Handbook of mathematics for engineers and scienteists part 50 pps

7.2. DEFINITE INTEGRAL 311 7.2.11-2. Principal value of a singular integral. 1 ◦ . Consider the integral  b a dx x – c , a < c < b. Evaluating this integral as an improper integral, we obtain  b a dx x – c = lim ε 1 →0 ε 2 →0  –  c–ε 1 a dx c – x +  b c+ε 2 dx x – c  =ln b – c c – a + lim ε 1 →0 ε 2 →0 ln ε 1 ε 2 .(7.2.11.2) The limit of the last expression obviously depends on the way in which ε 1 and ε 2 tend to zero. Hence, the improper integral does not exist. This integral is called a singular integral. However, this integral can be assigned a meaning if we assume that there is some relationship between ε 1 and ε 2 . For example, if the deleted interval is symmetric with respect to the point c, i.e., ε 1 = ε 2 = ε,(7.2.11.3) we arrive at the notion of the Cauchy principal value of a singular integral. The Cauchy principal value of the singular integral  b a dx x – c , a < c < b is the number lim ε→0   c–ε a dx x – c +  b c+ε dx x – c  . With regard to formula (7.2.11.2), we have  b a dx x – c =ln b – c c – a .(7.2.11.4) 2 ◦ . Consider the more general integral  b a f(x) x – c dx,(7.2.11.5) where f(x) [a, b] is a function satisfying the H ¨ older condition. Let us understand this integral in the sense of the Cauchy principal value, which we define as follows:  b a f(x) x – c dx = lim ε→0   c–ε a f(x) x – c dx +  b c+ε f(x) x – c dx  . We have the identity  b a f(x) x – c dx =  b a f(x)–f(c) x – c dx + f(c)  b a dx x – c ; moreover, the first integral on the right-hand side is convergent as an improper integral, because it follows from the H ¨ older condition that     f(x)–f(c) x – c     < A |x – c| 1–λ , 0 < λ ≤ 1, and the second integral coincides with (7.2.11.4). 312 INTEGRALS Thus, we see that the singular integral (7.2.11.5), where f(x) satisfies the H ¨ older condition, exists in the sense of the Cauchy principal value and is equal to  b a f(x) x – c dx =  b a f(x)–f(c) x – c dx + f(c)ln b – c c – a . Some authors denote singular integrals by special symbols like v.p.  (valeur principale). However, this is not necessary because, on the one hand, if an integral of the form (7.2.11.5) exists as a proper or an improper integral, then it exists in the sense of the Cauchy principal value, and their values coincide; on the other hand, we shall always understand a singular integral in the sense of the Cauchy principal value. For this reason, we denote a singular integral by the usual integral sign. 7.2.12. Stieltjes Integral 7.2.12-1. Basic definitions. Let f(x)andϕ(x) be functions definedonaninterval[a, b]. Let us partition this interval into n elementary subintervals defined by a set of points {x 0 , x 1 , , x n } such that a = x 0 < x 1 < ···< x n = b. Each subinterval [x k–1 , x k ] will be characterized by its length Δx k = x k – x k–1 and an arbitrarily chosen point ξ k [x k–1 , x k ]. Let us make up a Stieltjes integral sum s n = n  k=1 f(ξ k )Δ k ϕ(x), where Δ k ϕ(x)=ϕ(x k )–ϕ(x k–1 ) is the increment of the function ϕ(x)onthekth elementary subinterval. If there exists a limit of the integral sums s n , as the number of subintervals n increases indefinitely so that the length of every subinterval Δx k vanishes, and this limit depends on neither the way the interval [a, b] was partitioned nor the way the points ξ k were selected, then this limit is called the Stieltjes integral of the function f(x) with respect to the function ϕ(x) over the interval [a, b]:  b a f(x) dϕ(x) = lim λ→0 s n  max 1≤k≤n Δx k → 0 as n →∞  . Then f(x) is called an integrable function with respect to ϕ(x), and ϕ(x) is called an integrating function. The Stieltjes integral is a generalization of the Riemann integral; the latter corresponds to the special case ϕ(x)=x + const. 