Handbook of mathematics for engineers and scienteists part 47 ppt

If the density is continuous, thenΠf satisfies the Laplace equation ΔΠ =0outside z [–1,1] and vanishes at infinity... Watson’s formula also holds for improper integrals with a = ∞ if the

 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–1πn

 π/2

0 f (sin x) dx if f (–x) = –f (x);

 2π

0 f (a sin x + b cos x) dx =

 2π

2+ b2 sin x

dx=2 π

0 f a

2+ b2 cos x

dx;

 π

0 f



sin2x

1+2a cos x + a2



dx=

 π

0 f(sin

2x ) dx if |a| ≥ 1;

 π

0 f



sin2x

1+2a cos x + a2



dx=

 π

0 f



sin2x

a2



dx if 0<|a|<1

7.2.3-3 Integrals involving logarithmic functions

 b

a f (x) ln

n x dx= 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)]

λlnn 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–1f (x) dx

(x + ε) α ,

where0< a < ∞, β >0, f (0)≠ 0, and ε >0is a small parameter

The integral diverges as ε →0for α≥β, that is, lim

ε→0I (ε) = ∞ In this case, the leading

term of the asymptotic expansion of the integral I(ε) is given by

I (ε) = Γ(β)Γ(α – β)

Γ(α) f(0)ε β–α + O(ε σ ) if α > β,

I (ε) = –f (0) ln ε + O(1) if α = β,

whereΓ(β) is the gamma function and σ = min[β – α +1,0]

2◦ The leading term of the asymptotic expansion, as ε →0, of the more general integral

I (ε) =

 a 0

x β–1f (x) dx

(x k + ε k)α with0< a < ∞, β > 0, k >0, ε >0, and f (0)≠ 0is expressed as

I (ε) = f(0)

k Γ(α)Γ



β k

 Γ



α– β

k



ε β–αk + O(ε σ) if αk > β,

where σ = min[β – αk +1,0]

3◦ The leading terms of the asymptotic expansion, as ε →0, of the integral

I (ε) =

 ∞

αexp –εx β

f (x) dx with a >0, β >0, ε >0, and f (0)≠ 0has the form

I (ε) = 1

β f(0)Γ



α+1

β



ε–α+ β1

if α> –1,

I (ε) = –1

4◦ Now consider potential-type integrals

Π(f) =

 1

– 1

f (ξ) dξ

(ξ – z)2+ r2, 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, thenΠ(f) satisfies the Laplace equation ΔΠ =0outside z [–1,1] and vanishes at infinity

Asymptotics of the integral as r →0:

Π(f) = –2f (z) ln r + O(1), where|z| ≤ 1– δ with0< 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–1exp –λx α

f (x) dx,

where0< a < ∞, α >0, and β >0

The following formula, called Watson’s asymptotic formula, holds as λ → ∞:

I (λ) = 1

α

n



k=0

f(k)(0)

k! Γk + β

α



λ–(k+β)/α + O λ–(n+β+1 )/α

Remark 1. Watson’s formula also holds for improper integrals with a = ∞ if the original integral converges

absolutely for some λ0> 0

Remark 2. Watson’s formula remains valid in the case of complex parameter λ as |λ| → ∞, where

|arg λ| ≤ π

2 – ε < π2 (ε >0can 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

with0< a < ∞, β >0, and f (0)≠ 0is 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

where [a, b] is a finite 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 point x0[a, b] and is differentiable

in a neighborhood of it, with g  (x0) = 0, g  (x0)≠ 0, and f (x0) ≠ 0 Then the leading term

of the asymptotic expansion of the integral (7.2.4.1), as λ → ∞, is expressed as

I (λ) = f (x0)

!

λg  (x0) exp[λg(x0)] if a < x0< b,

I (λ) = 1

2f (x0)

!

λg  (x0) exp[λg(x0)] if x0= a or x0= b.

