A practical approach to linear algebra by Choudhary P.

By Choudhary P.

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Determine Pad6 approximants for p , ( t ) et el. , and differentiate to obtain recurrence relations. + + + + 3. Hint: Uniformize by means of trignometric functions and count zeros and poles. 4. Determine Pad6 approximants for (1 + a,t a,t2 u $ ~ ) ‘ / Hint: ~. Uniformize by means of elliptic functions. For the foregoing and additional references, see R. Bellman and E. G. Straus, “Algebraic Functions, Continued Fractions and the Pad6 Table,” Proc. Nut. Acad. , Vol. 35, 1949, pp. 472476. + + 5. Consider the differential equation U” au’ bu = 0, a, b > 0.

Let us consider first the case u” + a2(t)u = 0. T h e equation with the minus sign is easier to consider. Exercises 1. Use the Liouville transformation to establish the following result of Gusarov: If u” ( I + f ( t ) u = 0 with 1 + f ( t ) >, a2 > 0, J“ lf”(t)\ dt < 00, then u is bounded as t + co. + 2. Hence, discuss the boundedness of the solutions of U“ for 0 + (1 + (cos t”)/tb) u = 0 < a, b < 1. See R. Bellman, “Boundedness of Solutions of Second Order Linear Differential Equations,” Duke Math.

Are the expansions convergent ? + 4. What is the eventual behavior of the solution of v' v 2 - 1 -f(t) = 0 with v(0) > 1 iff + 0 as t + 03 ? Hint: Consider the behavior of v in the (v, t)-plane. 20. 1) 0 appears to have a solution that approaches -1 as t + GO, provided that f -+ 0 as t -+ 03. We can establish this indirectly as above by using the properties of u p . Can we demonstrate the existence of this solution directly ? Set v = - 1 + w , obtaining the equation w' = 2w - w2 + f ( t ) . 3) we see that the presence of ept causes some trouble.

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