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Problem 503

# Develop the definition of the Fourier transform of a function $$\mathrm{f}(\mathrm{x})$$ be extending the definition of the Fourier series of $$\mathrm{f}$$ to the case where the discrete spectrum of Fourier coefficients becomes a continuous spectrum.

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In order to develop the Fourier Transform from the Fourier Series, we start by considering the limit as the period $$T$$ goes to infinity, which leads to a continuous spectrum with angular frequency $$\omega$$. Then, we replace the trigonometric functions with the exponential function using Euler's identity. This gives us the Fourier Transform definition: $F(\omega) = \int_{-\infty}^{\infty} f(x) e^{-i\omega x} dx$ And the inverse Fourier Transform: $f(x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{i\omega x} d\omega$
See the step by step solution

## Step 1: Review the Fourier Series Definition

The Fourier series represents a periodic function $$f(x)$$ as a sum of sine and cosine functions. The general formula for a Fourier series is given by: $f(x) = a_0 + \sum_{n=1}^{\infty} [a_n \cos(\frac{2\pi nx}{T}) + b_n \sin(\frac{2\pi nx}{T})]$ where $$T$$ is the period of function $$f(x)$$, and the coefficients $$a_0$$, $$a_n$$, and $$b_n$$ are defined as follows: $a_0 = \frac{1}{T} \int_{-T/2}^{T/2} f(x) dx$ $a_n = \frac{2}{T} \int_{-T/2}^{T/2} f(x) \cos(\frac{2\pi nx}{T}) dx$ $b_n = \frac{2}{T} \int_{-T/2}^{T/2} f(x) \sin(\frac{2\pi nx}{T}) dx$

## Step 2: Introduce a continuous spectrum

Now, we will extend the definition of the discrete Fourier series to a continuous spectrum by considering the limit as the period $$T$$ goes to infinity. In this case, the spacing between frequencies will become infinitesimally small and we can replace the discrete frequencies with a continuous variable, usually denoted as $$\omega$$ (angular frequency). Instead of summing over discrete coefficients, we will integrate over the continuous variable $$\omega$$. Also, note that $$\omega$$ is defined as $$\omega = \frac{2\pi n}{T}$$, which simplifies to $$n = \frac{T\omega}{2\pi}$$.

## Step 3: Develop the Fourier Transform Definition

Given the continuous spectrum, the Fourier series will be transformed into the Fourier transform. The formula for the Fourier transform is given by: $F(\omega) = \int_{-\infty}^{\infty} f(x) e^{-i\omega x} dx$ Where $$F(\omega)$$ is the Fourier transform of the function $$f(x)$$, and $$i$$ is the imaginary unit ($$i^2 = -1$$). Note that we replaced the trigonometric functions with the exponential function using Euler's identity: $e^{ix} = \cos(x) + i\sin(x)$ And the inverse Fourier transform is defined as: $f(x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{i\omega x} d\omega$ The Fourier transform and its inverse allow us to move back and forth between the time domain (represented by the function $$f(x)$$) and the frequency domain (represented by its Fourier transform, $$F(\omega)$$).

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