is an asymptotic expansion if for any

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Asymptotic sequence and expansions

Lesson 6

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Definition of asymptotic expansion as x → ∞

The expansion

^f^{(x) =} â0 + â_x¹ + _xâ²² + . . . + _xâ^N_N + R_N

is an asymptotic expansion if for any

^N

R_N = O( 1

x^N⁺¹) x → ∞

In this case we write

f(x) ∼

∞

X

n=0

a_n

xⁿ

as

^x ^{→ ∞} ⁽

not

ⁿ ^{→ ∞)}

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Finding the coefficients

•

How to find

^a0, a₁, a₂, . . .

? Since

^f^{(x) =} ^a0 + O

1 x

, then

^a0 = lim

x→∞

f(x)

•

From

^f^{(x) =} ^a0 + ^a_x¹ + O

1 x²

we get

x(f(x) − a₀) = a₁ + O

x1

. So

a₁ = lim

x→∞

x(f(x) − a₀)

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Finding the coefficients

•

How to find

^a0, a₁, a₂, . . .

? Since

^f^{(x) =} ^a0 + O

1 x

, then

^a0 = lim

x→∞

f(x)

•

From

^f^{(x) =} ^a0 + ^a_x¹ + O

1 x²

we get

x(f(x) − a₀) = a₁ + O

x1

. So

a₁ = lim

x→∞

x(f(x) − a₀)

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Finding the coefficients

•

Further, from

^f^{(x) =} â0 + â_x¹ + â_x²² + O

x1³

a₂ = lim

x→∞

x²

f(x) − a₀ − ^a_x¹

•

In general

a_N = lim

x→∞

x^N

f(x) −

N−1

X

n=0

a_n xⁿ

• {1, _x¹, _x¹² , _x¹³ , . . .}

is an asymptotic sequence

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Finding the coefficients

•

Further, from

^f^{(x) =} â0 + â_x¹ + â_x²² + O

x1³

a₂ = lim

x→∞

x²

f(x) − a₀ − ^a_x¹

•

In general

a_N = lim

x→∞

x^N

f(x) −

N−1

X

n=0

a_n xⁿ

• {1, _x¹, _x¹² , _x¹³ , . . .}

is an asymptotic sequence

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Finding the coefficients

•

Further, from

^f^{(x) =} â0 + â_x¹ + â_x²² + O

x1³

a₂ = lim

x→∞

x²

f(x) − a₀ − ^a_x¹

•

In general

a_N = lim

x→∞

x^N

f(x) −

N−1

X

n=0

a_n xⁿ

• {1, _x¹, _x¹² , _x¹³ , . . .}

is an asymptotic sequence

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Asymptotic sequences

If we write

^ε ⁼ _x¹

for

^x ^{∼ ∞}

, then

{1, ε, ε¹, ε², ε³, . . .}

{1, ε^1/2, ε¹, ε^3/2, ε², . . .}

en general if

^δn(ε)

is a term of asymptotic sequence then

{δ₀(ε), δ₁(ε), δ₂(ε), . . .}

δ_n+1(ε) = o(δ_n(ε)) ε → 0

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Examples of asymptotic sequences

{1, sin ε, sin² ε, sin³ ε, . . . , sinⁿ ε, . . .}

{1, ln(1 + ε), ln(1 + ε²), . . . ln(1 + εⁿ), . . .}

{tan(ε^1/2), tan(ε), tan(ε^3/2), . . . , tan(ε^n/2), . . .}

{ln ε ε , 1

ε, ln ε, 1, ε ln ε, ε, ε² ln ε, ε², . . .}

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Expansion in terms of δ

ⁿ

(ε)

f(ε) ∼ P^∞

n=0 a_nδ_n(ε)

that means

f(ε) =

N

X

n=0

a_nδ_n(ε) + R_N, R_N = O(δ_n+1(ε)) ε → 0 a_N = lim

ε→0

f(ε) − P_N⁻₁

n=0 a_nδ_n(ε) δ_N(ε)

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Expansion in terms of δ

ⁿ

(ε)

f(ε) ∼ P^∞

n=0 a_nδ_n(ε)

that means

f(ε) =

N

X

n=0

a_nδ_n(ε) + R_N, R_N = O(δ_n+1(ε)) ε → 0

a_N = lim

ε→0

f(ε) − P_N⁻₁

n=0 a_nδ_n(ε) δ_N(ε)

