Question

Solve the integral equation $$ \int_{0}^{\infty} \cos (x v) y(v) d v=\exp \left(-x^{2} / 2\right) $$ for the function \(y=y(x)\) for \(x>0\). Note that for \(x<0, y(x)\) can be chosen as is most convenient.

Short answer

\(y(v) = \text{exp}(-v^2 / 2)\).

Step-by-step solution

Identify the Fourier Transform

Recognize that the integral equation involves a cosine transform, which is related to the Fourier transform. The equation can be rewritten using the inverse Fourier cosine transform of the given function.

Apply the Inverse Fourier Cosine Transform

Use the inverse Fourier cosine transform on both sides of the equation to isolate the function y(v). The transform of \(\text{exp}(-x^2 / 2)\) must be found.

Calculate the Inverse Transform

Find the inverse Fourier cosine transform of \(\text{exp}(-x^2 / 2)\). This requires knowledge of standard Fourier transform pairs. The result is obtained by recognizing that the transform of a Gaussian function results in another Gaussian function.

Write the Solution

Using the property that the inverse Fourier cosine transform of \(\text{exp}(-x^2 / 2)\) gives a Gaussian function \(\text{exp}(-v^2 / 2)\), rewrite the solution explicitly.

Key concepts

Integral Equation

Integral equations are vital tools in mathematics and physics. They involve finding a function that satisfies an equation where the unknown function appears under an integral sign. In this exercise, you're given an integral involving a cosine function and the goal is to solve for the function \(y(v)\).

Understanding how to handle these equations often involves recognizing known transforms and applying inverse operations. In the given problem, recognizing the relationship to the Fourier transform, specifically the cosine transform, is key. This allows us to convert the problem into something more manageable. By using these transforms, we essentially break down the integral into simpler parts that can be more easily manipulated and solved.

Cosine Transform

The cosine transform is an integral transformation closely related to the Fourier transform. It is particularly useful when dealing with even functions. In simpler terms, it transforms a function into another function that is often easier to handle in various mathematical problems.

In its general form, the cosine transform \(F(k)\) of a function \(f(x)\) is given by:

\[ F(k) = \frac{2}{\root{2}} \times \root{2} \times 2 \times 2 \]

Similarly, the inverse Fourier cosine transform allows us to revert back to the original function, which is critical in solving our exercise. By applying the inverse Fourier cosine transform to both sides of the given integral equation, we isolate the function y(v). This step is essential because it converts our problem back from the frequency domain to the time domain.

Gaussian Function

The Gaussian function, often denoted as \(G(x)=\text{exp}(-x^2/2)\), is a smooth and bell-shaped function that appears frequently in statistics and signal processing. Its Fourier transform properties make it exceptionally useful in solving integral equations.

In the exercise, recognizing that the given function \(\text{exp}(-x^2/2)\) is a Gaussian was crucial. A notable property is that the Fourier transform of a Gaussian function is another Gaussian function. This recognition simplifies the process because we know the behavior of the Gaussian under Fourier transforms.

By applying this knowledge, we determine that the inverse Fourier cosine transform of \(\text{exp}(-x^2/2)\) yields \(\text{exp}(-v^2/2)\). Thus, \(y(v)=\text{exp}(-v^2/2)\) provides the explicit solution to the integral equation. For \(x<0\), \(y(x)\) can be chosen as convenient, often reflecting symmetry or specific boundary conditions in practical applications.