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

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

General Result Example Specific Function Direct Computation of the Radius of Convergence May Not be Possible, But ... logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

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 x 0 if and only if f equals its Taylor series expansion in some open interval about x 0 . ∞ ∑ c n ( x − x 0 ) n That is, there is an ε > 0 such that f ( x ) = n = 0 for all x with | x − x 0 | < ε . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

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 x 0 if and only if f equals its Taylor series expansion in some open interval about x 0 . ∞ ∑ c n ( x − x 0 ) n That is, there is an ε > 0 such that f ( x ) = n = 0 for all x with | x − x 0 | < ε . 2. The same definition works for a function of a complex variable, and we will need to mind complex numbers throughout. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

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 x 0 if and only if f equals its Taylor series expansion in some open interval about x 0 . ∞ ∑ c n ( x − x 0 ) n That is, there is an ε > 0 such that f ( x ) = n = 0 for all x with | x − x 0 | < 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 x 0 is called an ordinary point if and only if both P and Q are analytic at x 0 . A point that is not ordinary will be called a singular point . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

General Result Example Specific Function Direct Computation of the Radius of Convergence May Not be Possible, But ... logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

General Result Example Specific Function Direct Computation of the Radius of Convergence May Not be Possible, But ... 4. If x 0 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 x 0 . That is, there are two linearly independent solutions of the form ∞ c n ( x − x 0 ) n . Moreover, the radius of ∑ y ( x ) = n = 0 convergence of the power series is at least the distance from x 0 to the closest singular point in the complex plane. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

General Result Example Specific Function Lower Bound for the Radius of Convergence of 1 + x 2 � 2 y ′′ + 3 x y ′ + 2 y = 0 1 + x 2 � � � Solutions of about x 0 = 0 and x 0 = 2 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

General Result Example Specific Function Lower Bound for the Radius of Convergence of 1 + x 2 � 2 y ′′ + 3 x y ′ + 2 y = 0 1 + x 2 � � � Solutions of about x 0 = 0 and x 0 = 2 1 + x 2 � 2 y ′′ + 3 x y ′ + 2 y � � 1 + x 2 � = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

General Result Example Specific Function Lower Bound for the Radius of Convergence of 1 + x 2 � 2 y ′′ + 3 x y ′ + 2 y = 0 1 + x 2 � � � Solutions of about x 0 = 0 and x 0 = 2 1 + x 2 � 2 y ′′ + 3 x y ′ + 2 y � � 1 + x 2 � = 0 3 x 2 y ′′ + 1 + x 2 y ′ + = ( 1 + x 2 ) 2 y 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

General Result Example Specific Function Lower Bound for the Radius of Convergence of 1 + x 2 � 2 y ′′ + 3 x y ′ + 2 y = 0 1 + x 2 � � � Solutions of about x 0 = 0 and x 0 = 2 1 + x 2 � 2 y ′′ + 3 x y ′ + 2 y � � 1 + x 2 � = 0 3 x 2 y ′′ + 1 + x 2 y ′ + = ( 1 + x 2 ) 2 y 0 Singular points at ± i . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

General Result Example Specific Function Lower Bound for the Radius of Convergence of 1 + x 2 � 2 y ′′ + 3 x y ′ + 2 y = 0 1 + x 2 � � � Solutions of about x 0 = 0 and x 0 = 2 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

General Result Example Specific Function Lower Bound for the Radius of Convergence of 1 + x 2 � 2 y ′′ + 3 x y ′ + 2 y = 0 1 + x 2 � � � Solutions of about x 0 = 0 and x 0 = 2 ✻ ✲ ❞ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

General Result Example Specific Function Lower Bound for the Radius of Convergence of 1 + x 2 � 2 y ′′ + 3 x y ′ + 2 y = 0 1 + x 2 � � � Solutions of about x 0 = 0 and x 0 = 2 ✻ i t ✲ ❞ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

General Result Example Specific Function Lower Bound for the Radius of Convergence of 1 + x 2 � 2 y ′′ + 3 x y ′ + 2 y = 0 1 + x 2 � � � Solutions of about x 0 = 0 and x 0 = 2 ✻ i t ✲ ❞ t − i logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

General Result Example Specific Function Lower Bound for the Radius of Convergence of 1 + x 2 � 2 y ′′ + 3 x y ′ + 2 y = 0 1 + x 2 � � � Solutions of about x 0 = 0 and x 0 = 2 ✻ i t ✲ ❞ t − i logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

General Result Example Specific Function Lower Bound for the Radius of Convergence of 1 + x 2 � 2 y ′′ + 3 x y ′ + 2 y = 0 1 + x 2 � � � Solutions of about x 0 = 0 and x 0 = 2 ✻ i t � ✒ � � ✲ R = 1 ❞ t − i logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

General Result Example Specific Function Lower Bound for the Radius of Convergence of 1 + x 2 � 2 y ′′ + 3 x y ′ + 2 y = 0 1 + x 2 � � � Solutions of about x 0 = 0 and x 0 = 2 ✻ i t ✒ � � � ✲ R = 1 ❞ ❞ 2 t − i logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

General Result Example Specific Function Lower Bound for the Radius of Convergence of 1 + x 2 � 2 y ′′ + 3 x y ′ + 2 y = 0 1 + x 2 � � � Solutions of about x 0 = 0 and x 0 = 2 ✻ i t ✒ � � � ✲ R = 1 ❞ ❞ 2 t − i logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

General Result Example Specific Function Lower Bound for the Radius of Convergence of 1 + x 2 � 2 y ′′ + 3 x y ′ + 2 y = 0 1 + x 2 � � � Solutions of about x 0 = 0 and x 0 = 2 ✻ i t � ✒ � � ✲ R = 1 ❞ ❞ 2 √ R = 5 t − i ✠ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

General Result Example Specific Function 1 f ( x ) = 1 + x 2 Solves 1 + x 2 � 2 y ′′ + 3 x y ′ + 2 y = 0 � � 1 + x 2 � logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

General Result Example Specific Function 1 f ( x ) = 1 + x 2 Solves 1 + x 2 � 2 y ′′ + 3 x y ′ + 2 y = 0 � � 1 + x 2 � logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution

General Result Example Specific Function 1 f ( x ) = 1 + x 2 Solves 1 + x 2 � 2 y ′′ + 3 x y ′ + 2 y = 0 � � 1 + x 2 � logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Radius of Convergence of a Series Solution