)

By combination of the foregoing results of this subsection with the linear transformations of §15.8(i) and the connection formulas of §15.10(ii), similar asymptotic approximations for can be obtained with or 0, . For more details see Olde Daalhuis (2010). For other extensions, see Wagner (1986) and Temme (2003).