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21: Bibliography O
  • A. B. Olde Daalhuis and F. W. J. Olver (1994) Exponentially improved asymptotic solutions of ordinary differential equations. II Irregular singularities of rank one. Proc. Roy. Soc. London Ser. A 445, pp. 39–56.
  • A. B. Olde Daalhuis (1998a) Hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one. Proc. Roy. Soc. London Ser. A 454, pp. 1–29.
  • F. W. J. Olver and F. Stenger (1965) Error bounds for asymptotic solutions of second-order differential equations having an irregular singularity of arbitrary rank. J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2 (2), pp. 244–249.
  • F. W. J. Olver (1965) On the asymptotic solution of second-order differential equations having an irregular singularity of rank one, with an application to Whittaker functions. J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2 (2), pp. 225–243.
  • F. W. J. Olver (1997a) Asymptotic solutions of linear ordinary differential equations at an irregular singularity of rank unity. Methods Appl. Anal. 4 (4), pp. 375–403.
  • 22: 14.21 Definitions and Basic Properties
    §14.21(iii) Properties
    This includes, for example, the Wronskian relations (14.2.7)–(14.2.11); hypergeometric representations (14.3.6)–(14.3.10) and (14.3.15)–(14.3.20); results for integer orders (14.6.3)–(14.6.5), (14.6.7), (14.6.8), (14.7.6), (14.7.7), and (14.7.11)–(14.7.16); behavior at singularities (14.8.7)–(14.8.16); connection formulas (14.9.11)–(14.9.16); recurrence relations (14.10.3)–(14.10.7). …
    23: 15.17 Mathematical Applications
    The three singular points in Riemann’s differential equation (15.11.1) lead to an interesting Riemann sheet structure. …
    24: 31.6 Path-Multiplicative Solutions
    This denotes a set of solutions of (31.2.1) with the property that if we pass around a simple closed contour in the z -plane that encircles s 1 and s 2 once in the positive sense, but not the remaining finite singularity, then the solution is multiplied by a constant factor e 2 ν π i . …
    25: 14.8 Behavior at Singularities
    §14.8 Behavior at Singularities
    14.8.16 𝑸 n ( 1 / 2 ) μ ( x ) π 1 / 2 Γ ( μ + n + 1 2 ) n ! Γ ( μ n + 1 2 ) ( 2 x ) n + ( 1 / 2 ) , n = 1 , 2 , 3 , , μ n + 1 2 0 , 1 , 2 , .
    26: 31.4 Solutions Analytic at Two Singularities: Heun Functions
    §31.4 Solutions Analytic at Two Singularities: Heun Functions
    27: 36.6 Scaling Relations
    Indices for k -Scaling of Magnitude of Ψ K or Ψ ( U ) (Singularity Index)
    Table 36.6.1: Special cases of scaling exponents for cuspoids.
    singularity K β K γ 1 K γ 2 K γ 3 K γ K
    28: 29.2 Differential Equations
    This equation has regular singularities at the points 2 p K + ( 2 q + 1 ) i K , where p , q , and K , K are the complete elliptic integrals of the first kind with moduli k , k ( = ( 1 k 2 ) 1 / 2 ) , respectively; see §19.2(ii). In general, at each singularity each solution of (29.2.1) has a branch point (§2.7(i)). …
    Figure 29.2.1: z -plane: singularities × × × of Lamé’s equation.
    29: 31.2 Differential Equations
    This equation has regular singularities at 0 , 1 , a , , with corresponding exponents { 0 , 1 γ } , { 0 , 1 δ } , { 0 , 1 ϵ } , { α , β } , respectively (§2.7(i)). All other homogeneous linear differential equations of the second order having four regular singularities in the extended complex plane, { } , can be transformed into (31.2.1). The parameters play different roles: a is the singularity parameter; α , β , γ , δ , ϵ are exponent parameters; q is the accessory parameter. …
    30: 10.25 Definitions
    This equation is obtained from Bessel’s equation (10.2.1) on replacing z by ± i z , and it has the same kinds of singularities. … …