tailieunhanh - Advanced Mathematical Methods for Scientists and Engineers Episode 3 Part 2

Tham khảo tài liệu 'advanced mathematical methods for scientists and engineers episode 3 part 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | The Point at Infinity Now we consider the behavior of first order linear differential equations at the point at infinity. Recall from complex variables that the complex plane together with the point at infinity is called the extended complex plane. To study the behavior of a function f z at infinity we make the transformation z 1 and study the behavior of f 1 z at z 0. Example Let s examine the behavior of sin z at infinity. We make the substitution z 1 z and find the Laurent expansion about z 0. Ỵ -1 n sin 1 z s 2n 1 z 2 1 Since sin 1 z has an essential singularity at z 0 sin z has an essential singularity at infinity. We use the same approach if we want to examine the behavior at infinity of a differential equation. Starting with the first order differential equation d w X - p z w 0 dz we make the substitution 1 d 2 d z dz -zd w z u z to obtain -z2 d p i z u 0 dz du p 1 z _n dz - 2 u 814 Result The behavior at infinity of dw o - p z w 0 dz is the same as the behavior at z 0 of du p 1 zi-a dZ u 0 Example We classify the singular points of the equation dw 7 dz 1 z2 9 w 0. We factor the denominator of the fraction to see that z 13 and z 13 are regular singular points. dw 1 dz z i3 z 13 We make the transformation z 1 Z to examine the point at infinity. du 11 dz z2 1 Z 2 9u u 0 dZ 9Z2 1 Since the equation for u has a ordinary point at z 0 z X is a ordinary point of the equation for w. .