(7.2.4.2)

Note that the latter formula differs from the former by the factor1/2only

Under the same conditions, if g(x) attains a maximum at either endpoint, x0 = a or

x0= b, but g  (x0)≠ 0, then the leading asymptotic term of the integral, as λ → ∞, is

I (λ) = f (x0)

|g  (x0)|

1

λ exp[λg(x0)], where x0 = a or x0 = b. (7.2.4.3) For more accurate asymptotic estimates for the Laplace integral (7.2.4.1), see below

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, x0(a < x0< b), with g  (x0) =0

and g  (x0) ≠ 0, and let the functions f (x) and g(x) be, respectively, n and n +1 times

differentiable in a neighborhood of x = x0 Then the asymptotic formula

I (λ) = exp[λg(x0)]

n–1

k=0

c k λ–k–1+ O(λ–n)

(7.2.4.4)

holds as λ → ∞, with

c k= 1

(2k)!2k+1 2Γ



k+ 1 2



lim

x→x0



d dx

k

f (x)



g (x0) – g(x) (x – x0)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) and g(x) are, respectively, n and n +1times 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)

where

c0= –g f  (a)

(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 λ0 > 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 < π2 (ε >0can be chosen arbitrarily small but independent of λ).

3◦ Some generalizations Let g(x) attain a maximum at only one internal point of the interval, x0 (a < x0 < b), with g  (x0) =· · · = g( 2m–1 )(x0) =0and g(2m) (x0) ≠ 0, m≥ 1

and f (x0) ≠ 0 Then the leading asymptotic term of the integral (7.2.4.1), as λ → ∞, is

expressed as

I (λ) = 1

mΓ

 1

2m



f (x0)



– (2m)!

g( 2m) (x0)

 1

2m

λ– 1

2m exp[λg(x0)]

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, where m ≥ 1 and f (a) ≠ 0 Then the leading asymptotic term of the

integral (7.2.4.1), as λ → ∞, has the form

I (λ) = 1

mΓ

 1

m



f (a)



– m!

g(m) (a)

1

m

λ–1

m exp[λg(a)].

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) for g(x):

I (λ) =

 b

a f (x)[g(x)]

where [a, b] is a finite 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 x0= [a, b] and to be differentiable in a neighborhood of x = x0,

with g  (x0) =0, g  (x0)≠ 0, and f (x0)≠ 0 Then the leading asymptotic term of the integral,

as λ → ∞, is expressed as

I (λ) = f (x0)

!

λg  (x0) [g(x0)]λ+1 2 if a < x0< b,

I (λ) = 1

2f (x0)

!

λg  (x0) [g(x0)]λ+1 2 if x0= a or x0= b.

Note that the latter formula differs from the former by the factor1/2only

Under the same conditions, if g(x) attains a maximum at either endpoint, x0 = a or

x0= b, but g  (x0)≠ 0, then the leading asymptotic term of the integral, as λ → ∞, is

I (λ) = f (x0)

|g  (x0)|

1

λ [g(x0)]λ+1 2, where x0= a or x0= 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 Let f (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) on a finite interval [a, b].

2◦ Dirichlet’s formula Let f (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

2f(+0).

7.2.4-7 Limiting properties of other integrals with parameter

Let f (x) and g(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 = maxx [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

THEOREM1 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].

THEOREM2 If f (x) is a continuous function on [a, b], and g(x) is integrable and of constant sign (g(x)≥ 0or g(x)≤ 0) on [a, b], then there exists at least one point c(a, b)

a f (x)g(x) dx = f (c)

 b

a g (x) dx.

THEOREM3 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.

THEOREM4 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]

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 If m≤f (x)≤M on [a, b], then

m (b – a)≤

 b

a f (x) dx≤M (b – a).

2 Inequality integration theorem If ϕ(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)≥ 0on [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)]

2dx1 2

≤ b

a f

2(x) dx1 2

+

 b

a g

2(x) dx1 2

6 Minkowski’s inequality (generalization of Cauchy’s inequality):

 b

a |f (x) + g(x)|p dx1

p

≤ b

a |f (x)|p dx1

p

+

 b

a |g (x)|p dx1

p

7.2.4-2 Asymptotic formulas for Laplace integrals of special form as λ → ∞.

1◦ Consider a Laplace integral of the special form

I... =0

and g  (x0) ≠ 0, and let the functions f (x) and g(x) be, respectively, n and n +1 times

differentiable in a neighborhood of x = x0... general form as λ → ∞.

Consider a Laplace integral of the general form

I (λ) =

 b

where [a, b] is a finite interval and f (x),