15.1 Error Function

INTRODUCTION

There are many functions in mathematics that are defined by means of an integral. For example, in many traditional calculus texts the natural logarithm is defined in the following manner: ln In earlier chapters we have already seen, albeit briefly, the error function erf (x), the complementary error function erfc(x), the sine integral function Si(x), the Fresnel sine integral S(x), and the gamma function Г(α); all of these functions are defined in terms of an integral. Before applying the Laplace transform to boundary-value problems, we need to know a little more about the error function and the complementary error function. In this section we examine the graphs and a few of the more obvious properties of erf (x) and erfc(x).

See Appendix A.

Properties and Graphs

Recall from (14) of Section 2.3 that the definitions of the error function erf (x) and complementary error function erfc(x) are, respectively,

(1)

With the aid of polar coordinates, it can be demonstrated that

(2)

We have already seen in (15) of Section 2.3 that when the second integral in (2) is written as we obtain an identity that relates the error function and the complementary error function:

erf (x) + erfc(x) = 1. (3)

For x > 0 it is seen in FIGURE 15.1.1 that erf(x) can be interpreted as the area of the blue region under the graph of on the interval [0, x] and erfc(x) is the area of the red region on [x, ). The graph of the function f is often referred to as a bell curve.

A bell curve plotted on a t, y coordinate plane is labeled f = (2 over root of pi) times e^(negative t square). The bell curve begins at the point (negative 2, 0), peaks at the estimated point (0, 1.1) and finally ends at the point (2, 0). A point x is marked out on the x axis such that 0 < x < 1. The region under the curve on the interval [0, x] is shaded in blue color. The region under the curve for all values of t > x is shaded in red color.

FIGURE 15.1.1 Bell curve

Because of the importance of erf(x) and erfc(x) in the solution of partial differential equations and in the theory of probability and statistics, these functions are built into computer algebra systems. So with the aid of Mathematica we get the graphs of erf(x) (in blue) and erfc(x) (in red) given in FIGURE 15.1.2. The y-intercepts of the two graphs give the values

Two graphs are plotted on an x y coordinate plane. The first curve traced in red color and labeled e r f c(x) begins in the second quadrant as a horizontal line at a height of y = 2 begins to dip at an approximate point when x = (negative 1) passes through the point (0, 1) coincides with the x axis at the point (2, 0) and remains horizontal thereafter. The second curve traced in blue color and labeled e r f(x) is symmetrical to the first curve in reference to the line y = 0.5. It begins in the third quadrant as a horizontal line at a height of y = (negative 1) begins to rise at an approximate point when x = (negative 1) passes through the point (0, 0) crosses the first line at an estimated point (0.5, 0.5) rises to the point (2, 1) and remains horizontal thereafter.

FIGURE 15.1.2 Graphs of erf(x) and erfc(x)

  • .

Other numerical values of erf(x) and erfc(x) can be obtained directly from a CAS. Further inspection of the two graphs in Figure 15.1.2 shows that:

  • the domains of erf(x) and erfc(x) are (−, ),
  • erf(x) and erfc(x) are continuous functions,
  • .

It should also be apparent that the graph of the error function is symmetric with respect to the origin and so erf(x) is an odd function:

erf (−x) = − erf(x). (4)

You are asked to prove (4) in Problem 14 of Exercises 15.1.

Table 15.1.1 contains Laplace transforms of some functions involving the error and complementary error functions. These results will be useful in the exercises in the next section.

TABLE 15.1.1
A table provides Laplace transforms of the following functions. 1. f(t), a > 0: 1 over, sqrt(pi times t), times e^negative a^2 over 4 t. Laplace transform of f(t) = F(s): e^negative a times sqrt(s) over, sqrt(s). 2. f(t), a > 0: a over, 2 times sqrt(pi times t^3), times e^negative a^2 over 4 t. Laplace transform of f(t) = F(s): e^negative a times sqrt(s). 3. f(t), a > 0: e r f c times, a over 2 times sqrt(t). Laplace transform of f(t) = F(s): e^negative a times sqrt(s) over, sqrt(s). 4. f(t), a > 0: 2 times, sqrt(t over pi), times e^negative a^2 over 4 t, minus a times e r f c times a over 2 times sqrt(t). Laplace transform of f(t) = F(s): e^negative a times sqrt(s) over, s times sqrt(s). 5. f(t), a > 0: e^a times b, times e^b^2 times t, time e r f c, times b times sqrt(t) + a over 2 times sqrt(t). Laplace transform of f(t) = F(s): e^negative a times sqrt(s) over, sqrt(s) times sqrt(s) + b. 6. f(t), a > 0: negative e^a times b, times e^b^2 times t, times e f r c, times times b times sqrt(t) + a over 2 times sqrt(t) + e r f c times, a over 2 times sqrt(t). Laplace transform of f(t) = F(s): b times e^negative a times sqrt(s) over, s times sqrt(s) + b.

REMARKS

The proofs of the results in Table 15.1.1 will not be given because they are long and somewhat complicated. For example, the proofs of entries 2 and 3 of the table require several changes of variables and the use of the convolution theorem. For those who are curious, see Introduction to the Laplace Transform, by Holl, Maple, and Vinograde (Appleton-Century-Crofts, 1959). A flavor of these kinds of proofs can be gotten by working Problem 1 in Exercises 15.1.

15.1 Exercises Answers to selected odd-numbered problems begin on page ANS-38.

    1. Show that
    2. Use part (a), the convolution theorem, and the result of Problem 47 in Exercises 4.1 to show that
  2. Use the result of Problem 1 to show that
  3. Use the result of Problem 1 to show that
  4. Use the result of Problem 2 to show that
  5. Use the result of Problem 4 to show that
  6. Find the inverse transform

    [Hint: Rationalize a denominator followed by a rationalization of a numerator.]

  7. Let C, G, R, and x be constants. Use Table 15.1.1 to show that
  8. Let a be a constant. Show that

    [Hint: Use the exponential definition of the hyperbolic sine. Expand in a geometric series.]

  9. Use the Laplace transform and Table 15.1.1 to solve the integral equation
  10. Use the third and fifth entries in Table 15.1.1 to derive the sixth entry.
  11. Show that
  12. Show that
  13. Show that
  14. Prove that erf(x) is an odd function.
  15. Show that erfc(−x) = 1 + erf(x).