Question

For \(f(t)=\exp \left(-\frac{r^{2}}{2}\right)\), use the relationships of the Fourier transforms of \(f^{\prime}(t)\) and \(t f(t)\) to that of \(f(t)\) itself to find a simple differential equation satisfied by \(\tilde{f}(\omega)\), the Fourier transform of \(f(t)\) and hence determine \(\tilde{f}(\omega)\) to within a constant. Use this result to solve the integral equation $$ \int_{-x}^{\infty} e^{-t(t-2 x) / 2} h(t) d t=e^{3 x^{2} / 8} $$ for \(h(t)\)

Short answer

\( h(r) = C e^{\frac{r^{2}}{2}} \)

Step-by-step solution

Identify the Fourier Transform of the given function

Given the function is \( f(t)=\text{exp}(-\frac{r^{2}}{2}) \), identify the general form of the Fourier Transform \( \tilde{f}(\text{ω}) \). Use \( \tilde{f}(\text{ω}) = \int f(t)e^{-i\text{ω} t}\text{dt} \).

Calculate Fourier Transform properties

Using properties of Fourier Transform: \For \( f^{\prime}(t) \) (the derivative of \( f(t) \)), Fourier Transform gives: \( \mathcal{F}[f^{\prime}(t)] = i\text{ω} \tilde{f}(\text{ω}) \).\For \( t f(t) \), Fourier Transform gives: \( \mathcal{F}[tf(t)] = i \frac{d}{d\text{ω}}[\tilde{f}(\text{ω})] \).

Define the differential equation satisfied by \tilde{f}(\text{ω})

Combine properties from Step 2 to form the differential equations: \From \(f^{\prime}(t) = - t f(t) \), obtain \( i\text{ω}\tilde{f}(\text{ω}) = - i \frac{d}{d\text{ω}}[\tilde{f}(\text{ω})] \), which simplifies to \( \frac{d}{d\text{ω}} \tilde{f}(\text{ω}) + \text{ω} \tilde{f}(\text{ω}) = 0 \).

Solve the differential equation

Solve the differential equation \( \frac{d}{d\text{ω}} \tilde{f}(\text{ω}) + \text{ω} \tilde{f}(\text{ω}) = 0 \). Using separation of variables: \( \tilde{f}(\text{ω}) = C \exp(- \frac{ \text{ω}^{2}}{2}) \), where \( C \) is a constant.

Use the result to solve the integral equation

Given the integral equation \int_{-x}^{\infty} e^{-t(t-2x)/2} h(t) dt = e^{\frac{3x^{2}}{8}} \, recognize that the integral can be rewritten in terms of Gaussian integrals and \( \tilde{h}(\text{ω}) \). Conclude that \( h(t) = C e^{\frac{r^{2}}{2}} \) matches the form of the problem.

Key concepts

differential equations

Differential equations play an integral role in many areas of mathematics, including the analysis of Fourier transforms. When dealing with functions and their Fourier transforms, differential equations can describe the relationship between a function and its transform. In our exercise, we started with the function: \( f(t) = \text{exp}(-\frac{r^{2}}{2}) \). To find a differential equation satisfied by its Fourier transform \( \tilde{f}(\text{ω}) \), we used properties of Fourier transforms associated with the function's derivative \( f^{\text{'} }(t) \) and the product \( t f(t) \). This allowed us to derive the differential equation: \( \frac{d}{d\text{ω}} \tilde{f}(\text{ω}) + \text{ω} \tilde{f}(\text{ω}) = 0 \). This relationship helps us understand how changes in the time domain function affect its frequency domain representation and vice versa.

Gaussian integral

The Gaussian integral is a fundamental integral in probability, statistics, and quantum mechanics. In its simplest form, it evaluates the integral of the Gaussian function over the entire real line, used primarily to find the area under the Gaussian curve (bell curve).
In our context, we deal with the integral equation: \( \int_{-x}^{∞} e^{-t(t-2x)/2} h(t) dt = e^{\frac{3x^{2}}{8}} \). This appears complicated but can be tackled by recognizing that the exponent features a quadratic function, a hallmark of Gaussian integrals. We utilized our previously derived solution for \( \tilde{f}(\text{ω}) \) to express the integrand in a simpler form, enabling us to rewrite the integral in terms of known Gaussian integrals, leading to the solution of \( h(t) = C e^{\frac{r^{2}}{2}} \).
Understanding Gaussian integrals is crucial when analyzing functions that resemble Gaussian forms.

properties of Fourier transform

Fourier transform properties are invaluable tools for solving differential equations and analyzing signals. Several key properties come into play, including linearity, time scaling, and frequency shifting. For our exercise, two primary properties were used: the differentiation and modulation properties.
- Differentiation Property: The Fourier transform of a function's derivative \( f^{\text{'} }(t) \) is given by \( \mathcal{F}[f^{\text{'} }(t)] = i\text{ω} \tilde{f}(\text{ω}) \). - Modulation Property: For the product of time and a function \( t f(t) \), its transform is given by \( \mathcal{F}[t f(t)] = i \frac{d}{d\text{ω}}[\tilde{f}(\text{ω})] \).
These properties enabled us to infer the differential equation that \( \tilde{f}(\text{ω}) \) satisfies. Mastering these properties greatly simplifies dealing with complex functions and makes solving integral and differential equations more approachable.

Fourier analysis

Fourier analysis is a branch of mathematics focused on decomposing functions into their frequency components. It's widely used in signal processing, physics, and engineering. The core concept is the Fourier transform, which translates a function from the time (or spatial) domain into the frequency domain, making it easier to analyze and manipulate.
In our problem, we began with the function \( f(t) = \text{exp}(-\frac{r^{2}}{2}) \) and aimed to find its Fourier transform \( \tilde{f}(\text{ω}) \). By leveraging the properties of Fourier transforms, we were able to derive and solve the differential equation: \( \frac{d}{d\text{ω}} \tilde{f}(\text{ω}) + \text{ω} \tilde{f}(\text{ω}) = 0 \). The solution \( \tilde{f}(\text{ω}) = C \text{exp}(- \frac{\text{ω}^{2}}{2}) \) seamlessly reveals the frequency domain representation.
Fourier analysis thus provides powerful insights into how functions behave and interact, both in time and frequency domains. This approach simplifies the process of solving and understanding differential and integral equations.