APPENDIX A

Integral-Defined Functions

INTRODUCTION

Many important functions encountered in applied mathematics are defined in terms of an integral. There are several ways of doing this. In Sections 1.1, 2.3, and 3.2, we have already seen that a solution of a differential equation may lead to a function of the form

(1)

In calculus, you may have seen that one definition of the natural logarithm is given in terms of an integral of the form given in (1):

An integral-defined function can also be of the form

(2)

where it is understood that x is a parameter; that is, x is treated as a constant in the t-integration. In many cases, especially for functions with a name, the defining integral is nonelementary. The integrand g in (2) could also depend on several parameters, say, In this case, the integral (2) defines a function of two variables x and y. Moreover, (1) and (2) can also be improper integrals. For example, in (1) the lower limit of integration a could be , or the interval of integration could be chosen to be In (2), one or both of the limits of integration could be infinite, or the integrand could have an infinite discontinuity in the interval of integration The convergence of an improper integral will, in general, depend on values of the parameter x. See Examples 1, 2, and 3 in Section 4.1.

The Error Function

In Sections 2.3 and 15.1, we saw that the error function and complementary error function,

, (3)

are both of the type given in (1) with the identification As we have seen in Chapter 4, the Laplace transform of a function f(t) is the function The last integral is of the type given in (2) with the symbol s playing the part of x so that In Chapter 15, the Fourier transform is also of the type given in (2), with the symbols x and replacing t and x, respectively, and

We begin our discussion of specific integral-defined functions with one of the form given in (2).

The Gamma Function

On a short list of important functions in the study of special functions (such as the Bessel function), the gamma function would appear near the top. The integral definition of this function,

(4)

was first given by the Swiss mathematician Leonhard Euler in his text Institutiones calculi integralis, published in 1768.

Graphs and Properties

Convergence of the improper integral (4) requires that or Although the integral (4) does not converge for it can be shown by alternative definitions that the domain of the gamma function can be expanded to the set of real numbers except the nonpositive integers: See Problem 28 in Exercises for Appendix A. Considered as a function of a real variable x, the graph of is given in FIGURE A.1. As seen in the figure, the dashed lines are vertical asymptotes of the graph.

Four dashed line and five curves are graphed on an x gamma(x) plane. The dashed lines are vertical and pass through x equals negative 4, negative 3, negative 2, negative 1, respectively. The first curve starts at the top of the first quadrant just to the right of the positive gamma(x) axis, goes sharply down and to the right, and reaches a low point in the first quadrant. Then it goes up and to the right, and ends at the top of the first quadrant. The second curve follows the same pattern as the first curve. It lies between x equals negative 4 and x equals negative 3 in the second quadrant. The third curve lies between x equals negative 3 and x equals negative 2 in the third quadrant. It starts from the bottom of the third quadrant, goes sharply up and to the right, reaches a high point, then goes down sharply down and to the right, and ends at the bottom of the third quadrant. The fourth curve follows the same pattern as the second curve. It lies between x equals negative 2 and x equals negative 1 in the second quadrant. The fifth curve follows the same pattern as the third curve. It lies between x equals negative 1 and x equals 0 in the third quadrant.

FIGURE A.1 Graph of the gamma function

There is a remarkably simple and extremely useful connection between the value of the gamma function at a number x and its value at To obtain this relationship, we replace x by in (4) and integrate by parts:

     (5)

.(6)

In (5), follows from the assumption that and repeated applications of L’Hôpital’s rule. The limit in (6), is the definition of the improper integral (4). Hence we have shown that

(7)

Equation (7) is called a recursion formula. It is easy to find the value of when x is a positive integer. For example, when we see that (4) is an elementary integral:

From this single numerical value, the repeated application of formula (7) gives:

(8)

and so on. In this manner, it is seen that when n is a positive integer,

(9)

Recall, (read “n factorial”) is defined as the product of consecutive integers from 1 to n; that is, and is called the factorial function. For convenience, it is customary to define Because is a special case of (4), the gamma function is sometimes referred to as the generalized factorial function.

In the discussion of the Bessel functions of half-integral order in Section 5.3, we used the value of With we see that (4) becomes

(10)

Note that the foregoing integral is improper for two reasons: an infinite limit of integration, and the integrand has an infinite discontinuity at 0. Nonetheless, (10) can be explicitly evaluated. To that end, we begin with the substitutions

(11)

But so

Switching to polar coordinates enables us to evaluate the double integral:

Hence (12)

We used (11), in the form to derive the fundamental identity that relates the error function and complementary error function: See (16) in Section 2.3.

Because of (7) and (12), the value of the gamma function for x equal to one half an odd integer, that is, where can be expressed in terms of The recursion formula (7) written as

(13)

is one way to extend the definition of the gamma function to negative real numbers with the negative integers being the only exception. Part (b) of the next example illustrates this idea.

