∞ summationdisplay n = −∞ F n e i k n x ; F m = 1 L integraldisplay L/ 2 − L/ 2 d x e − i k m x f ( x ) . (C . 5) At this stage it proves expedient to replace the F n with rescaled coefficients tildewide f ( k n ) ≡ LF n . (C . 6) so our equations become f ( x ) = ∞ summationdisplay n = −∞ 1 L tildewide f ( k n ) e i k n x ; tildewide f ( k m ) = integraldisplay L/ 2 − L/ 2 d x e − i k m x f ( x ) . (C . 7) 1 After dropping out from a seminary Joseph Fourier (1768–1830) joined the Auxerre Revolutionary Committee. The Revolution’s fratricidal violence led to his arrest but he avoided the guillotine by virtue of Robespierre’s fall in 1794. He invented Fourier series while serving Napoleon as Prefect of Grenoble. His former teachers Laplace and Lagrange were not convinced. 2 Marshall Stone strengthened a theorem proved by Karl Weierstrass in 1885. 3 You can check that the integration can be over any interval of length L . We have chosen the interval ( − 1 2 L, 1 2 L ) for later convenience.
Operators in classical statistical mechanics 289 Now we eliminate L from the first equation in favour of the difference d k ≡ k n +1 − k n = 2 π/L and have f ( x ) = ∞ summationdisplay n = −∞ d k 2 π tildewide f ( k n ) e i k n x ; tildewide f ( k m ) = integraldisplay L/ 2 − L/ 2 d x e − i k m x f ( x ) . (C . 8) Finally we imagine the period getting longer and longer without limit. As L grows the difference d k between successive values of k n becomes smaller and smaller, so k n becomes a continuous variable k , and the sum in the first equation of (C.8) becomes an integral. Hence in the limit of infinite L we are left with f ( x ) = integraldisplay ∞ −∞ d k 2 π tildewide f ( k ) e i kx ; tildewide f ( k ) = integraldisplay ∞ −∞ d x e − i kx f ( x ) . (C . 9) These are the basic formulae of Fourier transforms. The original restriction to periodic functions has been lifted because any function can be considered to repeat itself after an infinite interval. The only restriction on f for these formulae to be valid is that it vanishes sufficiently fast at infinity for the integral in the second of equations (C.9) to converge: the requirement proves to be that integraltext ∞ −∞ d x | f | 2 exists, which requires that asymptotically | f | < | x | − 1 / 2 . Physicists generally don’t worry too much about this restriction. Using the second of equations (C.9) to eliminate the Fourier transform tildewide f from the first equation, we have f ( x ) = integraldisplay ∞ −∞ d k 2 π integraldisplay ∞ −∞ d x ′ e i k ( x − x ′ ) f ( x ′ ) . (C . 10) Mathematicians stop here because our next step is illegal. 4 Physicists reverse the order of the integrations in equation (C.10) and write f ( x ) = integraldisplay ∞ −∞ d x ′ f ( x ′ ) integraldisplay ∞ −∞ d k 2 π e i k ( x − x ′ ) . (C . 11) Comparing this equation with equation (2.41) we see that the inner integral on the right satisfies the defining condition of the Dirac delta function, and we have δ ( x − x ′ ) = integraldisplay ∞ −∞ d k 2 π e i k ( x − x ′ ) . (C . 12) Appendix D: Operators in classical statistical mechanics