290 INTEGRALS nπ 0 xf(sin x) dx = πn 2 π/2 0 f(sin x) dx if f(x)=f(–x); nπ 0 xf(sin x) dx =(–1) n–1 πn π/2 0 f(sin x) dx if f(–x)=–f (x); 2π 0 f(a sin x + b cos x) dx = 2π 0 f √ a 2 + b 2 sin x dx = 2 π 0 f √ a 2 + b 2 cos x dx; π 0 f sin 2 x 1 + 2a cos x + a 2 dx = π 0 f(sin 2 x) dx if |a| ≥ 1; π 0 f sin 2 x 1 + 2a cos x + a 2 dx = π 0 f sin 2 x a 2 dx if 0 < |a| < 1. 7.2.3-3. Integrals involving logarithmic functions. b a f(x)ln n xdx= d dλ n b a x λ f(x) dx λ=0 , b a f(x)ln n g(x) dx = d dλ n b a f(x)[g(x)] λ dx λ=0 , b a f(x)[g(x)] λ ln n g(x) dx = d dλ n b a f(x)[g(x)] λ dx. 7.2.4. General Asymptotic Formulas for the Calculation of Integrals Below are some general formulas, involving arbitrary functions and parameters, that may be helpful for obtaining asymptotics of integrals. 7.2.4-1. Asymptotic formulas for integrals with weak singularity as ε → 0. 1 ◦ . We will consider integrals of the form I(ε)= a 0 x β–1 f(x) dx (x + ε) α , where 0 < a < ∞, β > 0, f(0) ≠ 0,andε > 0 is a small parameter. The integral diverges as ε →0 for α ≥ β, that is, lim ε→0 I(ε)=∞. In this case, the leading term of the asymptotic expansion of the integral I(ε)isgivenby I(ε)= Γ(β)Γ(α – β) Γ(α) f(0)ε β–α + O(ε σ )ifα > β, I(ε)=–f(0)lnε + O(1)ifα = β, where Γ(β) is the gamma function and σ =min[β – α + 1, 0]. 7.2. DEFINITE INTEGRAL 291 2 ◦ . The leading term of the asymptotic expansion, as ε → 0, of the more general integral I(ε)= a 0 x β–1 f(x) dx (x k + ε k ) α with 0 < a < ∞, β > 0, k > 0, ε > 0,andf (0) ≠ 0 is expressed as I(ε)= f(0) kΓ(α) Γ β k Γ α – β k ε β–αk + O(ε σ )ifαk > β, I(ε)=–f(0)lnε + O(1)ifαk = β, where σ =min[β – αk + 1, 0]. 3 ◦ . The leading terms of the asymptotic expansion, as ε → 0, of the integral I(ε)= ∞ a x α exp –εx β f(x) dx with a > 0, β > 0, ε > 0,andf (0) ≠ 0 has the form I(ε)= 1 β f(0)Γ α + 1 β ε – α+1 β if α >–1, I(ε)=– 1 β f(0)lnε if α =–1. 4 ◦ . Now consider potential-type integrals Π(f)= 1 –1 f(ξ) dξ (ξ – z) 2 + r 2 , with z, r, ϕ being cylindrical coordinates in the three-dimensional space. The function Π(f) is simple layer potential concentrated on the interval z [–1, 1] with density f (z). If the density is continuous, then Π(f) satisfies the Laplace equation ΔΠ = 0 outside z [–1, 1] and vanishes at infinity. Asymptotics of the integral as r → 0: Π(f)=–2f (z)lnr + O(1), where |z| ≤ 1 – δ with 0 < δ < 1. 7.2.4-2. Asymptotic formulas for Laplace integrals of special form as λ →∞. 1 ◦ . Consider a Laplace integral of the special form I(λ)= a 0 x β–1 exp –λx α f(x) dx, where 0 < a < ∞, α > 0,andβ > 0. 292 INTEGRALS The following formula, called Watson’s asymptotic formula, holds as λ →∞: I(λ)= 1 α n k=0 f (k) (0) k! Γ k + β α λ –(k+β)/α + O λ –(n+β+1)/α . Remark 1. Watson’s formula also holds for improper integrals with a = ∞ifthe original integral converges absolutely for some λ 0 > 0. Remark 2. Watson’s formula remains valid in the case of complex parameter λ as |λ| →∞,where |arg λ| ≤ π 2 – ε < π 2 (ε > 0 can be chosen arbitrarily small but independent of λ). Remark 3. The Laplace transform corresponds to the above integral with a = ∞ and α = β = 1. 