Proving Continuous Function: Expectation Values of Periodic Functions Over Time

In summary, the conversation discusses the need to prove the continuity of the expectation value of a periodic function over time, specifically the exponential of the angle. The formula provided includes a complex exponential factor and a Fourier series with Fourier coefficients. The suggested approach is to use an ordinary ##\varepsilon-\delta## proof, possibly with the complex exponential factor taken out of the sum.
  • #1
thegaussian
8
0
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!
 
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  • #2
An ordinary ##\varepsilon-\delta## prove should do. You can even take ##\exp(- i t)## out of the sum.
 

FAQ: Proving Continuous Function: Expectation Values of Periodic Functions Over Time

What is a continuous function?

A continuous function is a mathematical function that has no sudden jumps or breaks in its graph. This means that the output of the function changes gradually as the input changes.

How do you prove that a function is continuous?

To prove that a function is continuous, you need to show that it satisfies the three criteria for continuity: 1) the function is defined at the point in question, 2) the limit of the function as the input approaches the point exists, and 3) the limit is equal to the function value at that point.

What are periodic functions?

Periodic functions are mathematical functions that repeat their values at regular intervals. This means that the function will have the same output for certain inputs, and then repeat that same output after a certain interval.

How do you calculate the expectation value of a periodic function over time?

To calculate the expectation value of a periodic function over time, you need to integrate the function over one period and then divide by the length of the period. This will give you the average value of the function over time.

Why are expectation values important in the study of periodic functions over time?

Expectation values are important because they provide a way to quantify the average behavior of a periodic function over time. This can help us understand the overall trend or behavior of the function, rather than just looking at specific points or intervals.

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