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Why X0 equals zero?

Electrical Engineering Asked by OMAR on January 25, 2021

The question is as shown below:
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The answer is as shown below:
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Everything is clear except based on what did he assume that X0 = 0?

Below are the reference for formulas
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Below is my trial:
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One Answer

He is not assuming X0 to be 0, he is defining X0 to be 0.

The formula for the amplitude of the coefficients is undefined at coefficient 0. If you try to calculate it from the formula given, you will arrive at zero, infinity, or arguably any number in between.

It's rather like the behaviour of the Fourier transform of a pulse response. You can fit the ratio sin(kx)/kx to the entire function, except at x=0, where you again have a 0/0. The important thing here is that the thing you are modelling is defined at x=0, and the arithmetical formula apparently doesn't fit, so is usually defined for x=0 when you first meet this formulation.

There are techniques to determine the value of f(x)/g(x) where both f and g tend to zero at the same value of x on to a more thorough footing, by expanding f and g in polynomials in x, and cancelling terms. However, that detail is beyond the scope of the question you've been set. You might however try doing it for sin(x)/x as a fun exercise for yourself.

Answered by Neil_UK on January 25, 2021

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