7.2.12-2. Properties of the Stieltjes integral. The Stieltjes integral has properties analogous to those of the definite Riemann integral: 1)  b a dϕ(x)=ϕ(b)–ϕ(a); 2)  b a  Af(x) Bg(x)  dϕ(x)=A  b a f(x) dϕ(x) B  b a g(x) dϕ(x); 7.2. DEFINITE INTEGRAL 313 3)  b a f(x) d[Aϕ(x) Bψ(x)] = A  b a f(x) dϕ(x) B  b a f(x) dψ(x); 4)  b a f(x) dϕ(x)=  c a f(x) dϕ(x)+  b c f(x) dϕ(x)(a < c < b). It is assumed that all integrals on the left- and right-hand sides exist. T HEOREM (MEAN VALUE). If a function f(x) satisfies inequalities m ≤ f (x) ≤ M on an interval [a, b] and is integrable with respect to an increasing function ϕ(x) ,then  b a f(x) dϕ(x)=μ[ϕ(b)–ϕ(a)], where m < μ < M . 7.2.12-3. Existence theorems for the Stieltjes integral. The existence of the Stieltjes integral and its reduction to the Riemann integral is established by the following theorem. T HEOREM 1. If f(x) is continuous on [a, b] and ϕ(x) has a bounded variation* on [a, b] , then the integral  b a f(x) dϕ(x) exists. THEOREM 2. Let f(x) be integrable on [a, b] in the sense of Riemann and let ϕ(x) satisfy the Lipschitz condition |ϕ(x 2 )–ϕ(x 1 )| < K|x 2 – x 1 |, where x 1 and x 2 are arbitrary points of the interval [a, b] and K is a fixed positive constant. Then the function f(x) is integrable with respect to the function ϕ(x) . THEOREM 3. Let f(x) be integrable on [a, b] in the sense of Riemann and let ϕ(x) be differentiable and have an integrable derivative on [a, b] . Then the function f(x) is integrable with respect to the function ϕ(x) and, moreover,  b a f(x) dϕ(x)=  b a f(x)ϕ  (x) dx, where the integral on the right-hand side is understood in the sense of Riemann. Remark. If a function f(x) is integrable on an interval [a, b] with respect to a function ϕ(x), then, vice versa, the function ϕ(x) is also integrable with respect to the function f (x)on[a, b]. Owing to this property, the functions f(x)andϕ(x) are interchangeable in Theorems 1 and 2. THEOREM 4. Let f(x) be continuous on [a, b] and let ϕ(x) have an absolutely integrable derivative ϕ  (x) everywhere on [a, b] , except, perhaps, finitely many points. Let, in addition, the function ϕ(x) undergo a jump discontinuity at finitely many points a = c 0 < c 1 < ··· < c m = b. * A function ϕ(x)issaidtohaveabounded variation on an interval [a, b] if there exists a number M > 0 such that for any set of points a = x 0 < x 1 < ···< x n = b the inequality n  k=1 |ϕ(x k+1 )–ϕ(x k )| < M holds (see also Subsection 6.1.7). 314 INTEGRALS Then the Stieltjes integral exists and is calculated as  b a f(x) dϕ(x)=  b a f(x)ϕ  (x) dx + f(a)[ϕ(a + 0)–ϕ(a)] + m–1  k=1 f(c k )[ϕ(c k + 0)–ϕ(c k – 0)] + f(b)[ϕ(b)–ϕ(b – 0)], where the right-hand side contains a Riemann integral. Note the presence of terms outside the integral on the right-hand side, where, apart from the ordinary jumps of the function ϕ(x) at the internal points of discontinuity, there are terms with one-sided jumps at the endpoints (if there is no jump at either endpoint, the corresponding term vanishes). The Stieltjes integral is useful for finding static moments, moments of inertia, and some other distributed quantities on an interval [a, b], where, apart from continuous distributions, there are concentrated quantities like point masses that correspond to a discontinuous function ϕ(x) with finite jumps. 7.2.13. Square Integrable Functions 7.2.13-1. Definitions. A function f(x)issaidtobesquare integrable on an interval [a, b]iff 2 (x) is integrable on [a, b]. The set of all square integrable functions is denoted by L 2 (a, b)or,briefly, L 2 .* Likewise, the set of all integrable functions on [a, b] is denoted by L 1 (a, b)or,briefly, L 1 . 7.2.13-2. Basic properties of functions from L 2 . 1 ◦ . The sum of two square integrable functions is a square integrable function. 2 ◦ . The product of a square integrable function by a constant is a square integrable function. 3 ◦ . The product of two square integrable functions is an integrable function. 