# turning points

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##### 1: 33.23 Methods of Computation
Inside the turning points, that is, when $\rho<\rho_{\operatorname{tp}}\left(\eta,\ell\right)$, there can be a loss of precision by a factor of approximately $|G_{\ell}|^{2}$. … WKBJ approximations (§2.7(iii)) for $\rho>\rho_{\operatorname{tp}}\left(\eta,\ell\right)$ are presented in Hull and Breit (1959) and Seaton and Peach (1962: in Eq. … Hull and Breit (1959) and Barnett (1981b) give WKBJ approximations for $F_{0}$ and $G_{0}$ in the region inside the turning point: $\rho<\rho_{\operatorname{tp}}\left(\eta,\ell\right)$.
##### 2: 33.3 Graphics Figure 33.3.3: F ℓ ⁡ ( η , ρ ) , G ℓ ⁡ ( η , ρ ) with ℓ = 0 , η = 2 . The turning point is at ρ tp ⁡ ( 2 , 0 ) = 4 . Magnify Figure 33.3.4: F ℓ ⁡ ( η , ρ ) , G ℓ ⁡ ( η , ρ ) with ℓ = 0 , η = 10 . The turning point is at ρ tp ⁡ ( 10 , 0 ) = 20 . Magnify Figure 33.3.5: F ℓ ⁡ ( η , ρ ) , G ℓ ⁡ ( η , ρ ) , and M ℓ ⁡ ( η , ρ ) with ℓ = 0 , η = 15 / 2 . The turning point is at ρ tp ⁡ ( 15 / 2 , 0 ) = 30 = 5.47 ⁢ … . Magnify Figure 33.3.6: F ℓ ⁡ ( η , ρ ) , G ℓ ⁡ ( η , ρ ) , and M ℓ ⁡ ( η , ρ ) with ℓ = 5 , η = 0 . The turning point is at ρ tp ⁡ ( 0 , 5 ) = 30 (as in Figure 33.3.5). Magnify
##### 3: 9.15 Mathematical Applications
Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turning point. …
##### 4: 33.14 Definitions and Basic Properties
###### §33.14(i) Coulomb Wave Equation
When $\epsilon>0$ the outer turning point is given by
33.14.3 $r_{\operatorname{tp}}\left(\epsilon,\ell\right)=\left(\sqrt{1+\epsilon\ell(% \ell+1)}-1\right)\Bigm{/}\epsilon;$
##### 5: 33.2 Definitions and Basic Properties
###### §33.2(i) Coulomb Wave Equation
There are two turning points, that is, points at which $\ifrac{{\mathrm{d}}^{2}w}{{\mathrm{d}\rho}^{2}}=0$2.8(i)). …
33.2.2 $\rho_{\operatorname{tp}}\left(\eta,\ell\right)=\eta+(\eta^{2}+\ell(\ell+1))^{1% /2}.$
##### 6: 10.72 Mathematical Applications
###### Simple TurningPoints
These expansions are uniform with respect to $z$, including the turning point $z_{0}$ and its neighborhood, and the region of validity often includes cut neighborhoods (§1.10(vi)) of other singularities of the differential equation, especially irregular singularities. …
##### 7: 12.16 Mathematical Applications
PCFs are used as basic approximating functions in the theory of contour integrals with a coalescing saddle point and an algebraic singularity, and in the theory of differential equations with two coalescing turning points; see §§2.4(vi) and 2.8(vi). …
##### 8: 13.27 Mathematical Applications
For applications of Whittaker functions to the uniform asymptotic theory of differential equations with a coalescing turning point and simple pole see §§2.8(vi) and 18.15(i). …
##### 9: 9.16 Physical Applications
The frequent appearances of the Airy functions in both classical and quantum physics is associated with wave equations with turning points, for which asymptotic (WKBJ) solutions are exponential on one side and oscillatory on the other. The Airy functions constitute uniform approximations whose region of validity includes the turning point and its neighborhood. … This reference provides several examples of applications to problems in quantum mechanics in which Airy functions give uniform asymptotic approximations, valid in the neighborhood of a turning point. …
##### 10: 33.12 Asymptotic Expansions for Large $\eta$
When $\ell=0$ and $\eta>0$, the outer turning point is given by $\rho_{\operatorname{tp}}\left(\eta,0\right)=2\eta$; compare (33.2.2). …