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Positive example

Let us decompose the function

^e^ε

using

δ_n(ε) = ln(1 + εⁿ)

, e. g.

e^ε ∼ a₀ + a₁ ln(1 + ε) + a₂ ln(1 + ε²) + . . . a₀ = 1

e^ε − 1 = a₁ ln(1 + ε) + o(ln(1 + ε))

,

εlim→0

e^ε − 1 ln(1 + ε)

= a₁ + lim

ε→0

o(ln(1 + ε)) ln(1 + ε)

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Positive example

e^ε − 1 − ln(1 + ε) = a₂ ln(1 + ε²) + o(ln(1 + ε²))

,

a₂ = lim

ε→0

e^ε − 1 − ln(1 + ε) ln(1 + ε²)

= lim

ε→0

1 + ε + ε²/2 + O(ε³) − 1 − ε + ε²/2 ε² + O(ε³)

= 1

e^ε = 1 + ln(1 + ε) + ln(1 + ε²) − 1

6 ln(1 + ε³) + o(ln(1 + ε³))

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Negative example

Let us attempt to expand

^cos(ε^1/2 ⁺ ^ε)

using the same asymptotic equation

^δn(ε) = ln(1 + εⁿ)

cos(ε^1/2 + ε) ∼ a₀ + a₁ ln(1 + ε) + a₂ ln(1 + ε²) + . . . a₀ = cos(ε^1/2 + ε) = 1

a₁ = lim

ε→0

cos(ε^1/2 + ε) − 1 ln(1 + ε)

= lim 1 − (ε^1/2 + ε)²/2 + O(ε²) − 1

= −1

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Negative example

a₂ = lim

ε→0

cos(ε^1/2 + ε) − 1 ln(1 + ε)

= lim

ε→0

1 − (ε + 2ε^3/2 + ε²)/2 + O(ε²) − 1 + ε/2 − ε²/4 ε² + O(ε⁴)

= lim

ε→0

− ε^3/2 + O(ε²) ε² + O(ε⁴)

= −∞

What is the solution of the problem? Add the term

ln(1 + ε^3/2)

and its subsequent powers.

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Uniqueness of expansions

•

Any function

^f^(x)

produce unique expansion.

•

Let us consider

^f1 = sin ε

and

f₂ = sin ε + exp(−1/ε²)

. If

sinε + exp(−1/ε²) = a₀ + a₁ε + a₂ε² + a₃ε³ + . . .

then

^a0 = 0

,

a₁ = lim

ε→0

sin ε + exp(−1/ε²) ε

= lim

ε→0

sinε ε

+ lim

ε→0

+ exp(−1/ε²) ε

= 1 + 0 = 1

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Uniqueness of expansions

•

Any function

^f^(x)

produce unique expansion.

•

Let us consider

^f1 = sin ε

and

f₂ = sin ε + exp(−1/ε²)

. If

sinε + exp(−1/ε²) = a₀ + a₁ε + a₂ε² + a₃ε³ + . . .

then

^a0 = 0

,

a₁ = lim

ε→0

sin ε + exp(−1/ε²) ε

= lim

ε→0

sinε ε

+ exp(−1/ε²)

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Uniqueness of expansions

a₂ = lim

ε→0

sin ε + exp(−1/ε²) − ε ε²

= 0

a₃ = lim

ε→0

sin ε + exp(−1/ε²) − ε ε³

= −1 6 exp(−1/ε²) = 0 · 1 + 0 · ε + 0 · ε² + 0 · ε³ + . . .

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Taylor series is asymptotic expansion

•

Why the Taylor series at

⁰

are asymptotic expansions?

• f(x) = P_N

n=0 a_nxⁿ + R_N

with

R_N = x^N⁺¹

(N + 1)!f^(N⁺¹⁾(z).

So

R_N = O(x^N⁺¹) x → 0.

xlim→0

R_N

x^N⁺¹ = 1

(N + 1)!f^(N⁺¹⁾(0) = 0

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Taylor series is asymptotic expansion

•

Why the Taylor series at

⁰

are asymptotic expansions?

• f(x) = P_N

n=0 a_nxⁿ + R_N

with

R_N = x^N⁺¹

(N + 1)!f^(N⁺¹⁾(z).

So

R_N = O(x^N⁺¹) x → 0.

R 1

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