Problem 11 - Advanced Engineering Mathematics

Question

Show that \(\int_{a}^{b} e^{-u^{2}} d u=\frac{\sqrt{\pi}}{2}[\operatorname{erf}(b)-\operatorname{erf}(a)]\)

Step-by-Step Solution

Answer

The integral \( \int_{a}^{b} e^{-u^{2}} du \) equals \( \frac{\sqrt{\pi}}{2} [\operatorname{erf}(b) - \operatorname{erf}(a)] \) using the error function definition.

Step 1: Identify the Function and Its Integral

The given exercise involves the integral \( \int_{a}^{b} e^{-u^{2}} du \). We need to show that this integral equals \( \frac{\sqrt{\pi}}{2}[\operatorname{erf}(b) - \operatorname{erf}(a)] \). This integral is not elementary, meaning it cannot be expressed in terms of basic algebraic functions. Instead, it is related to the error function, denoted as \( \operatorname{erf}(x) \), which is a special function used to define such integrals.

Step 2: Expression of Error Function

The error function, \( \operatorname{erf}(x) \), is defined as \( \operatorname{erf}(x) = \frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^2} dt \). This shows the relationship between the error function and the integral of the Gaussian function \( e^{-t^2} \).

Step 3: Rewriting the Given Integral in Terms of the Error Function

To express \( \int_{a}^{b} e^{-u^{2}} du \) in terms of the error function, note that the error function involves an integral from 0 to a certain limit. We can express the integral from \( a \) to \( b \) using the limits of integration in the definition of the error function: \[ \int_{a}^{b} e^{-u^2} du = \int_{0}^{b} e^{-u^2} du - \int_{0}^{a} e^{-u^2} du. \] Therefore, based on the definition of the error function, \[ \int_{0}^{b} e^{-u^2} du = \frac{\sqrt{\pi}}{2} \operatorname{erf}(b) \] and \[ \int_{0}^{a} e^{-u^2} du = \frac{\sqrt{\pi}}{2} \operatorname{erf}(a). \]

Step 4: Simplifying the Result

Substitute the expressions from the error function back into the difference we found in step 3: \[ \int_{a}^{b} e^{-u^2} du = \frac{\sqrt{\pi}}{2} \operatorname{erf}(b) - \frac{\sqrt{\pi}}{2} \operatorname{erf}(a). \] This simplifies to the desired expression: \[ \int_{a}^{b} e^{-u^2} du = \frac{\sqrt{\pi}}{2} [\operatorname{erf}(b) - \operatorname{erf}(a)]. \]

Key Concepts

Gaussian Integral

The Gaussian Integral is a fundamental concept in advanced calculus, often associated with the bell curve shape of the Gaussian function, which is the graph of the exponential function \( e^{-x^2} \). The integral \( \int_{-\infty}^{\infty} e^{-x^2} \,dx \) is known as the Gaussian integral and is significant because it has a closed value: it equals \( \sqrt{\pi} \). This result is noteworthy in mathematical physics and probability theory.

Calculating the Gaussian integral over indefinite limits is complex, as it cannot be computed using elementary functions. However, within definite bounds, like from zero to some value \( x \), it becomes accessible using special functions like the error function. The standard method of computing this integral involves changing variables and completing the square, which are advanced calculus techniques necessary for evaluating non-elementary integrals.

Special Functions

Special functions are a group of functions that arise in mathematical applications, including differential equations, and notably, they frequently appear in the context of calculus and integration. The error function, \( \operatorname{erf}(x) \), is one of the most central special functions in calculus, particularly when dealing with Gaussian distributions.

• The error function is defined as \( \operatorname{erf}(x) = \frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^2} \,dt \).
• It effectively measures the probability of a random variable falling within a certain range in a normal distribution.

Understanding such functions is essential because they provide solutions that cannot be otherwise expressed through simple algebraic forms. In solving integrals that can't be reduced to elementary operations, special functions provide an edge, simplifying expressions and integrations that arise in practical applications and engineering.

Definite Integration

Definite integration is the process of finding the value of an integral between two specific points, \( a \) and \( b \). This type of integration calculates the net area under the curve of a function within these bounds. In the context of the error function and the Gaussian integral, definite integration is pivotal.

• Here, we consider the integral \( \int_{a}^{b} e^{-u^2} \,du \), which evaluates the area under the curve of \( e^{-u^2} \) from \( a \) to \( b \).
• To handle such integrals, substitute expressions involving special functions, like \( \operatorname{erf}(x) \), because the primitive function of \( e^{-x^2} \) isn’t straightforward.

Definite integration often links with areas in physics, related to cumulative probabilities and heat distributions in statistics and thermodynamics, making it invaluable in scientific computations.

Advanced Calculus

Advanced calculus incorporates a wide variety of techniques and theories for handling more complex mathematical problems, often involving non-elementary functions and integrals. Understanding and applying concepts from advanced calculus are crucial for solving real-world problems.

Key areas where advanced calculus applies include:

• Changing variables in integrals to simplify complex expressions.
• Using techniques such as integration by parts or substitution to handle special functions.

Deeper calculus studies often involve non-standard functions, like the error function, which don't have simple primitive forms but are crucial for computers and engineers when dealing with probabilities and other analytical science fields.