Principles of Mathematics in Operations Research
176 13 Power Series and Special Functions Proposition 13.1.2 (Cauchy's criterion) The series oo k-0 is convergent if and only if ...
13.1 Series 177 13.1.2 Operations on Series Proposition 13.1.6 IfJ2T=o uk and SfcLo Vk are convergent series and a € C, then the ...
178 13 Power Series and Special Functions (a) We have u U\ U2 U"1 u-2 «nn w nil n = "o^1 U Vra = 0,1,2,... 0 U\ Un_! < q < ...
13.3 Power Series Remark 13.2.2 Every uniformly convergent sequence is pointwise conver- gent. If (/„) converges pointwise on E, ...
180 13 Power Series and Special Functions Theorem 13.3.2 Suppose the series J2^=ocnxn converges for \x\ < R, and define oo f( ...
13.4 Exponential and Logarithmic Functions 181 (^00) yn °° ,.,m °° n yk..,n-k E(z)E(w)v ; v ' ^ n\ ^ = E^E^ TO! ^ ^ = EE kUn - k ...
182 13 Power Series and Special Functions x = 0 => L{1) — 0. Thus, we have rv dx L(y) = — = logy- Ji x Let u — E(x), v = E(y) ...
13.5 Trigonometric Functions 183 We assert that there exists positive numbers x such that C(x) = 0. Let XQ be the smallest among ...
184 13 Power Series and Special Functions The right hand side of the above inequality tends to oo as R —)• oo. Hence, 3i?o B \P( ...
13.7 Gamma Function Definition 13.6.3 A trigonometric series is a series of the form oo f(x) = Y, c " einx > xeR - — oo If f ...
186 13 Power Series and Special Functions Remark 13.7.6 Let t = s^2 in the definition of T. />oo r(x) = 2 s2x-le-a2 ds, 0 < ...
13.7 Problems 187 of solutions in a general instance r. Use generating functions to a) Prove the binomial theorem u«>-£ (I) i ...
188 13 Power Series and Special Functions our fourth objective function (linear!) is min _3z wz _3i @iz (0"')> where (^0) iz ...
13.8 Web material 189 http://planetmath.org/encyclopedia/PowerSeries.html http://planetmath.org/encyclopedia/SlowerDivergentSeri ...
14 Special Transformations In functional analysis, the Laplace transform is a powerful technique for ana- lyzing linear time-inv ...
192 14 Special Transformations homogeneous, otherwise it is non-homogeneous. If we assume 0 6/, and 2/(0) = 2/o, y'(0) = y'o,... ...
14.2 Laplace Transforms 193 Proposition 14.2.2 //J/:RHR satisfies (i) y{t) = 0fort<0, (ii) y(t) is piecewise continuous, (Hi) ...
194 14 Special Transformations y(t)^V(s). Solve the resulting linear algebraic equation, call the solution n(s) the formal Lapl ...
14.2 Laplace Transforms 195 // we relax y'{t) = f(t), then we have (^1 1) V(s) = Vo + s — a (s) and y(t) = eaty 0 + ...
196 14 Special Transformations Then, the unique solution is y(t) = etAyo +p(t), where etAyo 2e* + e" 2e< - e~ and p(t) = / 0 ...
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