Classification of singularity

1. POWER SERIES & CLASSIFICATION
OF SINGULARITY
NAME: RAJ PAREKH
ENROLLMENT NO: 140990119029
SUBJECT: AEM

2. Introduction To Power Series
 A power series of (x – x0) is an infinite series of the form :
 If a = 0 then
 Consider Second Order Linear Homogeneous Differential Equation
Then,
𝑑2 𝑦
𝑑𝑥2 + 𝑃 𝑥
𝑑𝑦
𝑑𝑥
+ 𝑄 𝑥 𝑦 = 0 Where 𝑃 𝑥 =
𝑃1(𝑥)
𝑃0(𝑥)
& 𝑄 𝑥 =
𝑃2(𝑥)
𝑃0(𝑥)
𝑘=0
∞
𝜒 − 𝜒0
𝑘
𝑎0 + 𝑎1(𝑥 − 𝑥0) + 𝑎2 𝑥 − 𝑥0
2
+. . . . . . . . . +𝑎 𝑛 𝑥 − 𝑥0
𝑛
P0 𝑥
𝑑2 𝑦
𝑑𝑥2 +P1 𝑥
𝑑𝑦
𝑑𝑥
+P2 ( 𝑥)𝑦 = 0 where Po ( 𝑥) ≠ 0
𝑑2 𝑦
𝑑𝑥2 +
P1 𝑥
𝑃0 𝑥
𝑑𝑦
𝑑𝑥
+
𝑃2 𝑥
𝑃0 𝑥
𝑦 = 0

3. Classification Of Singularity
 Analytic Function: A function 𝑓(𝑥) is said to be analytic at point 𝑥0 if 𝑓(𝑥) has
Taylor Series Expansion about point 𝑥0 given by
 𝑓 𝑥 = 𝑛=0
∞ 𝑓 𝑛 𝑥0
𝑛!
𝑥 − 𝑥0
𝑛 exists and converges to 𝑓 𝑥 .
 If a function 𝑓(𝑥) is not analytic at point 𝑥0 , then it is called a Singular Point.
 Ordinary Point : A point 𝑥 = 𝑥0 is said to be an ordinary point of equation:
𝑑2 𝑦
𝑑𝑥2 + 𝑃 𝑥
𝑑𝑦
𝑑𝑥
+ 𝑄 𝑥 𝑦 = 0 , if both P(𝑥) and Q(𝑥) are analytic at point 𝑥0.

4.  Singular Point: A point 𝑥0 is said to be a singular point of equation:
𝑑2 𝑦
𝑑𝑥2 + 𝑃 𝑥
𝑑𝑦
𝑑𝑥
+ 𝑄 𝑥 𝑦 = 0 , if both P(𝑥) and Q(𝑥) are analytic at point 𝑥0.
A point 𝑥 is 𝑥0 that is not an ordinary point ,then it is called a Singular Point.
 SINGULAR POINTS CLASSIFICATION
 Singular points are classified as:
 1. Regular Singular Point 2. Irregular Singular Point

5.  1.Regular Singular Point: A singular point 𝑥 = 𝑥0 is said to be regular singular point of
equation :
𝑑2 𝑦
𝑑𝑥2 + 𝑃 𝑥
𝑑𝑦
𝑑𝑥
+ 𝑄 𝑥 𝑦 = 0 ,
 If both term 𝑥 − 𝑥0 𝑃(𝑥) and 𝑥 − 𝑥0
2
𝑄 𝑥 are analytic at point 𝑥0.
 2.Irregular Singular Point: If 𝑥 − 𝑥0 𝑃(𝑥) and 𝑥 − 𝑥0
2 𝑄(𝑥) or both are not analytic,
then it is called Irregular singular point.