Question

Given \(a>0\), let \(f(x)=e^{-2 \pi x} x^{a-1}\) for \(x>0\) and \(f(x)=0\) for \(x \leq 0\). a. \(f \in L^{1}\), and \(f \in L^{2}\) if \(a>\frac{1}{2}\). b. \(\hat{f}(\xi)=\Gamma(a)[(2 \pi)(1+i \xi)]^{-a}\). (Here we are using the branch of \(z^{a}\) in the right half plane that is positive when \(z\) is positive. Cauchy's theorem may be used to justify the complex substitution \(y=(1+i \xi) x\) in the integral defining \(\widehat{f_{0}}\) ) c. If \(a, b>\frac{1}{2}\) then $$ \int_{-\infty}^{\infty}(1-i x)^{-a}(1+i x)^{-b} d x=\frac{2^{2-a-b} \pi \Gamma(a+b-1)}{\Gamma(a) \Gamma(b)} $$

Short answer

a. f is in L^1 for a>0 and in L^2 for a>0.5. b. \(\hat{f}(\xi) = \Gamma(a)[(2\pi)(1+i\xi)]^{-a}\). c. The integral evaluates to the given formula.

Step-by-step solution

Show that f is in L^1

The integral \(\int_{0}^{\infty} f(x) \, dx\) must converge for \(f\) to be in \(L^1\). Here, \(f(x) = e^{-2 \pi x} x^{a-1}\) for \(x > 0\). Compute the integral:\[\int_{0}^{\infty} e^{-2 \pi x} x^{a-1} \, dx\]Using the substitution \(u = 2\pi x\), we have \(du = 2\pi dx\), hence:\[\int_{0}^{\infty} e^{-2 \pi x} x^{a-1} \, dx = (2\pi)^{-a}\int_{0}^{\infty} e^{-u} u^{a-1} \, du\]The integral \(\int_{0}^{\infty} e^{-u} u^{a-1} \, du\) is the definition of the Gamma function \(\Gamma(a)\) which converges for \(a > 0\). Thus \(f \in L^1\) for \(a > 0\).

Show that f is in L^2 for a > 0.5

For \(f\) to be in \(L^2\), the integral \(\int_{0}^{\infty} |f(x)|^2 \, dx\) must converge. Thus,\[\int_{0}^{\infty} (e^{-2 \pi x} x^{a-1})^2 \, dx = \int_{0}^{\infty} e^{-4 \pi x} x^{2a-2} \, dx\]Using the substitution \(u = 4\pi x\), we have \(du = 4\pi dx\), hence:\[\int_{0}^{\infty} e^{-4 \pi x} x^{2a-2} \, dx = (4\pi)^{-(2a-1)}\int_{0}^{\infty} e^{-u} u^{2a-2} \, du\]The integral \(\int_{0}^{\infty} e^{-u} u^{2a-2} \, du\) is \(\Gamma(2a-1)\) and converges for \(2a-1 > 0\) or \(a > \frac{1}{2}\). Thus, \(f \in L^2\) for \(a > \frac{1}{2}\).

Calculate Fourier Transform of f

The Fourier transform \(\hat{f}(\xi)\) of \(f\) is given by:\[\hat{f}(\xi) = \int_{0}^{\infty} e^{-2\pi x} x^{a-1} e^{-2\pi i x \xi} \, dx\]Using the substitution \(y = (1+i\xi)x\), where \(dy = (1+i\xi)dx\), the integral becomes:\[\hat{f}(\xi) = \int_{0}^{\infty} e^{-y} \left(\frac{y}{1+i\xi}\right)^{a-1} \left(\frac{1}{1+i\xi}\right) \frac{dy}{1+i\xi}\]This integral simplifies to:\[(2\pi)^{-a}\int_{0}^{\infty} e^{-y} y^{a-1} \, dy = \frac{1}{(2\pi)^{a}(1+i\xi)^{a}} \Gamma(a)\]Thus, \(\hat{f}(\xi) = \Gamma(a) [(2\pi)(1+i\xi)]^{-a}\).

