# second order

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## 1—10 of 149 matching pages

##### 1: 10.45 Functions of Imaginary Order
and $\widetilde{I}_{\nu}\left(x\right)$, $\widetilde{K}_{\nu}\left(x\right)$ are real and linearly independent solutions of (10.45.1): … The corresponding result for $\widetilde{K}_{\nu}\left(x\right)$ is given by … In consequence of (10.45.5)–(10.45.7), $\widetilde{I}_{\nu}\left(x\right)$ and $\widetilde{K}_{\nu}\left(x\right)$ comprise a numerically satisfactory pair of solutions of (10.45.1) when $x$ is large, and either $\widetilde{I}_{\nu}\left(x\right)$ and $(1/\pi)\sinh\left(\pi\nu\right)\widetilde{K}_{\nu}\left(x\right)$, or $\widetilde{I}_{\nu}\left(x\right)$ and $\widetilde{K}_{\nu}\left(x\right)$, comprise a numerically satisfactory pair when $x$ is small, depending whether $\nu\neq 0$ or $\nu=0$. … For graphs of $\widetilde{I}_{\nu}\left(x\right)$ and $\widetilde{K}_{\nu}\left(x\right)$ see §10.26(iii). For properties of $\widetilde{I}_{\nu}\left(x\right)$ and $\widetilde{K}_{\nu}\left(x\right)$, including uniform asymptotic expansions for large $\nu$ and zeros, see Dunster (1990a). …
##### 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: Bibliography O
• A. B. Olde Daalhuis (1995) Hyperasymptotic solutions of second-order linear differential equations. II. Methods Appl. Anal. 2 (2), pp. 198–211.
• F. W. J. Olver (1967a) Numerical solution of second-order linear difference equations. J. Res. Nat. Bur. Standards Sect. B 71B, pp. 111–129.
• F. W. J. Olver (1967b) Bounds for the solutions of second-order linear difference equations. J. Res. Nat. Bur. Standards Sect. B 71B (4), pp. 161–166.
• F. W. J. Olver (1975a) Second-order linear differential equations with two turning points. Philos. Trans. Roy. Soc. London Ser. A 278, pp. 137–174.
• F. W. J. Olver (1977c) Second-order differential equations with fractional transition points. Trans. Amer. Math. Soc. 226, pp. 227–241.
• ##### 5: 10.74 Methods of Computation
For the computation of the functions $\widetilde{I}_{\nu}\left(x\right)$ and $\widetilde{K}_{\nu}\left(x\right)$ defined by (10.45.2) see Temme (1994c) and Gil et al. (2002d, 2003a, 2004b).
##### 6: Mark J. Ablowitz
ODEs with the Painlevé property contain the well-known Painlevé equations which are special second order scalar equations; their solutions are often called Painlevé transcendents. …
##### 8: Bibliography T
• E. C. Titchmarsh (1946) Eigenfunction Expansions Associated with Second-Order Differential Equations. Clarendon Press, Oxford.
• E. C. Titchmarsh (1958) Eigenfunction Expansions Associated with Second Order Differential Equations, Part 2, Partial Differential Equations. Clarendon Press, Oxford.
• E. C. Titchmarsh (1962a) Eigenfunction expansions associated with second-order differential equations. Part I. Second edition, Clarendon Press, Oxford.
• S. A. Tumarkin (1959) Asymptotic solution of a linear nonhomogeneous second order differential equation with a transition point and its application to the computations of toroidal shells and propeller blades. J. Appl. Math. Mech. 23, pp. 1549–1565.
• ##### 9: 31.14 General Fuchsian Equation
The general second-order Fuchsian equation with $N+1$ regular singularities at $z=a_{j}$, $j=1,2,\dots,N$, and at $\infty$, is given by … An algorithm given in Kovacic (1986) determines if a given (not necessarily Fuchsian) second-order homogeneous linear differential equation with rational coefficients has solutions expressible in finite terms (Liouvillean solutions). …
##### 10: Bibliography D
• D. Ding (2000) A simplified algorithm for the second-order sound fields. J. Acoust. Soc. Amer. 108 (6), pp. 2759–2764.
• T. M. Dunster, D. A. Lutz, and R. Schäfke (1993) Convergent Liouville-Green expansions for second-order linear differential equations, with an application to Bessel functions. Proc. Roy. Soc. London Ser. A 440, pp. 37–54.
• T. M. Dunster (1990b) Uniform asymptotic solutions of second-order linear differential equations having a double pole with complex exponent and a coalescing turning point. SIAM J. Math. Anal. 21 (6), pp. 1594–1618.
• T. M. Dunster (1996a) Asymptotic solutions of second-order linear differential equations having almost coalescent turning points, with an application to the incomplete gamma function. Proc. Roy. Soc. London Ser. A 452, pp. 1331–1349.
• A. J. Durán and F. A. Grünbaum (2005) A survey on orthogonal matrix polynomials satisfying second order differential equations. J. Comput. Appl. Math. 178 (1-2), pp. 169–190.