Is 0.5sin2x the Fourier Series of sinx*cosx?

In summary, a Fourier Series is a mathematical representation of a periodic function as an infinite sum of sinusoidal functions. It is calculated using the Fourier coefficients, which are determined by integrating the function over one period and solving for the coefficients. The formula for the Fourier Series of sinx*cosx is 0.5sin2x. However, a Fourier Series can only accurately represent periodic functions and is unique for a given function, but different functions can have the same Fourier Series if they have the same values over one period.
  • #1
fiziksfun
78
0
My professor said the Fourier series of sinx*cosx is simply 0.5sin2x.

Is this right?? Is there no summation??

I don't understand why .. Help!
 
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  • #2
She's right, by uniqueness of Fourier series.

Trig identity for sin2x shows that sin x*cos x=0.5sin2x.

This is the form of a Fourier series, hence the unique. (All the other infinitely many terms are 0.)
 

1. What is a Fourier Series?

A Fourier Series is a mathematical representation of a periodic function as an infinite sum of sinusoidal functions.

2. How is a Fourier Series calculated?

A Fourier Series is calculated using the Fourier coefficients, which are determined by integrating the function over one period and solving for the coefficients.

3. What is the formula for the Fourier Series of sinx*cosx?

The formula for the Fourier Series of sinx*cosx is 0.5sin2x. This is because the function can be written as a sum of two sinusoidal functions with different frequencies, and the coefficients are determined by the product of the two frequencies.

4. Can a Fourier Series accurately represent any function?

No, a Fourier Series can only accurately represent periodic functions. If a function is not periodic, the Fourier Series will have an infinite number of terms and will not converge.

5. Is the Fourier Series unique for a given function?

Yes, the Fourier Series is unique for a given function. However, different functions can have the same Fourier Series if they have the same values over one period.

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