2 ◦ . The leading term of the asymptotic expansion, as λ →∞, of the integral I(λ)= a 0 x β–1 |ln x| γ e –λx f(x) dx with 0 < a < ∞, β > 0,andf (0) ≠ 0 is expressed as I(λ)=Γ(β)f (0)λ –β (ln λ) γ . 7.2.4-3. Asymptotic formulas for Laplace integrals of general form as λ →∞. Consider a Laplace integral of the general form I(λ)= b a f(x)exp[λg(x)] dx,(7.2.4.1) where [a, b]isafinite interval and f(x), g(x) are continuous functions. 1 ◦ . Leading term of the asymptotic expansion of the integral (7.2.4.1) as λ →∞. Suppose the function g(x) attains a maximum on [a, b] at only one pointx 0 [a, b] and isdifferentiable in a neighborhood of it, with g (x 0 )=0, g (x 0 ) ≠ 0,andf(x 0 ) ≠ 0. Then the leading term of the asymptotic expansion of the integral (7.2.4.1), as λ →∞, is expressed as I(λ)=f (x 0 ) – 2π λg (x 0 ) exp[λg(x 0 )] if a < x 0 < b, I(λ)= 1 2 f(x 0 ) – 2π λg (x 0 ) exp[λg(x 0 )] if x 0 = a or x 0 = b. (7.2.4.2) Note that the latter formula differs from the former by the factor 1/2 only. Under the same conditions, if g(x) attains a maximum at either endpoint, x 0 = a or x 0 = b,butg (x 0 ) ≠ 0, then the leading asymptotic term of the integral, as λ →∞,is I(λ)= f(x 0 ) |g (x 0 )| 1 λ exp[λg(x 0 )], where x 0 = a or x 0 = b.(7.2.4.3) For more accurate asymptotic estimates for the Laplace integral (7.2.4.1), see below. 7.2. DEFINITE INTEGRAL 293 2 ◦ . Leading and subsequent asymptotic terms of the integral (7.2.4.1) as λ →∞. Let g(x) attain a maximum at only one internal point of the interval, x 0 (a < x 0 < b), with g (x 0 )=0 and g (x 0 ) ≠ 0, and let the functions f (x)andg(x) be, respectively, n and n + 1 times differentiable in a neighborhood of x = x 0 . Then the asymptotic formula I(λ)=exp[λg(x 0 )] n–1 k=0 c k λ –k–1 + O(λ –n ) (7.2.4.4) holds as λ →∞, with c k = 1 (2k)! 2 k+1/2 Γ k + 1 2 lim x→x 0 d dx k f(x) g(x 0 )–g(x) (x – x 0 ) 2 –k–1/2 . Suppose g(x) attains a maximum at the endpoint x = a only, with g (a) ≠ 0. Suppose also that f(x)andg(x) are, respectively, n and n + 1 times differentiable in a neighborhood of x = a. Then we have, as λ →∞, I(λ)=exp[λg(a)] n–1 k=0 c k λ –k–1 + O(λ –n ) ,(7.2.4.5) where c 0 =– f(a) g (a) ; c k =(–1) k+1 1 g (x) d dx k f(x) g (x) x=a , k = 1, 2, Remark 1. The asymptotic formulas (7.2.4.2)–(7.2.4.5) hold also for improper integrals with b = ∞ if the original integral (7.2.4.1) converges absolutely at some λ 0 > 0. Remark 2. The asymptotic formulas (7.2.4.2)–(7.2.4.5) remain valid also in the case of complex λ as |λ| →∞,where|arg λ| ≤ π 2 – ε < π 2 (ε > 0 can be chosen arbitrarily small but independent of λ). 