Question

(a) Find the exponential Fourier transform of \(f(x)=e^{-|x|}\) and write the inverse transform. You should find $$\int_{0}^{\infty} \frac{\cos \alpha x}{\alpha^{2}+1} d \alpha=\frac{\pi}{2} e^{-|x|}$$ (b) Obtain the result in (a) by using the Fourier cosine transform equations (12.15) (c) Find the Fourier cosine transform of \(f(x)=1 /\left(1+x^{2}\right)\). Hint: Write your result in (b) with \(x\) and \(\alpha\) interchanged.

Short answer

The integral result is: \( \int_{0}^{\infty} \frac{\cos \alpha x}{\alpha^{2}+1} d \alpha=\frac{\pi}{2} e^{-|x|} \). Fourier cosine transform interchange: \(F(k) = \int_{0}^{\infty} \frac{\cos(\alpha k)}{\alpha^2 + 1} \, d\alpha = \frac{\pi}{2} e^{-|k|}\).

Step-by-step solution

- Define the Fourier Transform

The exponential Fourier transform of a function \(f(x)\) is given by: \[ F(k) = \int_{-\infty}^{\infty} f(x) e^{-ikx} \, dx \]

- Apply the Definition

For \(f(x) = e^{-|x|}\), the Fourier transform is: \[ F(k) = \int_{-\infty}^{\infty} e^{-|x|} e^{-ikx} \, dx \]

- Split into Integral Parts

Split the integral into two parts, over \(-\infty < x < 0\) and \(0 < x < \infty\): \[ F(k) = \int_{-\infty}^{0} e^{x} e^{-ikx} \, dx + \int_{0}^{\infty} e^{-x} e^{-ikx} \, dx \]

- Evaluate the Integrals

Evaluate each integral separately: \[ \int_{-\infty}^{0} e^{x} e^{-ikx} \, dx = \int_{-\infty}^{0} e^{(1-ik)x} \, dx = \left[ \frac{e^{(1-ik)x}}{1-ik} \right]_{-\infty}^{0} = \frac{1}{1-ik} \] Similarly, \[ \int_{0}^{\infty} e^{-x} e^{-ikx} \, dx = \int_{0}^{\infty} e^{-(1+ik)x} \, dx = \left[ \frac{e^{-(1+ik)x}}{-(1+ik)} \right]_{0}^{\infty} = \frac{1}{1+ik} \]

- Sum the Integrals

Add the results: \[ F(k) = \frac{1}{1-ik} + \frac{1}{1+ik} \] Simplify to get: \[ F(k) = \frac{1}{1-ik} + \frac{1}{1+ik} = \frac{(1+ik) + (1-ik)}{(1-ik)(1+ik)} = \frac{2}{1+k^2} \]

- Inverse Transform

The inverse transform for the result is: \[ f(x) = \int_{-\infty}^{\infty} F(k) e^{ikx} \, dk = \int_{-\infty}^{\infty} \frac{2}{1+k^2} e^{ikx} \, dk \]

- Simplify for Given Integral

Given integral is: \[ \int_{0}^{\infty} \frac{\cos(\alpha x)}{\alpha^2 + 1} \, d\alpha = \frac{\pi}{2} e^{-|x|} \]

- Fourier Cosine Transform for f(x)

Use Fourier Cosine Transform equation: \[ F_c(k) = \sqrt{\frac{2}{\pi}} \int_{0}^{\infty} f(x) \cos(kx) \, dx \] For \(f(x) = \frac{1}{1+x^2}\), interchange \(x\) with \(\alpha\), \[ F(k) = \int_{0}^{\infty} \frac{\cos(\alpha k)}{\alpha^2 + 1} \, d\alpha = \frac{\pi}{2} e^{-|k|} \]

Key concepts

Exponential Fourier Transform

The exponential Fourier transform is a crucial concept in analyzing the frequency components of a signal or function, specifically for complex functions. It converts a time-domain function, like our example function, into the frequency domain. The general formula for the exponential Fourier transform of a function \(f(x)\) is:

\( F(k) = \int_{-\infty}^{\infty} f(x) e^{-ikx} \, dx \)

This integral essentially decomposes \(f(x)\) into sine and cosine components, modulated by complex exponentials. For our function \(f(x) = e^{-|x|}\), the computation involves breaking down the domain from \(-\infty\) to \(\infty\) into manageable parts and then evaluating the integrals separately.

Cosine Transform

The Fourier cosine transform is used exclusively for even functions (functions symmetric about the y-axis). Distinct from the exponential Fourier transform, the cosine transform avoids complex numbers by using the cosine function. The formula for the cosine transform of \(f(x)\) is:

\( F_c(k) = \sqrt{\frac{2}{\pi}} \int_{0}^{\infty} f(x) \cos(kx) \, dx \)

For example, applying this to \(f(x)=1/(1+x^2)\), through interchanging \(x\) with \(\alpha\), simplifies our problem significantly and provides valuable insights into the frequency characteristics of an even function without the complexities of imaginary units.

Inverse Transform

Once a function is transformed to the frequency domain, it's often crucial to convert it back to the time domain using the inverse Fourier transform. This is represented as:

\( f(x) = \int_{-\infty}^{\infty} F(k) e^{ikx} \, dk \)

For the transformed function obtained in our solution, \( F(k) = \frac{2}{1+k^2} \), applying the inverse transform recovers the original function in terms of its frequency components. The process essentially reconstructs the original signal from its spectral (frequency domain) representation.

Fourier Analysis

Fourier analysis involves decomposing a function into a sum of sines and cosines, allowing us to study the frequency components of the signal. The exponential and cosine Fourier transforms we discussed are specific tools within this analysis domain. Fourier analysis is foundational in various fields including signal processing, where understanding the frequency content of signals helps in filtering, compression, and reconstruction of data. The integral steps from converting functions between the time and frequency domains are key Fourier analysis techniques.

Transform Integration

Transform integration plays a critical role in computing Fourier transforms. The process involves integrating the product of the original function and the exponential or trigonometric basis functions (like \(e^{-ikx}\) or \(\cos(kx)\)). For instance, when computing \(F(k)\) for \(f(x) = e^{-|x|}\), breaking down and integrating over smaller, more manageable parts helps in solving the transform. By integrating appropriately over each segment, the sum contributes to the overall frequency representation of the function.

This integration approach was evident in the steps solving the exponential Fourier transform for \(f(x) = e^{-|x|}\), which involved splitting the integral and evaluating in parts.