Erf x=(2)/(√(π)) ∫0^x e^-x^2 d w This function is important...

$$\operatorname{erf} x=\frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-x^{2}} d w$$ This function is important in applied mathematics and physics (probability theory and statistics, thermodynamics, etc.) and fits our present discussion. Regarding it as a typical case of a special function defined by an integral that cannot be evaluated es in elementary calculus, do the following. (a) Sketch or graph the bell-shaped curve [the curve of the integrand in (21) ). Show that erf $x$ is odd. Show that $$\begin{array}{c} \int_{a}^{b} e^{-w^{a}} d w=\frac{\sqrt{\pi}}{2}(\operatorname{crf} b-\cos a) \\ \int_{-b}^{b} e^{-x^{2}} d w=\sqrt{\pi} \operatorname{erf} b \end{array}$$ (b) Obtain the Maciaurin series of erf $x$ from that of the integrand. Use that series to compute a table of erf $x$ for $x=0(0,01) 3$ (meaning $x=0,0.01,0.02$ $\cdots \times 3$) (c) Obtain the values required in (b) by an integration command of your CAS. Compare accuracy. (d) It can be shown that erf $(\infty)=1 .$ Confirm this experimentally by computing erf $x$ for large $x$ (e) Let $f(x)=1$ when $x>0$ and 0 when $x<0 .$ Using erf $(\infty)=1,$ show that (12) then gives $$\begin{aligned} u(x, t) &=\frac{1}{\sqrt{\pi}} \int_{-\pi t 2 \epsilon \sqrt{t}} e^{-x^{2}} d z \\ &=\frac{1}{2}-\frac{1}{2} \operatorname{erf}\left(-\frac{x}{2 c \sqrt{t}}\right) \quad(t>0) \end{aligned}$$ (f) Express the temperature (13) in terms of the error function $$\begin{aligned} &\text { (g) Show that } \Phi(x)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} e^{-s^{2} \pi k} d s\\ &=\frac{1}{2}+\frac{1}{2} \operatorname{erf}\left(\frac{x}{\sqrt{2}}\right) \end{aligned}$$ Here, the integral is the definition of the "distribution function of the normal distribution" to be discussed in $\sec 24.8$

$$\operatorname{erf} x=\frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-x^{2}} d w$$
This function is important in applied mathematics and physics (probability theory and statistics, thermodynamics, etc.) and fits our present discussion. Regarding it as a typical case of a special function defined by an integral that cannot be evaluated es in elementary calculus, do the following.
(a) Sketch or graph the bell-shaped curve [the curve of the integrand in (21) ). Show that erf $x$ is odd. Show that
$$\begin{array}{c}
\int_{a}^{b} e^{-w^{a}} d w=\frac{\sqrt{\pi}}{2}(\operatorname{crf} b-\cos a) \\
\int_{-b}^{b} e^{-x^{2}} d w=\sqrt{\pi} \operatorname{erf} b
\end{array}$$
(b) Obtain the Maciaurin series of erf $x$ from that of the integrand. Use that series to compute a table of erf $x$ for $x=0(0,01) 3$ (meaning $x=0,0.01,0.02$
$\cdots \times 3$)
(c) Obtain the values required in (b) by an integration command of your CAS. Compare accuracy.
(d) It can be shown that erf $(\infty)=1 .$ Confirm this experimentally by computing erf $x$ for large $x$
(e) Let $f(x)=1$ when $x>0$ and 0 when $x<0 .$ Using erf $(\infty)=1,$ show that (12) then gives
$$\begin{aligned}
u(x, t) &=\frac{1}{\sqrt{\pi}} \int_{-\pi t 2 \epsilon \sqrt{t}} e^{-x^{2}} d z \\
&=\frac{1}{2}-\frac{1}{2} \operatorname{erf}\left(-\frac{x}{2 c \sqrt{t}}\right) \quad(t>0)
\end{aligned}$$
(f) Express the temperature (13) in terms of the error function
$$\begin{aligned}
&\text { (g) Show that } \Phi(x)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} e^{-s^{2} \pi k} d s\\
&=\frac{1}{2}+\frac{1}{2} \operatorname{erf}\left(\frac{x}{\sqrt{2}}\right)
\end{aligned}$$
Here, the integral is the definition of the "distribution function of the normal distribution" to be discussed in $\sec 24.8$

Key Concepts

Normal Distribution and Its Cumulative Distribution Function (CDF)

The cumulative distribution function of the normal distribution can be expressed in terms of the error function. Understanding this relationship is essential in probability theory and statistics, as it provides a direct method to compute probabilities and percentiles for normally distributed data.

Maclaurin Series Expansion

The Maclaurin series is a power series expansion of a function centered at zero. It is an essential tool in approximating functions that are defined by complicated integrals or other non-elementary forms. In the context of the error function, the series provides a means for numerical computation and insight into the function’s behavior near the origin.

Computational Methods and Computer Algebra Systems (CAS)

CAS tools are utilized to perform complex integrations, series expansions, and numerical evaluations of special functions. They provide efficient means to verify analytical results, compare the accuracy of approximations, and explore the behavior of functions like the error function across various regimes.

Asymptotic Behavior

The study of asymptotic behavior involves analyzing the behavior of functions as their arguments approach large values. For the error function, the property that erf(?) = 1 is significant because it links the function to probability distribution functions, ensuring that total probability is conserved in stochastic processes.

Error Function (erf)

The error function, denoted as erf, is a special function defined by an integral that quantifies the probability integral of the Gaussian distribution. It appears in solutions to problems in heat conduction, diffusion, and probability theory, and is characterized by its odd symmetry and asymptotic behavior, approaching unity as its argument tends to infinity.

Special Functions

Special functions are functions that arise in applied mathematics contexts and are often defined by integral representations that are not expressible in terms of elementary functions. Their study involves properties, such as recurrence relations and asymptotics, which are vital in solving differential equations and modeling physical phenomena.

Symmetry of Functions

Understanding the symmetry properties of functions, such as odd or even symmetry, is crucial for simplifying computational tasks and theoretical analysis. The error function is an example of an odd function, meaning that it satisfies f(-x) = -f(x), which has implications in both integration limits and physical interpretations.

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