Fourier series coefficients

In summary, the conversation discusses the use of complex identities and the fundamental theorem of calculus to solve a problem involving sin(pi*k/3). It is suggested to also use elemental algebra and complex analysis, and to multiply by 1 in the final answer to simplify the equation.
  • #1
lottotlyl
4
0
Homework Statement
how did my prof get the last term after the third equal sign
Relevant Equations
fourier series coeffecient equation
i tired using complex identity equation for sin(pi*k/3) but it doesn't work out
 

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  • #2
Applying the fundamental theorem of calculus which says that
$$\int_a^b f(t)dt=F(t)\Big|_a^b=F(b)-F(a)$$ where ##F(t)## is a function satisfying $$\frac{dF(t)}{dt}=f(t).$$
Then you also need to use elemental algebra and complex analysis.
 
  • #3
I got this, but I don't know the rest of the steps
 

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  • #4
Well, notice that in the final answer you have the term $$e^{\frac{-j\pi k}{3}}$$ so it would be a good idea to multiply by $$1=e^{\frac{-j\pi k}{3}}e^{\frac{j\pi k}{3}}.$$
 

1. What are Fourier series coefficients?

Fourier series coefficients are the numerical values used to represent a periodic function as an infinite sum of simple sine and cosine functions. They are used in Fourier series to describe the amplitude and phase of the individual sine and cosine functions that make up the periodic function.

2. How are Fourier series coefficients calculated?

Fourier series coefficients are calculated using the Fourier series formula, which involves integrating the periodic function over one period and dividing by the period. This process is repeated for each term in the series to determine the coefficients for each sine and cosine function.

3. What do Fourier series coefficients tell us about a function?

Fourier series coefficients provide information about the frequency content of a periodic function. They tell us the amplitude and phase of each individual sine and cosine function that makes up the function, which can help us understand its behavior and make predictions about its future values.

4. Can Fourier series coefficients be negative?

Yes, Fourier series coefficients can be negative. The coefficients represent the amplitude of the sine and cosine functions, which can have both positive and negative values. Negative coefficients indicate that the corresponding sine or cosine function is inverted or flipped over the x-axis.

5. How do Fourier series coefficients relate to the Fourier transform?

The Fourier transform is a mathematical operation that decomposes a function into its frequency components. Fourier series coefficients are a special case of the Fourier transform, where the function is assumed to be periodic. The Fourier transform can be seen as the continuous version of the Fourier series coefficients.

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