Complex Fourier Series Calculator

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Example

Created on 2024-06-20Asked by Logan King (Solvelet student)

Find the first term of the complex Fourier series for

f(x) = e^{ix}

on the interval

[-\pi, \pi]

Solution

Step 1: Identify the function

f(x) = e^{ix}

. Step 2: Compute the complex Fourier coefficient

c_n

c_n = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) e^{-inx} \, dx = \frac{1}{2\pi} \int_{-\pi}^{\pi} e^{ix} e^{-inx} \, dx = \frac{1}{2\pi} \int_{-\pi}^{\pi} e^{i(1-n)x} \, dx.

Step 3: Evaluate the integral for

n = 1

c_1 = \frac{1}{2\pi} \int_{-\pi}^{\pi} e^{i(1-1)x} \, dx = \frac{1}{2\pi} \int_{-\pi}^{\pi} 1 \, dx = \frac{1}{2\pi} [x]_{-\pi}^{\pi} = \frac{1}{2\pi} ( \pi - (-\pi) ) = \frac{1}{2\pi} \cdot 2\pi = 1.

Step 4: Conclusion. The first term of the complex Fourier series for

f(x) = e^{ix}

c_1 = 1

. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Samuel Hall on Solvelet

1. Find the complex Fourier series representation of the function

f(t) = t

, defined on the interval

[-\pi, \pi]

2. Determine the convergence of the complex Fourier series of the function

f(t) = t^2

, defined on the interval

[-\pi, \pi]

DefinitionThe Fourier series is a concept that serves as a mathematical tool in numerous fields of application, ranging from signal processing and telecommunications to quantum mechanics and partial differential equations that have periodic boundary conditions. The complex Fourier series is an alternative to the real Fourier series, which is defined by the classical trigonometric functions and employs complex exponentials instead. The term “complex” is used in the title to indicate that complex exponentials are used, even though the terminology of “complex” looks unnecessary today; still, the concept itself is closely related to this particular term. A remarkable fact is that the projection of any periodic function onto the subspace spanned by functions e i k x, k integer, corresponds to a series that includes complex exponential functions. Let ‘f’ be a periodic function with complex Fourier series representing it: when Fourier series converges to f(x) in mean to the sum of its Fourier series.

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