EXAMPLE 1 Using (7), (12), and (13)

Evaluate

SOLUTION
  1. With and the result in (12), the recursion formula (7) yields

  2. If we choose then formula (13) gives

    ≡

The Beta Function

Analogous to (2), an integral of the type defines a function of two variables x and y. An example of this kind of integral is

(14)

The foregoing integral is known as the beta function and is closely related to the gamma function. The integrand of (14) can be discontinuous at 0 and at 1 for particular choices of x and y, but it can be shown that the integral exists for Although we are not going to prove it, the beta function can be expressed in terms of the gamma function:

(15)

EXAMPLE 2 Using (12) and (15)

Note that the integrand has an infinite discontinuity at t = 1.

Evaluate

SOLUTION

Even though the given integral is improper, we can evaluate it straight away using (15). By rewriting the integral in the equivalent form

we recognize it as (14) in the case and From (15) it follows that

where we have used the previously obtained values and ≡

Other Integral-Defined Functions

We have barely touched the subject of integral-defined functions. Here are a few more functions that occur in mathematics, physics, and engineering:

Sine and cosine integral functions:

(16)

Fresnel sine and cosine integral functions:

(17)

Exponential integral function:

(18)

Logarithmic integral function:

(19)

Airy functions of first and second kind:

(20)

The definitions of the functions in (16)–(20) vary slightly throughout the literature and websites. For example, the Fresnel integrals, which first appeared in the study of optics, are often defined without the factor in the integrands. In the computer algebra system Mathematica, the exponential integral is written Using the substitution it is easily shown that this latter form is equivalent to (18). Finally, it should be noted that the sine integral function is not regarded as an improper integral because the integrand has a removable discontinuity at ; that is, the integrand should be interpreted as

The graphs of the functions (16)–(20) are given in FIGURE A.2.

Five graphs are given. The first graph is titled: (a) Sine and cosine integral functions. Two curves are graphed on an x y plane. The first curve labeled Si(x) starts from the approximate point (negative 10, negative 1.75), goes slightly up and down and to the right to the approximate point (negative 4, negative 2), goes up and to the right through the point (0, 0), and reaches the approximate point (3, 1.75). Then it goes down and up and to the right, and ends at the approximate point (10, 1.5). The second curve labeled Ci(x) starts from the approximate point (0, negative 2), goes sharply up and to the right through the positive x axis, and reaches the approximate point (2, 0.5). Then it goes down and to the right through the positive x axis to the approximate point (5, negative 0.25), goes up and to the right through the positive x axis, again goes down and to the right, ends at the approximate point (10, 0). The second graph is titled: (b) Fresnel sine and cosine integral functions. Two curves are graphed on an x y plane. The first curve labeled S(x) starts from the left of the third quadrant, follows an increasing oscillatory pattern, and reaches a point in the negative x axis. Then it goes to the right to the origin, reaches a point in the positive x axis, goes up and to the right in the first quadrant, follows a decaying oscillatory pattern, and ends at the right of the first quadrant. The second curve labeled C(x) is same as the first curve except that it is slightly delayed than the first curve from the start till the end. It starts from the left of the third quadrant, follows an increasing oscillatory pattern, and reaches the origin. Then it goes up and to the right in the first quadrant, follows a decaying oscillatory pattern, and ends at the right of the first quadrant. The third graph is titled: (c) Exponential integral function. Two parts of a curve are graphed on an x y plane. The curve is labeled Ei(x). The first part starts from the point (negative 2, 0), goes to the right and slightly down to the approximate point (negative 1, negative 0.2), goes smoothly and then sharply down and to the right, and ends at the approximate point (0, negative 5). The second part starts from the approximate point (0, negative 5), goes up and to the right through the positive x axis, goes up and to the right, and ends at the approximate point (2, 5). The fourth graph is titled: (d) Logarithmic integral function. Two parts of a curve are graphed on an x y plane. The curve is labeled Li(x). The first part starts from the point (0, negative 1), goes smoothly and then sharply down and to the right, and ends at the approximate point (1, negative 6). The second part starts from the approximate point (1, negative 6), goes sharply and then smoothly up and to the right through the point (1, 0), goes gradually up and to the right, and ends at the approximate point (2, 2). The fifth graph is titled: (e) Airy functions of first and second kind. Two curves are graphed on an x y plane. The first curve labeled Ai(x) starts from the left of the second quadrant, follows an increasing oscillatory pattern, and reaches a point in the positive y axis. Then it goes down and to the right to a point in the positive x axis, and ends at the approximate point (5, 0). The second curve labeled Bi(x) is same as the first curve except that it is slightly delayed than the first curve. It starts from the left of the third quadrant, follows an increasing oscillatory pattern, and reaches a point in the positive y axis. Then it goes up and to the right, and ends at the top of the first quadrant.

