Download A course in mathematical physics. Classical dynamical by Walter E Thirring PDF

By Walter E Thirring

Mathematical Physics, Nat. Sciences, Physics, arithmetic

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Additional info for A course in mathematical physics. Classical dynamical systems

Example text

43) the cosine series converges to f (0) at x = 0 and to f (π) at x = π. The situation is different for the sine series, however. 41) is zero; thus, the series is equal to the functions f (0) and f (π), respectively, only when these values are zero. 46) f (x) sin nπx dx. 47) 0 or l 2 bn = l 0 If we then wish to extend these expansions to the entire axis, −∞ < x < +∞, it is necessary in both cases to consider the interval (−l, l), which can be expanded to the entire x-axis with period T = 2l. For example, suppose we have the function f (x) = 2x + 1 on the interval [0, 1].

85) a or more commonly (because it is closer to the method of least squares), the total square deviation b r 2 (x)dx. 86) a Here, r(x) = f (x) − σn (x), and n γk ϕk (x) σn (x) = γ1 ϕ1 (x) + γ2 ϕ2 (x) + . . 87) k=1 ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ 46 1. Fourier Series is a linear combination the first n functions from the infinite set {ϕn (x)} (usually the contribution of the states decreases when n → ∞). 87) that gives the best total square deviation; that is, we want to minimize the value b 2 f (x) − σn (x) ∆n = dx.

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