The Radius of Convergence of a Series Solution Bernd Schr oder - - PowerPoint PPT Presentation

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the radius of convergence of a series solution
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The Radius of Convergence of a Series Solution Bernd Schr oder - - PowerPoint PPT Presentation

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General Result Example Specific Function The Radius of Convergence of a Series Solution Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

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SLIDE 1

logo1 General Result Example Specific Function

The Radius of Convergence of a Series Solution

Bernd Schr¨

  • der

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

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SLIDE 2

logo1 General Result Example Specific Function

Direct Computation of the Radius of Convergence May Not be Possible, But ...

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

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SLIDE 3

logo1 General Result Example Specific Function

Direct Computation of the Radius of Convergence May Not be Possible, But ...

  • 1. A function f is called analytic at x0 if and only if f equals

its Taylor series expansion in some open interval about x0. That is, there is an ε > 0 such that f(x) =

∞

∑

n=0

cn(x−x0)n for all x with |x−x0| < ε.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

slide-4
SLIDE 4

logo1 General Result Example Specific Function

Direct Computation of the Radius of Convergence May Not be Possible, But ...

  • 1. A function f is called analytic at x0 if and only if f equals

its Taylor series expansion in some open interval about x0. That is, there is an ε > 0 such that f(x) =

∞

∑

n=0

cn(x−x0)n for all x with |x−x0| < ε.

  • 2. The same definition works for a function of a complex

variable, and we will need to mind complex numbers throughout.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

slide-5
SLIDE 5

logo1 General Result Example Specific Function

Direct Computation of the Radius of Convergence May Not be Possible, But ...

  • 1. A function f is called analytic at x0 if and only if f equals

its Taylor series expansion in some open interval about x0. That is, there is an ε > 0 such that f(x) =

∞

∑

n=0

cn(x−x0)n for all x with |x−x0| < ε.

  • 2. The same definition works for a function of a complex

variable, and we will need to mind complex numbers throughout.

  • 3. For the differential equation y′′ +P(x)y′ +Q(x)y = 0 the

point x0 is called an ordinary point if and only if both P and Q are analytic at x0. A point that is not ordinary will be called a singular point.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

slide-6
SLIDE 6

logo1 General Result Example Specific Function

Direct Computation of the Radius of Convergence May Not be Possible, But ...

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

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SLIDE 7

logo1 General Result Example Specific Function

Direct Computation of the Radius of Convergence May Not be Possible, But ...

  • 4. If x0 is an ordinary point of the differential equation

y′′ +P(x)y′ +Q(x)y = 0, then there exist two linearly independent solutions of the equation that are power series about x0. That is, there are two linearly independent solutions of the form y(x) =

∞

∑

n=0

cn(x−x0)n. Moreover, the radius of convergence of the power series is at least the distance from x0 to the closest singular point in the complex plane.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

slide-8
SLIDE 8

logo1 General Result Example Specific Function

Lower Bound for the Radius of Convergence of Solutions of

  • 1+x22y′′ +3x
  • 1+x2

y′ +2y = 0 about x0 = 0 and x0 = 2

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

slide-9
SLIDE 9

logo1 General Result Example Specific Function

Lower Bound for the Radius of Convergence of Solutions of

  • 1+x22y′′ +3x
  • 1+x2

y′ +2y = 0 about x0 = 0 and x0 = 2

  • 1+x22

y′′ +3x

  • 1+x2

y′ +2y =

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

slide-10
SLIDE 10

logo1 General Result Example Specific Function

Lower Bound for the Radius of Convergence of Solutions of

  • 1+x22y′′ +3x
  • 1+x2

y′ +2y = 0 about x0 = 0 and x0 = 2

  • 1+x22

y′′ +3x

  • 1+x2

y′ +2y = y′′ + 3x 1+x2y′ + 2 (1+x2)2y =

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

slide-11
SLIDE 11

logo1 General Result Example Specific Function

Lower Bound for the Radius of Convergence of Solutions of

  • 1+x22y′′ +3x
  • 1+x2

y′ +2y = 0 about x0 = 0 and x0 = 2

  • 1+x22

y′′ +3x

  • 1+x2

y′ +2y = y′′ + 3x 1+x2y′ + 2 (1+x2)2y = Singular points at ±i.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

