Question

Determine the Fourier transform of each of the following functions: (a) \(f(x)=\frac{1}{a^{2}+x^{2}}\) (b) \(f(x)=\frac{\cos a x}{a^{2}+x^{2}}\) (c) \(f(x)=\frac{\sin b x}{a^{2}+x^{2}}\)

Short answer

(a) \(\frac{\pi}{a} e^{-2\pi a |k|}\), (b) \(\frac{\pi}{a} e^{-2\pi a |k - \frac{a}{2\pi}|} + \frac{\pi}{a} e^{-2\pi a |k + \frac{a}{2\pi}|}\), (c) \(\frac{\pi}{ia} (e^{-2\pi a |k - \frac{b}{2\pi}|} - e^{-2\pi a |k + \frac{b}{2\pi}|})\)."}

Step-by-step solution

Define the Fourier Transform

The Fourier transform of a function \( f(x) \) is defined as \( F(k) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i k x} \, dx \). This will be used for all parts of the problem.

Solve for Part (a)

To find the Fourier transform of \( f(x) = \frac{1}{a^2 + x^2} \), substitute it into the Fourier transform integral to get:\[ F(k) = \int_{-\infty}^{\infty} \frac{1}{a^2 + x^2} e^{-2\pi i k x} \, dx \]Recognize this as a known result; the Fourier transform of \( \frac{1}{a^2 + x^2} \) is \( \frac{\pi}{a} e^{-2\pi a |k|} \).

Solve for Part (b)

For \( f(x) = \frac{\cos ax}{a^2 + x^2} \), use the property that \( \cos(ax) = \frac{1}{2}(e^{iax} + e^{-iax}) \) to rewrite the function:\[ f(x) = \frac{1}{2} \left( \frac{e^{iax}}{a^2 + x^2} + \frac{e^{-iax}}{a^2 + x^2} \right) \]Substitute this into the Fourier transform integral yielding two separate integrals. For each, use the result from Step 2:\[ F(k) = \frac{1}{2} \left( \frac{\pi}{a} e^{-2\pi a |k - \frac{a}{2\pi}|} + \frac{\pi}{a} e^{-2\pi a |k + \frac{a}{2\pi}|} \right) \].

Solve for Part (c)

For \( f(x) = \frac{\sin bx}{a^2 + x^2} \), use the identity \( \sin(bx) = \frac{1}{2i}(e^{ibx} - e^{-ibx}) \):\[ f(x) = \frac{1}{2i} \left( \frac{e^{ibx}}{a^2 + x^2} - \frac{e^{-ibx}}{a^2 + x^2} \right) \]Similar to Step 3, this splits into two integrals. Solving them using the result from Step 2, get:\[ F(k) = \frac{1}{2i} \left( \frac{\pi}{a} e^{-2\pi a |k - \frac{b}{2\pi}|} - \frac{\pi}{a} e^{-2\pi a |k + \frac{b}{2\pi}|} \right) \].

Key concepts

Integral Transforms

An integral transform is a mathematical operation that transforms a function into another function by integrating it against a kernel. In the context of the Fourier transform, the kernel is the complex exponential function, denoted as \( e^{-2\pi i k x} \). Integral transforms are powerful tools for simplifying complex differential equations and analyzing functions in different domains. They help in converting functions from the time domain to the frequency domain and enable easier manipulation and solution finding.

• The Fourier transform, in particular, is used to analyze the frequencies that make up a signal or function. It provides a frequency spectrum of the original function.
• This process can highlight periodicity, enable the identification of patterns, and allow the handling of signals in a more manageable form.

The essential idea is to decompose a function into its constituent frequencies. Integral transforms, including Fourier transforms, are invaluable in engineering, physics, and many applied sciences for solving real-world problems and analyzing data.

Fourier Series

A Fourier series is a way to represent a function as a sum of simple sine and cosine waves. It's particularly useful for periodic functions and serves as a foundation for understanding the Fourier transform.

• The Fourier series breaks down a periodic function into a sum of sine and cosine components, each with specific coefficients.
• Each component corresponds to a harmonic of the function, and the sum represents the original function in the time or spatial domain.

The relationship between Fourier series and Fourier transform is significant. While a Fourier series applies to periodic functions, the Fourier transform extends this idea to non-periodic functions allowing analysis over the frequency domain.
Through Fourier series, students can understand the concept of frequency decomposition better, ultimately leading to a deeper grasp of how Fourier transforms work to analyze various functions in terms of their frequencies.

Complex Analysis

Complex analysis deals with functions that involve complex numbers. In the context of Fourier transforms, complex numbers play a vital role in expressing oscillatory behavior, like waves and periodic signals.

• Complex numbers, expressed as \( a + ib \), where \( i \) is the imaginary unit, are essential for representing exponential and trigonometric functions.
• Using complex exponentials (like \( e^{i\theta} \)) allows the combination of sine and cosine functions into a singular form, simplifying calculations in the Fourier transform.

The exponential form \( e^{-2\pi i k x} \) in the Fourier transform is rooted in complex analysis, enabling concise representation of frequency and oscillation. This mathematical elegance allows the transition from time to frequency domains smoothly, and it facilitates the computation and understanding of complex oscillatory phenomena. Complex analysis thus equips students with necessary tools for engaging with integral transforms effectively, rendering advanced topics far more accessible.