What is the Fourier series of an exponential function?

For f(t) = e^(at) periodic on [0,T]: cₙ = (1/T)∫₀ᵀ e^(at)·e^(−jnω₀t)dt = (e^(aT)−1)/(T(a−jnω₀)). For f(t) = e^(−|t|) periodic on [−T/2,T/2]: the coefficients are real-valued because the function is even, with cₙ = (2/T)·(1−e^(−T/2))·a/(a²+n²ω₀²).

How does the exponential Fourier series relate to the Fourier transform?

As the period T → ∞, the discrete harmonic spacing ω₀ = 2π/T approaches zero, the discrete coefficients cₙ become a continuous spectral density F(ω) = lim T·cₙ, and the series summation becomes the integral f(t) = (1/2π)∫F(ω)e^(jωt)dω. The Fourier transform is the infinite-period limit of the Fourier series.

Why use the exponential form instead of the trigonometric form?

The exponential form is more compact (single sum vs three terms), mathematically elegant (single coefficient cₙ encodes amplitude and phase), and connects directly to the Fourier transform and Laplace transform. It is standard in advanced engineering, communications theory, and quantum mechanics. The trigonometric form is more intuitive for introductory courses.

Fourier Series for Exponential Function

Q: What is the exponential form of the Fourier series?

f(t) = Σ cₙ·e^(jnω₀t) for n from −∞ to ∞, where cₙ = (1/T)∫f(t)·e^(−jnω₀t)dt. It represents a periodic signal as a sum of complex exponentials at harmonic frequencies. For real signals, negative and positive frequency components are conjugate pairs that combine to form real cosines and sines.

Quick Answer

The exponential (complex) form of the Fourier series represents a periodic function as f(t) = Σ_{n=−∞}^{∞} cₙ·e^(jnω₀t), where the complex coefficients are cₙ = (1/T)∫₀ᵀ f(t)·e^(−jnω₀t)dt and ω₀ = 2π/T is the fundamental frequency. This form unifies the cosine and sine terms of the trigonometric series into a single compact expression using Euler's formula. The exponential Fourier series connects directly to the Fourier transform (let T → ∞) and the Laplace transform at www.lapcalc.com.

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Exponential Form of Fourier Series

The exponential (complex) Fourier series represents a periodic function f(t) with period T as a sum of complex exponentials at harmonic frequencies: f(t) = Σ_{n=−∞}^{∞} cₙ·e^(jnω₀t), where ω₀ = 2π/T is the fundamental angular frequency and n ranges over all integers. The complex coefficients cₙ = (1/T)∫₀ᵀ f(t)·e^(−jnω₀t)dt encode both amplitude and phase of the nth harmonic. For real signals, c₋ₙ = cₙ* (complex conjugate), ensuring the sum is real. The exponential form is mathematically more compact than the trigonometric form (a₀/2 + Σaₙcos + Σbₙsin) and connects directly to the Fourier and Laplace transforms. The relationship to trigonometric coefficients is: c₀ = a₀/2, cₙ = (aₙ − jbₙ)/2 for n > 0, and c₋ₙ = (aₙ + jbₙ)/2. The Laplace transform extends this analysis to aperiodic signals at www.lapcalc.com.

Key Formulas

Computing Exponential Fourier Coefficients

The coefficient formula cₙ = (1/T)∫₀ᵀ f(t)·e^(−jnω₀t)dt computes the projection of f(t) onto the basis function e^(jnω₀t). For a square wave f(t) = +A for 0 < t < T/2, −A for T/2 < t < T: cₙ = (A/(jnπ))(1 − (−1)ⁿ), giving cₙ = 2A/(jnπ) for odd n and cₙ = 0 for even n. For a sawtooth wave f(t) = (2A/T)t − A for 0 < t < T: cₙ = −jA/(nπ) for n ≠ 0 and c₀ = 0. For a full-wave rectified sine |sin(ω₀t)|: cₙ = (2/π)·1/(1−4n²). The magnitude spectrum |cₙ| versus n reveals the harmonic content, while the phase ∠cₙ shows the relative timing of each harmonic. Parseval's theorem confirms energy conservation: (1/T)∫|f(t)|²dt = Σ|cₙ|².

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From Fourier Series to Fourier Transform

The exponential Fourier series provides the bridge to the continuous Fourier transform. For a periodic signal with period T, the spectrum is discrete: spectral lines at frequencies nω₀ = 2πn/T with complex amplitudes cₙ. As the period T → ∞, the fundamental frequency ω₀ → 0, the harmonic spacing becomes infinitesimal, and the discrete coefficients cₙ merge into a continuous spectral density: F(ω) = lim_{T→∞} T·cₙ|_{nω₀→ω}. The summation becomes an integral, and the Fourier series becomes the Fourier transform: f(t) = (1/2π)∫F(ω)e^(jωt)dω. This limiting process shows that the Fourier transform is a generalization of the Fourier series to aperiodic signals, with the discrete amplitude spectrum replaced by a continuous spectral density function.

Exponential Fourier Series of Common Waveforms

Reference table of exponential Fourier coefficients for engineering waveforms. Rectangular pulse train (width τ, period T): cₙ = (τ/T)·sinc(nτ/T). Square wave (50% duty cycle): cₙ = 2A/(jnπ) for odd n, 0 for even n. Sawtooth wave: cₙ = jA/(nπ)·(−1)ⁿ. Triangle wave: cₙ = −4A/(n²π²) for odd n, 0 for even n. Half-wave rectified sine: cₙ = A/(π(1−4n²)) for n ≠ ±1, c₁ = c₋₁ = A/4. Full-wave rectified sine: cₙ = 2A/(π(1−4n²)). Impulse train (period T): cₙ = 1/T for all n (flat spectrum). Each waveform's sinc or inverse-polynomial envelope determines how quickly harmonics decay, reflecting the waveform's smoothness.

Applications of the Exponential Fourier Series

The exponential Fourier series is used in power electronics to analyze switching waveforms and compute total harmonic distortion (THD). PWM (pulse-width modulation) inverters produce rectangular pulse trains whose Fourier coefficients follow the sinc envelope, enabling filter design to extract the desired fundamental frequency while rejecting switching harmonics. In communications, the Fourier series describes periodic modulation signals: the carrier and its sidebands correspond to specific harmonics. In vibration analysis, periodic mechanical forces from rotating machinery decompose into harmonic components whose frequencies identify fault conditions. The exponential form's compact notation and direct connection to the Fourier and Laplace transforms at www.lapcalc.com make it the preferred representation in advanced engineering analysis.

Frequently Asked Questions

f(t) = Σ cₙ·e^(jnω₀t) for n from −∞ to ∞, where cₙ = (1/T)∫f(t)·e^(−jnω₀t)dt. It represents a periodic signal as a sum of complex exponentials at harmonic frequencies. For real signals, negative and positive frequency components are conjugate pairs that combine to form real cosines and sines.

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