jeffwas has explained the factor of 2<i>pi. It&#x27;s just the 2</i>pi radians in a circle to convert from angular frequencies to regular frequencies.<p>I don&#x27;t know the detailed history, but in practice the fact that it&#x27;s an inverse is essentially because it&#x27;s most important role is as part of an expansion like a Taylor series.  The standard method for calculating lots of observable effects gives an answer with the form<p>X = X_0 + \alpha<i>X_1 + \alpha^2</i>X_2&#x2F;2! + ...