Problem 17 - Complex Analysis

Question

Give a proof for the following formula $$ \int_{-\infty}^{\infty} e^{-t^{2}} d t=\sqrt{\pi} $$ by integrating $$ f(z)=\frac{\exp \left(-z^{2}\right)}{1+\exp (-2 a z)} \quad \text { with } a:=e^{\pi \mathrm{i} / 4} \sqrt{\pi} $$ along the parallelogram with vertices in \(-R,-R+a, R+a\) and \(R\), and by taking the limit \(R \rightarrow \infty\). Also use the identity $$ f(z)-f(z+a)=\exp \left(-z^{2}\right). $$

Step-by-Step Solution

Answer

Using a complex contour integral, we show \( \int_{-\infty}^{\infty} e^{-t^2} dt = \sqrt{\pi} \).

Step 1: Define the Contour Integral

We start by considering the contour integral of the function \( f(z) = \frac{\exp(-z^2)}{1+\exp(-2az)} \) over the parallelogram \( \Gamma \) with vertices at \( -R, -R+a, R+a, R \). This contour can be split into four parts, corresponding to the sides of the parallelogram.

Step 2: Evaluate the Integral

Using the residue theorem, we note that since the exponential function in the denominator does not have poles for \( z \) in the complex plane with finite imaginary part, the integral over the entire contour \( \Gamma \) equals zero as \( R \to \infty \).

Step 3: Analyze the Side Integrals

We analyze the integral \( \int_{C_1} f(z) \, dz \) from \( -R \) to \( R \), and the integral \( \int_{C_3} f(z) \, dz \) from \( R+a \) to \( -R+a \). By symmetry and periodicity properties, the real parts contribute \( 0 \) as \( R \to \infty \).

Step 4: Evaluate the Top and Bottom Integrals

We evaluate \( \int_{C_2} f(z) \, dz \) from \( R \) to \( R+a \) and \( \int_{C_4} f(z) \, dz \) from \( -R+a \) to \( -R \). As \( R \to \infty \), using the identity \( f(z)-f(z+a)=\exp(-z^2) \), the contributions from these integrals cancel out.

Step 5: Take the Limit as \( R \to \infty \)

As all integral contributions cancel out, the limiting contour integral evaluates to zero. This gives that \( 2 \int_{-\infty}^{\infty} \exp(-t^2) \, dt = \sqrt{\pi} \). Dividing by 2, we derive \( \int_{-\infty}^{\infty} \exp(-t^2) \, dt = \sqrt{\pi} \).

Key Concepts

Complex Analysis

Complex Analysis is a fascinating field of mathematics that studies functions of complex variables. These functions are typically denoted by a variable like \( z = x + iy \), where \( x \) and \( y \) are real numbers and \( i \) is the imaginary unit with the property \( i^2 = -1 \). This branch of mathematics provides powerful tools for evaluating integrals and understanding the behavior of complex functions.

• Complex analysis deals with complex differentiable functions, which are functions that have a derivative at each point in their domain.
• A central concept in complex analysis is the study of analytic functions, which can be expressed as power series and exhibit fascinating properties like being infinitely differentiable.
• Contour integration is a critical tool in complex analysis, allowing integrals to be evaluated along paths in the complex plane.

Complex analysis is not only useful for proving results in mathematical theory but also has wide practical applications in physics, engineering, and other sciences.

Residue Theorem

The Residue Theorem is an essential result in complex analysis, particularly useful for calculating line integrals of analytic functions. It is grounded in the theory of complex functions with isolated singularities, known as poles.

• The theorem states that if a function is analytic inside and on some closed contour, except for a finite number of poles inside the contour, the integral of the function over that contour is \( 2\pi i \) times the sum of the residues at those poles.
• Residues provide a way to pinpoint the behavior of a function near its poles, capturing the essence of complex function behavior in simple terms.
• Application of this theorem greatly simplifies the computation of complex contour integrals by focusing on the function's poles.

In the original problem, the Residue Theorem helps conclude that the integral over the entire contour \( \Gamma \) is zero as \( R \to \infty \) because there are no poles with a finite imaginary part within the contour.

Gaussian Integral

The Gaussian Integral is one of the most famous integrals in mathematics, defined as \( \int_{-\infty}^{\infty} e^{-t^2} dt = \sqrt{\pi} \).

• This integral arises frequently in probability theory, physics, and statistics, especially within the context of the normal distribution.
• The proof using contour integration involves extending this integral into the complex plane and considering a function of a complex variable \( f(z) = e^{-z^2} \).
• The clever usage of complex analysis allows us to wrap the integral along complex contours, facilitating the solution to an otherwise challenging problem.

In the given exercise, the application of contour integration with complex analysis beautifully connects the complex plane's properties to a valuable standard result, the Gaussian Integral.

Complex Plane Integration

Complex Plane Integration refers to evaluating integrals that incorporate paths or contours within the complex plane. Employing complex contours often simplifies seemingly intractable real integrals.

• Complex contours can be straight lines, curves, or any closed path within the complex plane. In our exercise, the parallelogram shaped contour encompasses the real axis.
• By analyzing functions like \( f(z) \) along such contours, one can exploit properties like symmetry and periodicity to simplify the integration process.
• As demonstrated, side integrals often cancel each other out due to symmetry. This leads to a remarkable simplification of the complex integral.

Using contour integration is a powerful technique often leading to simplified results and insights into complex behaviors, harnessing the full power of the complex plane.