slide-12
SLIDE 12

logo1 General Result Example Specific Function

Lower Bound for the Radius of Convergence of Solutions of

  • 1+x22y′′ +3x
  • 1+x2

y′ +2y = 0 about x0 = 0 and x0 = 2

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

slide-13
SLIDE 13

logo1 General Result Example Specific Function

Lower Bound for the Radius of Convergence of Solutions of

  • 1+x22y′′ +3x
  • 1+x2

y′ +2y = 0 about x0 = 0 and x0 = 2

✻ ✲ ❞

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

slide-14
SLIDE 14

logo1 General Result Example Specific Function

Lower Bound for the Radius of Convergence of Solutions of

  • 1+x22y′′ +3x
  • 1+x2

y′ +2y = 0 about x0 = 0 and x0 = 2

✻ ✲ t ❞

i Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

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SLIDE 15

logo1 General Result Example Specific Function

Lower Bound for the Radius of Convergence of Solutions of

  • 1+x22y′′ +3x
  • 1+x2

y′ +2y = 0 about x0 = 0 and x0 = 2

✻ ✲ t t ❞

i −i Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

slide-16
SLIDE 16

logo1 General Result Example Specific Function

Lower Bound for the Radius of Convergence of Solutions of

  • 1+x22y′′ +3x
  • 1+x2

y′ +2y = 0 about x0 = 0 and x0 = 2

✻ ✲ t t ❞

i −i Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

slide-17
SLIDE 17

logo1 General Result Example Specific Function

Lower Bound for the Radius of Convergence of Solutions of

  • 1+x22y′′ +3x
  • 1+x2

y′ +2y = 0 about x0 = 0 and x0 = 2

✻ ✲ t t ❞

i −i

  • ✒

R = 1 Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

slide-18
SLIDE 18

logo1 General Result Example Specific Function

Lower Bound for the Radius of Convergence of Solutions of

  • 1+x22y′′ +3x
  • 1+x2

y′ +2y = 0 about x0 = 0 and x0 = 2

✻ ✲ t t ❞ ❞

2 i −i

  • ✒

R = 1 Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

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SLIDE 19

logo1 General Result Example Specific Function

Lower Bound for the Radius of Convergence of Solutions of

  • 1+x22y′′ +3x
  • 1+x2

y′ +2y = 0 about x0 = 0 and x0 = 2

✻ ✲ t t ❞ ❞

2 i −i

  • ✒

R = 1 Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

slide-20
SLIDE 20

logo1 General Result Example Specific Function

Lower Bound for the Radius of Convergence of Solutions of

  • 1+x22y′′ +3x
  • 1+x2

y′ +2y = 0 about x0 = 0 and x0 = 2

✻ ✲ t t ❞ ❞

2 i −i

  • ✒

R = 1 R = √ 5

✠

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

slide-21
SLIDE 21

logo1 General Result Example Specific Function

f(x) = 1 1+x2 Solves

  • 1+x22y′′ +3x
  • 1+x2

y′ +2y = 0

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

slide-22
SLIDE 22

logo1 General Result Example Specific Function

f(x) = 1 1+x2 Solves

  • 1+x22y′′ +3x
  • 1+x2

y′ +2y = 0

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

slide-23
SLIDE 23

logo1 General Result Example Specific Function

f(x) = 1 1+x2 Solves

  • 1+x22y′′ +3x
  • 1+x2

y′ +2y = 0

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

slide-24
SLIDE 24

logo1 General Result Example Specific Function

f(x) = 1 1+x2 Solves

  • 1+x22y′′ +3x
  • 1+x2

y′ +2y = 0

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

slide-25
SLIDE 25

logo1 General Result Example Specific Function

f(x) = 1 1+x2 Solves

  • 1+x22y′′ +3x
  • 1+x2

y′ +2y = 0

N = 0

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

slide-26
SLIDE 26

logo1 General Result Example Specific Function

f(x) = 1 1+x2 Solves

  • 1+x22y′′ +3x
  • 1+x2

y′ +2y = 0

N = 2

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

slide-27
SLIDE 27

logo1 General Result Example Specific Function

f(x) = 1 1+x2 Solves

  • 1+x22y′′ +3x
  • 1+x2

y′ +2y = 0

N = 4

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

slide-28
SLIDE 28

logo1 General Result Example Specific Function

f(x) = 1 1+x2 Solves

  • 1+x22y′′ +3x
  • 1+x2

y′ +2y = 0

N = 6

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

slide-29
SLIDE 29

logo1 General Result Example Specific Function

f(x) = 1 1+x2 Solves

  • 1+x22y′′ +3x
  • 1+x2

y′ +2y = 0

N = 8

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

slide-30
SLIDE 30

logo1 General Result Example Specific Function

f(x) = 1 1+x2 Solves

  • 1+x22y′′ +3x
  • 1+x2

y′ +2y = 0

N = 10

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

slide-31
SLIDE 31

logo1 General Result Example Specific Function

f(x) = 1 1+x2 Solves

  • 1+x22y′′ +3x
  • 1+x2

y′ +2y = 0

N = 12

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

slide-32
SLIDE 32

logo1 General Result Example Specific Function

f(x) = 1 1+x2 Solves

  • 1+x22y′′ +3x
  • 1+x2

y′ +2y = 0

N = 14

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

slide-33
SLIDE 33

logo1 General Result Example Specific Function

f(x) = 1 1+x2 Solves

  • 1+x22y′′ +3x
  • 1+x2

y′ +2y = 0

N = 16

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

slide-34
SLIDE 34

logo1 General Result Example Specific Function

f(x) = 1 1+x2 Solves

  • 1+x22y′′ +3x
  • 1+x2

y′ +2y = 0

N = 18

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

slide-35
SLIDE 35

logo1 General Result Example Specific Function

f(x) = 1 1+x2 Solves

  • 1+x22y′′ +3x
  • 1+x2

y′ +2y = 0

N = 20

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution