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Lecture Note Sketches Hermann Riecke - ESAM Home Page

Lecture Note Sketches Hermann Riecke - ESAM Home Page

a) R 2 R 2 K 2 b) Ω

a) R 2 R 2 K 2 b) Ω Figure 9: Bifurcation diagrams arising from cuts at fixed Ω (a) and at fixed K (b) in the phase diagram shown in Fig.8. 2 Asymptotic Evaluation of Integrals 2 Motivation: Many functions and solutions of differential equations are given or can be written in terms of definite integrals that depend on additional parameters. E.g. modified Bessel function K0(x) = e −x � ∞ 0 � t 2 + 2t � −1/2 e −xt dt To get analytical insight into the behavior of such a complicated solution it is often useful to extract the behavior of that solution in a limiting case like large argument. Often one could analyse the differential equation leading to that special function in that limiting case. But then not all boundary conditions are exploited and one looses information (prefactors). If one has an integral representation of that function one can analyze that in the limit, which retains the full boundary information. E.g. y ′ = xy + 1 y(0) = 0 For large x one can get the approximate solution via y ′ = xy d 1 ln y = x ⇒ y = y0e 2 dx x2 To obtain the prefactor y0 one needs to use the initial condition. However, the approximate solution is only valid for x → ∞ and not for x = 0. 2 This chapter follows quite closely Bender & Orszag. 42

Here we can get an integral expression for the solution using an integrating factor Using the initial condition y(0) = 0 we get d � e dx −1 2x2 � y = e −1 2x2 y(x) = e 1 2 x2 � x 0 e −1 2t2 dt From this integral one can get the asymptotic behavior of y(x) for x → ∞ including the prefactor y0. 2.1 Elementary Examples Consider I(x) = For large x one can easily approximate � 2 0 I(x) → � 1 � 2 3 cos t + t x � 2 0 � 1 4 cos [0] dt = 2 Not alway one can interchange the limit with the integral! If f(x, t) is asymptotic to f0(t) for x → x0, f(x, t) ∼ f0(t) x → x0 uniformly in the integration interval [a, b] then � dt � b � b I(x) = f(x, t)dt ∼ f0(t)dt if the integral on the r.h.s. is finite and nonzero. a In fact, if f(x, t) is given in terms of a series that is asymptotic to f(x, t) uniformly in [a, b] f(x, t) ∼ ∞� fn(t) (x − x0) αn n=0 then the integration can be performed term by term Notes: � b (f(x, t) dt ∼ a a x → x0, α > 0 ∞� (x − x0) αn � b fn(t) dt x → x0 n=0 43 a

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