By Mats Gyllenberg, Lars-Erik Persson
Offering the lawsuits of the twenty-first Nordic Congress of Mathematicians at Luleå college of know-how, Sweden, this striking reference discusses contemporary advances in research, algebra, stochastic procedures, and using pcs in mathematical study.
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Extra info for Analysis, algebra, and computers in mathematical research: proceedings of the Twenty-first Nordic Congress of Mathematicians
J. J. solutions originates from an appropriate change of the dependent variable. The given equation is where the parameter h is taken to be large and positive, and where the function q(z, h) tends to a limit as h → ∞ for fixed complex z. A prime denotes differentiation with respect to z. If we let with a constant lower limit of integration, then w′ = ø exp (∫ ødz) and w˝ = ø2 exp (∫ ødz) + ø′ exp(∫ ødz), yielding ø2 + ø′ + h2 q = 0. , where each øj(z, h) tends to a limit as h → ∞ for fixed α, then substitution yields as far as terms in h2, h, h0 are concerned.
Since many problems that arise in theoretical physics lead to differential equations of this type, attempts must be made to produce satisfactory methods of solution. If p and q are specially chosen, the various standard transcendental differential equations are produced. For example, if p = 1/ x and q = 1 − v2/ x2, we obtain Bessel's equation of order v; if p = 0 and q = − x, we obtain the Airy equation. These special cases are singled out because exact solutions are readily obtainable in terms of contour integrals, from which the power-series solutions in ascending powers of x and the asymptotic solutions in descending powers of large x may be obtained.
Solution. In other words, we have traced the solution into a domain in which the original approximate representation is no longer valid. Boundaries across which such a change takes place are known as anti-Stokes lines; within such boundaries, the solution is uniformly asymptotic. J. 11). However, he did not elucidate the question of the boundaries in a systematic way. Olver  (1959) examined the equation with two transition points whose exact solutions are expressed in terms of Weber functions.
Analysis, algebra, and computers in mathematical research: proceedings of the Twenty-first Nordic Congress of Mathematicians by Mats Gyllenberg, Lars-Erik Persson
Categories: Differential Equations