Question

Let \(f(\theta)\) be a periodic function given by \(f(\theta)=\sum_{n=-\infty}^{\infty} a_{n} e^{i n \theta}\). Find its Fourier transform \(\tilde{f}(t)\)

Short answer

The Fourier Transform of the function \(f(\theta)=\sum_{n=-\infty}^{\infty} a_{n} e^{i n \theta}\) is \(\tilde{f}(t)=a_{2\pi t}\) if \(2\pi t\) is an integer, and 0 otherwise.

Step-by-step solution

Review Fourier Transform Definition

The Fourier transform \(\tilde{f}(t)\) of a function \(f(x)\) is defined as: \(\tilde{f}(t)=\int_{-\infty}^{\infty} f(x) e^{-2\pi i xt} dx\). So use this definition in the context of the given Fourier series of \(f(\theta)\).

Substitute Fourier Series into Fourier Transform Formula

Substitute the formula for \(f(\theta)\) into the Fourier transform equation: \(\tilde{f}(t)=\int_{0}^{1} (\sum_{n=-\infty}^{\infty} a_{n} e^{i n \theta}) e^{-2\pi i \theta t} d\theta\)

Swap the Summation and Integral

Use the linearity property of the integral to swap the summation and integral: \(\tilde{f}(t)=\sum_{n=-\infty}^{\infty} a_{n} \int_{0}^{1} e^{i n \theta} e^{-2\pi i \theta t} d\theta\)

Simplify the Integral

The integral now simplifies to \(\int_{0}^{1} e^{i \theta (n-2\pi t)} d\theta\)

Evaluate the Integral

Evaluating the integral results in \(\tilde{f}(t)=\sum_{n=-\infty}^{\infty} a_{n} \left[ \frac{1}{i(n-2\pi t)} \left( e^{i(n-2\pi t)}-1 \right) \right]\)

Simplify the Summation

The summation in the expression simplifies when you realize the terms for \(n \neq 2\pi t\) vanish due to the periodicity of the exponential term. So, the result is \(\tilde{f}(t)=a_{2\pi t}\) if \(2\pi t\) is an integer, and 0 otherwise.

Key concepts

Fourier Series

The concept of a Fourier series is a cornerstone in the analysis of periodic functions, and understanding it is crucial for tackling problems in fields like signal processing and electrical engineering. A Fourier series is a way to represent a periodic function as an infinite sum of sines and cosines or, equivalently, using complex exponentials. This series breaks down a complex periodic waveform into simpler components, with each term representing a single sinusoidal wave with a specific frequency – these frequencies being integer multiples of a fundamental frequency. The coefficients of the series reflect the amplitude and phase of the corresponding frequencies, effectively encoding the shape of the original function.

The Fourier series can be written as:<\[ f(\theta) = \sum_{n=-\infty}^{\infty} a_n e^{i n \theta} \]>where \( a_n \) are complex numbers known as the Fourier coefficients and \( e^{i n \theta} \) are basis functions formed using the complex exponential function.

Complex Exponential

The complex exponential function is vital to the understanding of periodic phenomena across physics, engineering, and mathematics. It is represented by the mathematical expression\( e^{ix} \), using Euler's formula, which states that \( e^{ix} = \cos(x) + i\sin(x) \). In the context of Fourier analysis, complex exponentials serve as the building blocks for Fourier series and transforms. They are used instead of sines and cosines because they simplify many mathematical operations due to their properties under differentiation and integration. These exponentials make it easy to analyze and synthesize periodic signals as each term in the series corresponds to a specific frequency component of the overall function.

Periodic Functions

A periodic function is a function that repeats its values in regular intervals or periods. The most familiar examples of periodic functions are the trigonometric functions sine and cosine, which repeat every \( 2\pi \) radians. The importance of periodic functions in mathematics and science cannot be overemphasized—they model a wide array of time-recurring events, from the fluctuations of an audio signal to the seasonal length of days. The analysis of periodic functions is greatly aided by Fourier series, which allows any periodic function (under certain conditions) to be written as a sum of sines and cosines—thus revealing the frequency content of the function.

Linearity of the Integral

In the realm of calculus, the linearity of the integral is an essential property that allows for the simplification of complex integral expressions. It states that the integral of a sum of functions is equal to the sum of their integrals, and that a constant factor can be pulled out of the integral.<\[\int (af(x) + bg(x)) dx = a\int f(x) dx + b\int g(x) dx\]>When applied to Fourier transforms, this property enables us to interpose the operation of integration with that of summation. This is especially handy when working with Fourier series, since we can individually integrate each term of the series and then sum their contributions. This convenient swap between summation and integration is crucial in simplifying our mathematical expressions and leads to the identification and calculation of Fourier transform coefficients.

Mathematical Physics

The field of mathematical physics deals with the application of mathematical methods to solve physical problems. The Fourier transform specifically is a widely used tool in this field, providing a method for breaking down complex waveforms into their constituent parts. This is essential in the study of phenomena where different scales of analysis are involved—quantum mechanics, optics, and signal processing being prime examples. By using the Fourier transform, we can move between time (or space) representation and frequency representation. This is not only powerful for theoretical analyses like solving partial differential equations but also for practical data processing, like filtering out noise in signals or reconstructing images in medical diagnostics. The interplay between key concepts like Fourier series, complex exponentials, and the linearity of the integral allows mathematical physicists to parse through intricate systems and model the natural world with precision.