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Please write down the Fourier series expansion of v(t) in terms of complex exponential

User TigrouMeow
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Final answer:

The Fourier series expansion of a periodic function v(t) is represented using complex exponentials, summing terms of the form Cn*e^(inwt), where the coefficients Cn are calculated by integrating the product of v(t) and e^(-inwt) over one period.

Step-by-step explanation:

The student is asking for the Fourier series expansion of a function v(t) in terms of complex exponentials. In mathematics and physics, complex exponentials are often used to represent periodic functions using Fourier series. The Fourier expansion of a periodic function is given as a sum of sine and cosine terms or equivalently, as a sum of complex exponentials eiwt and e-iwt, taking advantage of Euler's formula, which relates the exponential function to trigonometric functions: eix = cos(x) + i sin(x).

The general form of a Fourier series in complex exponential form for a function v(t) over an interval T is:

v(t) = ∑n=-∞∞ Cn einwt

where n is an integer, ω = 2π/T is the angular frequency, and the coefficients Cn are given by:

Cn = (1/T) ∫0Tv(t) e-inwt dt

These coefficients are obtained by integrating the product of v(t) and e-inwt over one period of the function. To write the Fourier series expansion in complex exponential terms for a specific function v(t), we would need to evaluate these coefficients through integration.

User Ava
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