Green theorem
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1: 1.6 Vectors and Vector-Valued Functions
2: 2.7 Differential Equations
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►For irregular singularities of nonclassifiable rank, a powerful tool for finding the asymptotic behavior of solutions, complete with error bounds, is as follows:
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Liouville–Green Approximation Theorem
…3: 3.8 Nonlinear Equations
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►Initial approximations to the zeros can often be found from asymptotic or other approximations to , or by application of the phase principle or Rouché’s theorem; see §1.10(iv).
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►For describing the distribution of complex zeros of solutions of linear homogeneous second-order differential equations by methods based on the Liouville–Green (WKB) approximation, see Segura (2013).
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4: Bibliography G
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Superstring Theory: Introduction, Vol. 1.
2nd edition, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge.
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Superstring Theory: Loop Amplitudes, Anomalies and Phenomenolgy, Vol. 2.
2nd edition, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge.
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General form of the quantum-defect theory.
Phys. Rev. A 19 (4), pp. 1485–1509.
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Mathematics for the Analysis of Algorithms.
Progress in Computer Science, Vol. 1, Birkhäuser Boston, Boston, MA.
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Staircase polygons, elliptic integrals, Heun functions, and lattice Green functions.
Phys. Rev. E 47 (4), pp. R2233–R2236.
5: Bibliography D
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Note on the addition theorem of parabolic cylinder functions.
J. Indian Math. Soc. (N. S.) 4, pp. 29–30.
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Ramanujan’s master theorem for symmetric cones.
Pacific J. Math. 175 (2), pp. 447–490.
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On Vandermonde’s theorem, and some more general expansions.
Proc. Edinburgh Math. Soc. 25, pp. 114–132.
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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.
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