Mathematical Physics Vol 1

5. Series Solutions of Differential Equations. Spe cial functions

5.1 Functional series. Power series Let f 0 ( x ) , f 1 ( x ) , ··· , f k ( x ) , ··· be real functions defined for ∀ x ∈ [ a , b ] ⊂ R , where R is the set of real numbers.

Definition The infinite sum of functions

f 0 ( x )+ f 1 ( x )+ ··· + f k ( x )+ ··· = ∞ ∑ k = 0

f k ( x ) ,

(5.1)

whose terms are functions f k ( x ) defined for ∀ x ∈ [ a , b ] , is called a functional series (infinite functional series).

Definition The partial sum of a functional series (5.1) has the following form

n ∑ k = 0

S n ( x )=

f k ( x ) , ( n − a positive integer ) .

(5.2)

Definition The series (5.1) is convergent , for some x = x 1 ∈ [ a , b ] if lim n → ∞ S n ( x 1 )= S ( x 1 )̸ = ± ∞ .

(5.3)