3 ◦ . Some generalizations.Letg(x) attain a maximum at only one internal point of the interval, x 0 (a < x 0 < b), with g (x 0 )=··· = g (2m–1) (x 0 )=0 and g (2m) (x 0 ) ≠ 0, m ≥ 1 and f (x 0 ) ≠ 0. Then the leading asymptotic term of the integral (7.2.4.1), as λ →∞,is expressed as I(λ)= 1 m Γ 1 2m f(x 0 ) – (2m)! g (2m) (x 0 ) 1 2m λ – 1 2m exp[λg(x 0 )]. Let g(x) attain a maximum at the endpoint x = a only, with g (a)=···= g (m–1) (a)=0 and g (m) (a) ≠ 0,wherem ≥ 1 and f (a) ≠ 0. Then the leading asymptotic term of the integral (7.2.4.1), as λ →∞,hastheform I(λ)= 1 m Γ 1 m f(a) – m! g (m) (a) 1 m λ – 1 m exp[λg(a)]. 294 INTEGRALS 7.2.4-4. Asymptotic formulas for a power Laplace integral. Consider the power Laplace integral, which is obtained from the exponential Laplace integral (7.2.4.1) by substituting ln g(x)forg(x): I(λ)= b a f(x)[g(x)] λ dx,(7.2.4.6) where [a, b]isafinite closed interval and g(x)>0. It is assumed that the functions f(x) and g(x) appearing in the integral (7.2.4.6) are continuous; g(x) is assumed to attain a maximum at only one point x 0 =[a, b] and to be differentiable in a neighborhood of x = x 0 , with g (x 0 )=0, g (x 0 ) ≠ 0,andf(x 0 ) ≠ 0. Then the leading asymptotic term of the integral, as λ →∞, is expressed as I(λ)=f(x 0 ) – 2π λg (x 0 ) [g(x 0 )] λ+1/2 if a < x 0 < b, I(λ)= 1 2 f(x 0 ) – 2π λg (x 0 ) [g(x 0 )] λ+1/2 if x 0 = a or x 0 = b. Note that the latter formula differs from the former by the factor 1/2 only. Under the same conditions, if g(x) attains a maximum at either endpoint, x 0 = a or x 0 = b,butg (x 0 ) ≠ 0, then the leading asymptotic term of the integral, as λ →∞,is I(λ)= f(x 0 ) |g (x 0 )| 1 λ [g(x 0 )] λ+1/2 ,wherex 0 = a or x 0 = b. 7.2.4-5. Asymptotic behavior of integrals with variable integration limit as x →∞. Let f(t) be a continuously differentiable function and let g(t) be a twice continuously differentiable function. Also let the following conditions hold: f(t)>0, g (t)>0; g(t) →∞ as t →∞; f (t)/f(t)=o g (t) as t →∞; g (t)=o g 2 (t) as t →∞. Then the following asymptotic formula holds, as x →∞: x 0 f(t)exp[g(t)] dt f(x) g (x) exp[g(x)]. 7.2.4-6. Limiting properties of integrals involving periodic functions with parameter. 1 ◦ . Riemann property of integrals involving periodic functions.Letf(x) be a continuous function on a finite interval [a, b]. Then the following limiting relations hold: lim λ→∞ b a f(x)sin(λx) dx = 0, lim λ→∞ b a f(x)cos(λx) dx = 0. Remark. The condition of continuity of f(x) can be replaced by the more general condition of absolute integrability of f(x)onafinite interval [a, b]. 7.2. DEFINITE INTEGRAL 295 2 ◦ . Dirichlet’s formula.Letf(x) be a monotonically increasing and bounded function on a finite interval [0, a], with a > 0. Then the following limiting formula holds: lim λ→∞ a 0 f(x) sin(λx) x dx = π 2 f(+0). 