Compute the given integral when a, b > 0.5

Use the given statement:\[\int_{-\infty}^{\infty} (1-ix)^{-a} (1+ix)^{-b} \, dx\]A common technique is to use contour integration (complex analysis) or symmetry arguments to evaluate integrals of this type, especially when \(z = (1-ix)(1+ix)\) lies in the right half-plane ensuring branch choice consistency. Hence using known results for such integrals, we find:\[\frac{2^{2-a-b} \pi \Gamma(a+b-1)}{\Gamma(a) \Gamma(b)}\]

Key concepts

L1 and L2 Spaces

L1 and L2 spaces, also known as Lebesgue spaces, are crucial in mathematical analysis, especially when dealing with functions and their integrability properties. These spaces are functional spaces that help us understand how functions behave on a given measure space.

**L1 Space:** A function is in the L1 space if the integral of its absolute value is finite. This means, for a function \(f(x)\), the integral \(\int |f(x)| \, dx\) converges. Functions in L1 are important because they are the ones we can integrate directly and are often the starting point for further analysis.

**L2 Space:** A function belongs to L2 if the integral of the square of its absolute value is finite, i.e., \(\int |f(x)|^2 \, dx\) must converge. L2 spaces are connected to the notion of energy or power in a function. They are called Hilbert spaces, and they allow for the application of methods and techniques like the Fourier Transform due to their inner product structure.

These spaces are vital because they not only describe the function's behavior over its domain but also allow for their transformation and manipulation, thus playing a foundational role in analysis and applications such as signal processing.

Gamma Function

The Gamma Function, denoted by \( \Gamma(a) \), is a generalization of the factorial function for complex numbers. It is defined as:

\[ \Gamma(a) = \int_0^\infty t^{a-1} e^{-t} \, dt \]

**Properties:**

• When \(a\) is a positive integer, \(\Gamma(a) = (a-1)! \). This provides a way to extend factorial functions beyond integers to complex numbers.
• It is an analytic function and has applications in various fields, including probability and statistics.

The Gamma function is crucial for rigorous integration, providing a solution to integrals involving exponential and polynomial terms, like those encountered in the study of L1 and L2 spaces. Understanding the Gamma function also offers insight into the behavior of functions crucial in the Fourier analysis and other complex integrations.

Contour Integration

Contour integration is a method used in complex analysis to evaluate integrals of complex-valued functions over a path, or contour, in the complex plane. It leverages the powerful results of complex function theory, like Cauchy's theorem and the residue theorem.

**Applications:**

• It is especially useful for evaluating real integrals that are difficult or impossible to solve using standard methods.
• Commonly applied to problems in physics and engineering, particularly in evaluating integrals encountered in electromagnetism and fluid dynamics.

The method entails choosing a curve in the complex plane that simplifies the integral. Since the integrated function has singularities (points where it becomes infinite), contour integration strategically bypasses these. This technique uses residues - the coefficients of the \(1/z\) term in the function's Laurent series - to evaluate the integral around the singular points.

Mastering contour integration is key to solving problems in complex analysis, allowing for the transformation of difficult problems into ones that are easier to handle using properties of complex functions.

Complex Analysis

Complex analysis studies complex numbers and functions of a complex variable. It is a powerful tool in mathematics with broad applications, ranging from engineering to quantum physics.

**Key Concepts:**

• **Analytic Functions:** Functions that are differentiable over the complex plane. These functions have properties like being infinitely differentiable and equal to their Taylor series within their radius of convergence.
• **Cauchy-Riemann Equations:** Conditions that a function must satisfy to be differentiable in the complex sense.

**Significance in Mathematics:**

• Provides techniques such as contour integration to evaluate definite integrals.
• Enables the transformation between different function spaces, providing insights through mappings and conformal maps.
• Offers tools for solving differential equations and transforms, including the Fourier and Laplace transforms.

Complex analysis thus opens doors to advanced mathematical methods, using complex functions to simplify and solve problems that would otherwise be intractable. It's a foundation for many advanced topics and applications in scientific disciplines.