The complementary error function is defined as $$ \operatorname{erfc}(x)=\frac{2}{\sqrt{\pi}} \int_{x}^{\infty} \exp \left(-u^{2}\right) \mathrm{d} u $$ Show that the second term in the expansion of \(1 / r\) is given by $$ \frac{2}{\sqrt{\pi}} \int_{g}^{\infty} \exp \left(-r^{2} \rho^{2}\right) \mathrm{d} \rho=\frac{1}{r} \operatorname{erfc}(g r) $$ Use a spreadsheet or other programming tool to show that this function falls to zero much faster than \(1 / r\). A good numerical approximation is $$ \operatorname{erfc}(x)=\exp \left(-x^{2}\right) \sum_{n=1}^{5} a_{n} t^{n} $$ where \(t=(1+b x)^{-1}\), and \(a_{1}=0.25483, a_{2}=\) \(-0.28450, a_{3}=1.42141, a_{4}=1.45315, a_{5}=1.06141\) and \(b=0.32759\).

In this exercise, we encounter integral calculus, which is a fundamental part of calculus focusing on the concept of integration. Integration deals with finding the area under a curve or summing up a continuous function. Here, we specifically use integral calculus to define and work with the complementary error function, denoted as \(\text{erfc}(x)\). This function requires us to integrate an exponential function from a variable point to infinity. Understanding how to manipulate and solve these integrals is crucial to grasping the entire problem.

Numerical approximation is a technique used to find approximate solutions to mathematical problems. Sometimes, exact answers are difficult or impossible to obtain analytically, so we use numerical methods. In our exercise, we approximate the complementary error function \(\text{erfc}(x)\) using a series expansion. This helps us handle cases where calculating the exact value is impractical. By employing a series like\[\text{erfc}(x) = \text{exp}(-x^2) \sum_{n=1}^5 a_n t^n\], where \(t = (1 + bx)^{-1}\) and specific coefficients are given, we get an accurate approximation without performing complex integration.

Mathematical substitution is a valuable technique in integral calculus. It involves changing variables to simplify an integral. For instance, in our exercise, we convert the variable \(u\) to \(r\rho\). This substitution allows us to transform and simplify the given integral. Specifically, substituting \(u = r\rho\) and recognizing that \(du = r d\rho\), we can rewrite our complex integral into a more manageable form. Once simplified, integrating becomes more straightforward, and we can then relate it back to familiar functions like \(\text{erfc}(x)\).

Series expansion is another powerful tool in numerical approximation. It transforms functions into a sum of simpler terms. Our exercise uses a series expansion to approximate \(\text{erfc}(x)\). The expansion involves coefficients \(a_n\) and a variable \(t \((t = (1 + bx)^{-1})\)\). It simplifies the exponential function \(\text{exp}(-x^2)\) multiplied by a series sum. Series expansions make it easier to compute values for functions that are otherwise difficult to solve. Using the provided coefficients and the series formula helps us quickly estimate \(\text{erfc}(x)\) and understand its behavior.

Error functions, including the complementary error function \(\text{erfc}(x)\), are special functions in mathematics, particularly in statistics and probability. These functions measure the probability that a value falls within a range in a Gaussian distribution. The complementary error function is defined as \(\text{erfc}(x) = \frac{2}{\text{\text{sqrt}}{\text{\text{pi}}}} \int_{x}^{\text{\text{infinity}}} \text{\text{exp}}(-u^2) du\). This integral, spanning from a point \(x\) to infinity, simplifies calculations in probability and statistical analysis. The key insight is recognizing that \(\text{erfc}(x)\) falls to zero much faster than other functions like \(1/x\), highlighting its rapid decay and utility in various applications.