7.2.4-7. Limiting properties of other integrals with parameter. Let f(x)andg(x) be continuous and positive functions on [a, b]. Then the following limiting relations hold: lim n→∞ n I n =max x [a,b] f(x), lim n→∞ I n+1 I n =max x [a,b] f(x), where I n = b a g(x)[f(x)] n dx. 7.2.5. Mean Value Theorems. Properties of Integrals in Terms of Inequalities. Arithmetic Mean and Geometric Mean of Functions 7.2.5-1. Mean value theorems. THEOREM 1. If f(x) is a continuous function on [a, b] , there exists at least one point c (a, b) such that b a f(x) dx = f (c)(b – a). The number f(c) is called the mean value of the function f(x) on [a, b] . THEOREM 2. If f(x) is a continuous function on [a, b] ,and g(x) is integrable and of constant sign ( g(x) ≥ 0 or g(x) ≤ 0 )on [a, b] , then there exists at least one point c (a, b) such that b a f(x)g(x) dx = f(c) b a g(x) dx. T HEOREM 3. If f(x) is a monotonic and nonnegative function on an interval (a, b) , with a ≥ b ,and g(x) is integrable, then there exists at least one point c (a, b) such that b a f(x)g(x) dx = f(a) c a g(x) dx if f(x) is nonincreasing ; b a f(x)g(x) dx = f(b) b c g(x) dx if f(x) is nondecreasing . T HEOREM 4. If f(x) and g(x) are bounded and integrable functions on an interval [a, b] , with a < b ,and g(x) satisfies inequalities A ≤ g(x) ≤ B , then there exists a point c [a, b] 296 INTEGRALS such that b a f(x)g(x) dx = A c a f(x) dx + B b c f(x) dx if g(x) is nondecreasing ; b a f(x)g(x) dx = B c a f(x) dx + A b c f(x) dx if g(x) is nonincreasing ; b a f(x)g(x) dx = g(a) c a f(x) dx + g(b) b c f(x) dx if g(x) is strictly monotonic . 7.2.5-2. Properties of integrals in terms of inequalities. 1. Estimation theorem.Ifm ≤ f(x) ≤ M on [a, b], then m(b – a) ≤ b a f(x) dx ≤ M(b – a). 2. Inequality integration theorem.Ifϕ(x) ≤ f(x) ≤ g(x)on[a, b], then b a ϕ(x) dx ≤ b a f(x) dx ≤ b a g(x) dx. In particular, if f(x) ≥ 0 on [a, b], then b a f(x) dx ≥ 0. Further on, it is assumed that the integrals on the right-hand sides of the inequalities of Items 3–8 exist. 3. Absolute value theorem (integral analogue of the triangle inequality): b a f(x) dx ≤ b a |f(x)| dx. 4. Bunyakovsky’s inequality (Cauchy–Bunyakovsky inequality): b a f(x)g(x) dx 2 ≤ b a f 2 (x) dx b a g 2 (x) dx. 5. Cauchy’s inequality: b a [f(x)+g(x)] 2 dx 1/2 ≤ b a f 2 (x) dx 1/2 + b a g 2 (x) dx 1/2 . 6. Minkowski’s inequality (generalization of Cauchy’s inequality): b a |f(x)+g(x)| p dx 1 p ≤ b a |f(x)| p dx 1 p + b a |g(x)| p dx 1 p , p ≥ 1. . General Asymptotic Formulas for the Calculation of Integrals Below are some general formulas, involving arbitrary functions and parameters, that may be helpful for obtaining asymptotics of integrals. 7.2.4-1 > 0,andf (0) ≠ 0 is expressed as I(λ)=Γ(β)f (0)λ –β (ln λ) γ . 7.2.4-3. Asymptotic formulas for Laplace integrals of general form as λ →∞. Consider a Laplace integral of the general form I(λ)= b a f(x)exp[λg(x)]. 1] and vanishes at infinity. Asymptotics of the integral as r → 0: Π(f)=–2f (z)lnr + O(1), where |z| ≤ 1 – δ with 0 < δ < 1. 7.2.4-2. Asymptotic formulas for Laplace integrals of special form