4 ◦ .Iff(x) L 2 and g(x) L 2 , then the following Cauchy–Schwarz–Bunyakovsky inequal- ity holds: (f, g) 2 ≤ f 2 g 2 , (f, g)=  b a f(x)g(x) dx, f 2 =(f, f)=  b a f 2 (x) dx. The number (f, g) is called the inner product of the functions f (x)andg(x)andthe number f is called the L 2 -norm of f(x). 5 ◦ .Forf(x) L 2 and g(x) L 2 , the following triangle inequality holds: f + g ≤ f + g. 6 ◦ . Let functions f(x)andf 1 (x), f 2 (x), , f n (x), be square integrable on an interval [a, b]. If lim n→∞  b a  f n (x)–f(x)  2 dx = 0, * In the most general case the integral is understood as the Lebesgue integral of measurable functions. As usual, two equivalent functions—i.e., equal everywhere, or distinct on a negligible set (of zero measure)—are regarded as one and the same element of L 2 . 7.2. DEFINITE INTEGRAL 315 then the sequence f 1 (x), f 2 (x), is said to be mean-square convergent to f (x). Note that if a sequence of functions {f n (x)} from L 2 converges uniformly to f(x), then f(x) L 2 and {f n (x)} is mean-square convergent to f(x). 7.2.14. Approximate (Numerical) Methods for Computation of Definite Integrals 7.2.14-1. Rectangle, trapezoidal, and Simpson’s rules. For approximate computation of an integral like  b a f(x) dx, let us break up the interval [a, b]inton equal subintervals with length h = b – a n . Introduce the notation: x 0 = a, x 1 , , x n = b (the partition points), y i = f (x i ), i = 0, 1, , n. 1 ◦ . Rectangle rules:  b a f(x) dx ≈ h(y 0 + y 1 + ···+ y n–1 ),  b a f(x) dx ≈ h(y 1 + y 2 + ···+ y n ). The error of these formulas, R n , is proportional to h and is estimated using the inequality |R n | ≤ 1 2 h(b – a)M 1 , M 1 =max a≤x≤b   f  (x)   . 2 ◦ . Trapezoidal rule:  b a f(x) dx ≈ h  y 0 + y n 2 + y 1 + y 2 + ···+ y n–1  . The error of this formula is proportional to h 2 and is estimated as |R n | ≤ 1 12 h 2 (b – a)M 2 , M 2 =max a≤x≤b   f  (x)   . 3 ◦ . Simpson’s rule:  b a f(x) dx ≈ 1 3 h[y 0 + y n + 4(y 1 + y 3 + ···+ y n–1 )+2(y 2 + y 4 + ···+ y n–2 )], where n is even. The error of approximation by Simpson’s rule is proportional to h 4 : |R n | ≤ 1 180 h 4 (b – a)M 4 , M 4 =max a≤x≤b   f (4) (x)   . Simpson’s rule yields exact results for the case where the integrand function is a polynomial of degree two or three. Remark. Theaboveapproximation formulas are often used for numerical computation ofdefinite integrals; to achieve a higher accuracy, large n are normally taken. 316 INTEGRALS 7.2.14-2. Computation of integrals using uniformly distributed sequences. 1 ◦ . Consider a sequence of real numbers {x n } whose members all belong to the closed interval [0, 1]. Let ν n (a, b) be the number of the members of the sequence with subscript < n that belong to[a, b] [0, 1]. The sequence{x n } is calleduniformly distributed on the interval [0, 1]if lim n→∞ ν n (a, b) n = b – a. In the language of probability theory, this definition means that the probability of a randomly selected element of the sequence to fall into a subinterval [a, b] is equal to the length of this subinterval. A simple example of a uniformly distributed sequence is the sequence of all proper fractions m/n with n = 2, 3, and m = 1, 2, n– 1: 1 2 , 1 3 , 2 3 , 1 4 , 2 4 , 3 4 , The following theorem serves as an unlimited source for constructing uniformly dis- tributed sequences. T HEOREM 1. Let ϑ be an arbitrary irrational number. Then the sequence x n = nϑ–[nϑ] , n = 1, 2, , is uniformly distributed on the interval [0, 1] ; the symbol [z] stands for the integer part of z (the maximum integer not exceeding z ). 