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Hi. I'm looking to at how expectation values of periodic functions evolve in time, and i need to prove that ##\exp ( i \theta )## is continuous in time (this is the expectation of the exponential of the angle).

My formula is:

##\exp( i \theta) = \exp ( i t /2) \sum_{n=-\infty}^{\infty} a_n a^*_{n-1} \exp (- i n t )##

where ##a_n## are the fourier coefficients of the initial function, ##*## represents the complex conjugate. Now how do I go about proving it's continuous? We have basically a complex exponential factor (that's obviously continuous) multiplied by a fourier series, but I just have no idea really where to go from there.

Any help would be much appreciated.

Thanks!

My formula is:

##\exp( i \theta) = \exp ( i t /2) \sum_{n=-\infty}^{\infty} a_n a^*_{n-1} \exp (- i n t )##

where ##a_n## are the fourier coefficients of the initial function, ##*## represents the complex conjugate. Now how do I go about proving it's continuous? We have basically a complex exponential factor (that's obviously continuous) multiplied by a fourier series, but I just have no idea really where to go from there.

Any help would be much appreciated.

Thanks!

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