FIGURE A.2 Graphs of Si(x), Ci(x), S(x), C(x), Ei(x), Li(x), Ai(x), and Bi(x)

Derivative of an Integral-Defined Function

The integral (1) and its derivative are encountered in the first semester of a course in calculus. Recall, one form of the Fundamental Theorem of Calculus states that if g is continuous on an interval and a is a number in the interval, then is a differentiable function and its derivative is

(21)

In the most general case of (2) where the limits of integration are functions of x,

the derivative is given by Leibniz’s rule:

(22)

Without going into any details, it is usually assumed in (22) that are continuous on an interval and and are continuous on some region of the xt-plane.

Two special cases of (22) are of particular interest. First, if the integrand is replaced by then and (22) reduces to

This result also follows directly from (21) and the Chain Rule.

(23)

Second, if denote constants, then and (22) gives a rule for differentiating under an integral sign:

(24)

Under slightly more stringent conditions, the result in (24) is also valid when the interval of integration is The last result leads us back to differential equations. See Problems 49–52 in Exercises for Appendix A.

EXAMPLE 3 Using (22), (23), and (24)

Compute the derivative of the functions

  3. .
SOLUTION
  1. From (22) with and we have

  2. From (23),

  3. From (24),

    ≡

The result in (24) can sometimes be an aid in evaluating an integral that at first glance seems intractable. But the procedure usually involves formal manipulations and cleverness, which cannot be taught. Problems 53–56 in Exercises for Appendix A give you a small sample of the idea.

  Exercises for Appendix A Answers to odd-numbered problems begin on page ANS-51.

The Gamma Function

In Problems 1 and 2, evaluate the given quantity.

In Problems 3–6, use (7) and the known value to evaluate the given quantity.

In Problems 7–10, use (13) and the known value to evaluate the given quantity.

In Problems 11–14, use (7) and the given numerical value to evaluate the indicated quantity.

  5. Express in terms of
  6. For use a substitution to show that

In Problems 17–20, use the result in Problem 16 to express the given improper integral as a gamma function. Then evaluate the integral.

In Problems 21–24, use the substitution and the result in Problem 16 to express the given improper integral as a gamma function. Then evaluate the integral.

In Problems 25 and 26, use the indicated substitution to express the given improper integral as a gamma function. Then evaluate the integral using the indicated numerical value.

    1. For the gamma function possesses derivatives of all orders. Use Leibniz’s rule in the form given in (24) to show that

    2. For Without looking back at Figure A.1, what does this say about the graph of
  4. A definition of the gamma function given by Carl Friedrich Gauss in 1811 that is valid for all real numbers, except is

    Use this definition to give an alternative derivation of the recursion formula (7).

The Beta Function

In Problems 29 and 30, express the given improper integral as a beta function. Then evaluate the integral using (15) and known values of the gamma function.

In Problems 31 and 32, use the indicated substitution to express the given improper integral as a beta function. Then evaluate the integral using (15) and known values of the gamma function.

In Problems 33 and 34, use the substitution to express the given improper integral as a beta function. Then evaluate the integral using (15) and known values of the gamma function.

  3. Show that the beta function is symmetric in x and y; that is,
  4. Show that if and are positive integers, then (15) becomes

  5. Use the substitution to show that the beta function (14) can be expressed as

  6. If and then show that the integral in Problem 37 can be written

In Problems 39–42, use the result in Problem 38 to evaluate the given integral.

Leibniz’s Rule

In Problems 43–48, use Leibniz’s rule in the forms given in (22) and (23) to find the derivative of the given function.

In Problems 49 and 50, use Leibniz’s rule in the form given in (24) to show that the indicated integral-defined function is a solution of the given second-order differential equation. [Hint: After computing the first and second derivatives find a term in that can be evaluated using integration by parts.]

  3. Assume that Leibniz’s rule (24) is valid for the improper integral

    Show that this integral-defined function is a solution of the second-order differential equation [Hint: After computing the first and second derivatives find a term in that can be evaluated using integration by parts.]

    1. Find the derivative of the error function Then use the identity to find the derivative of the complementary error function
    2. Use the last result in part (a) to show that

      is a solution of the differential equation

    1. Show that for

      [Hint: See formula 36 in the table of integrals given on the right inside page of the front cover.]

    2. Use the result in part (a) and (24) to show that

  6. Consider the function

    1. Use (24) to find as an integral. Evaluate this integral.
    2. Use the result in part (a) to find where C is a constant.
    3. Use the result in part (b) and to find an explicit function
  7. Consider the function

    1. Use (24) to find as an integral. Evaluate this integral using a substitution and the known result (See page APP-4.)
    2. Use the result in part (a) to find where C is a constant.
    3. Use the result in part (b) and to find an explicit function
  8. Consider the function

    1. Use (24) to find as an integral. Evaluate this integral using integration by parts.
    2. Use the result in part (a) to find where C is a constant.
    3. Use the result in part (b) and to find an explicit function
  9. Express the function in terms of
  10. Evaluate the integral in terms of