By Jens Lorenz

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The operator R then multiplies the velocity vectors by −1. Time reversibility holds in the same way for n bodies, instead of two, as long as gravitational attraction is the only law of interaction. In fact, time reversibility holds more generally in micro physics. This goes under the name CPT symmetry: The dynamical equations of the standard model are symmetric under time reversal if one also applies parity and charge conjugation. See, for example, [6]. 3. t. an appropriate operator R. ) A natural question, then, is how irreversibility can arise, which we observe in the macro world.

17) if one deﬁnes the Bessel functions via the generating function ∞ ∑ 1 ) Jn (z)tn . 19) Here z and t are complex numbers, t ̸= 0. For ﬁxed z, the function t → g(z, t) is analytic in C \ {0} and, therefore, has a unique Laurent expansion about t = 0. The Bessel functions Jn (z) are deﬁned as the coeﬃcients. Note that for z and t real, the value g(z, t) is real. This implies that Jn (z) is real for real z. 1. Let f (t) = ∞ n=−∞ cn t denote a function which is analytic for t ̸= 0 and which is real for real t.

This implies that Jn (z) is real for real z. 1. Let f (t) = ∞ n=−∞ cn t denote a function which is analytic for t ̸= 0 and which is real for real t. Then the Laurent coeﬃcients cn are all real. ∑ n Proof: Set g(t) = c¯n t where c¯n denotes the complex conjugate of cn . Then g(t) is also analytic for t ̸= 0. Furthermore, for real t, f (t) = f¯(t) = g(t) . The identity theorem for analytic functions yields that f and g are identical. Uniqueness of the Laurent coeﬃcients cn implies that cn = c¯n is real.