2 ◦ . Uniformly distributed sequences can be used for the calculation of integrals on the basis of the following two theorems. T HEOREM 2. If a number sequence {x n } is uniformly distributed on the interval [0, 1] , then the following limiting relation holds for any function f(x) integrable on [0, 1] :  1 0 f(x) dx = lim n→∞ 1 n n  k=1 f(x k ). ( 7 .2.14.1) Conversely, if this relation holds, it follows that the sequence {x n } is uniformly distributed on [0, 1] . Remark. Formula (7.2.14.1) can be used for approximate calculation of integrals by defining a uniformly distributed sequence {x n } and computing the sum with a large n. The integrals definedonanarbitraryinterval [a, b]arefirst reduced by the change of variable z = x – a b – a to an integral over the interval [0, 1]. THEOREM 3. If a number sequence {x n } is uniformly distributed on [0, 1] , then the following limiting relation holds for any function f(x) integrable on [0, 1] and any mono- tonically decreasing sequence of positive numbers {α n } with a divergent sum:  1 0 f(x) dx = lim n→∞ α 1 f(x 1 )+α 2 f(x 2 )+···+ α n f(x n ) α 1 + α 2 + ···+ α n . 7.3. DOUBLE AND TRIPLE INTEGRALS 317 7.2.14-3. Computation of integrals by series expansion of the integrand function. THEOREM 4. If a sequence of functions, {f n (x)} , continuous on an interval [a, b] uni- formly converges on [a, b] to a function f(x) , then the sequence of functions  b a f n (x) dx  uniformly converges on [a, b] to the function   b a f(x) dx  . THEOREM 5. If the functions of a sequence {u n (x)} are continuous on an interval [a, b] and the series ∞  n=1 u n (x) uniformly converges on [a, b] to a function f(x) , then the series ∞  n=1  x a u n (t) dt is also uniformly convergent on [a, b] and ∞  n=1  b a u n (x) dx =  b a  ∞  n=1 u n (x) dx  =  b a f(x) dx; ∞  n=1  x a u n (t) dt =  x a  ∞  n=1 u n (t) dt  =  x a f(t) dt (a ≤ x ≤ b). This theorem is used for the calculation of integrals by expanding the integrand func- tions f (x) into a uniformly convergent series. Remark 1. Since a series of positive continuous functions with a continuous sum is always uniformly convergent, then this series can be integrated termwise. Remark 2. In general, the convergence of a sequence of continuous integrand functions, {f n (x)}, at each point of the integration interval does not guarantee that it is permitted to proceed to the limit under the sign of integral. 7.3. Double and Triple Integrals 7.3.1. Definition and Properties of the Double Integral 7.3.1-1. Definition and properties of the double integral. Suppose there is a bounded set of points defined on the plane, so that it can be placed in a minimal enclosing circle. The diameter of this circle is called the diameter of the set. Consider a domain D in the x, y plane. Let us partition D into n nonintersecting subdomains (cells). The largest of the cell diameters is called the partition diameter and is denoted λ = λ(D n ), where D n stands for the partition of the domain D into cells. Let a function z = f(x, y)bedefined in D. Select an arbitrary point in each cell (x i , y i ), i = 1, 2, , n, and make up an integral sum, s n = n  i=1 f(x i , y i ) ΔS i , where ΔS i is the area of the ith subdomain. If there exists a finite limit, J,ofthesumss n as λ → 0 and it depends on neither the partition D n nor the selection of the points (x i , y i ), this limit is denoted  D f(x, y) dx dy and is called the double integral of the function f(x, y) over the domain D:  D f(x, y) dx dy = lim λ→0 s n . This means that for any ε > 0 there exists a δ > 0 such that for all partitions D n such that λ(D n )<δ and for any selection of the points (x i , y i ), the inequality |s n – J| < ε holds. . is uniformly distributed on the interval [0, 1] ; the symbol [z] stands for the integer part of z (the maximum integer not exceeding z ). 2 ◦ . Uniformly distributed sequences can be used for the. of integral. 7.3. Double and Triple Integrals 7.3.1. Definition and Properties of the Double Integral 7.3.1-1. Definition and properties of the double integral. Suppose there is a bounded set of. for the case where the integrand function is a polynomial of degree two or three. Remark. Theaboveapproximation formulas are often used for numerical computation ofdefinite integrals; to achieve