Mathematical Physics Vol 1

Chapter 5. Series Solutions of Differential Equations. Special functions

226

2 ◦ Each power series, which is convergent for x ∈ ( − R , R ) , can be differentiated in the interval [ 0 , x ] , where | x | < R , and in that case the derivative of the sum is the sum of the derivatives, namely

k ! =

∞ ∑ k = 0

∞ ∑ k = 0

k )

d d x

d ( a k x

f ′ ( x )=

a k x

for | x | < R .

(5.11)

,

d x

3 ◦ By adding (subtracting) two convergent power series, a convergent power series is obtained, whose convergence radius is not less than the smaller convergence radius of the two given series. Namely, let

∞ ∑ k = 0 ∞ ∑ k = 0

k , | x | < R ,

f ( x )=

a k x

(5.12)

k , | x | < R ′ , R ′ ≤ R ,

g ( x )=

b k x

be two power series, then their sum (difference) is the following power series

∞ ∑ k = 0

k ,

f ( x ) ± g ( x )=

( a k ± b k ) x

(5.13)

which is convergent in the interval ( − R ′ , R ′ ) . 4 ◦ By multiplying two convergent power series, a convergent power series is obtained, whose convergence radius is not less than the smaller convergence radius of the two given series. Namely, let

∞ ∑ k = 0 ∞ ∑ k = 0

k , | x | < R ,

f ( x )=

a k x

(5.14)

k , | x | < R ′ , R ′ ≤ R ,

g ( x )=

b k x

be two power series, then their product is the following power series

∞ ∑ k = 0

k ,

f ( x ) · g ( x )=

( a 0 · b k + a 1 · b k

a k · b 0 ) x

(5.15)

− 1 + ··· +

which is convergent in the interval ( − R ′ , R ′ ) .

Theorem12 If a power series has a positive convergence radius ( R > 0), and its sum is equal to zero, then all terms of this series are equal to zero.

Definition Afunction f ( x ) is called analytical at point x = x 0 , if it can be represented by a power series of ( x − x 0 ) , with a convergence radius R > 0.