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Euler’s formula for fourier series
What is ‘Euler’s formula for fourier series’?
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Euler’s formula for the Fourier series relates the complex exponential function to the trigonometric functions used in the representation of periodic functions. It is given by:
e^(ix) = cos(x) + i * sin(x)
where e is the base of the natural logarithm, i is the imaginary unit (i.e., i^2 = -1), x is a real number, and cos(x) and sin(x) are the cosine and sine functions, respectively.
The Fourier series expresses a periodic function as an infinite sum of complex exponentials, and Euler’s formula allows us to rewrite these complex exponentials in terms of cosine and sine functions. By using Euler’s formula, we can simplify the representation of periodic functions and work with complex numbers instead.
The Fourier series representation of a periodic function f(x) with period 2π can be expressed as:
f(x) = a₀/2 + ∑[n=1 to ∞] (aₙ * cos(nx) + bₙ * sin(nx))
where a₀, aₙ, and bₙ are coefficients that depend on the function f(x). These coefficients can be calculated using integrals and the properties of the function. Euler’s formula allows us to rewrite the complex exponentials e^(inx) in terms of cosine and sine functions, leading to the familiar trigonometric form of the Fourier series.
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