Special Functions - Heat Conduction: Mathematical Models and Analytical Solutions - page 439

Laurent series, we have

+ ∞
n = 0 c n ( x ) t n

w ( x , t )= , | t | < r ,

1

t 2 −

1
2

− 2 xt +

1
2

c n ( x )= d t .

t n + 1

π i

C

Here C is a closed curve containing t = 0in | t | < r .

Consider the Euler transformation

− 1 t 2

2

1

t 2

1 − 2 xt +

2

− 2 xt + = 1 − tu or u = ,

t

x and curve C becomes a closed curve C that contains u when t → 0, u → = x .

Therefore

u 2

1 n

−

1
2

c n ( x )=

1 d u

2 n

n +

π i ( u − x )

C

n u = x =

d n
d u n (

1
2 n n !

u 2 = − 1 ) P n ( x ) .

Thus

)= 1 t 2 −

+ ∞
n = 0 P n ( x ) t n ;

2

w ( x , t − 2 xt + = (A.23)

and w ( x , t ) is called the generating function of the Legendre polynomials P n ( x ) .

When x = 1,

+ ∞
n = 0 P n ( x ) t n

1 1

t 2 t n − 1

w ( x , t )= = √ t 2

1 =

t =

1 + t + + ··· + + ··· .

1 −

− 2 t +

Thus P n ( 1 )= 1. Similarly, we have

n

)= ( −

1 ) ( 2 n ) !

n

P n ( − 1 )=( − 1 ) , P 2 n − 1 ( 0 )= 0 , P 2 n ( 0 .

2 2 n ( n ! )

2

By taking derivatives of

)= 1 t 2

2

w ( x , t − 2 xt +

with respect to t and x , we obtain
1

t 2 ∂

w
∂

− 2 xt +

t +(

t − x ) w = 0 , (A.24)

1

t 2 ∂

w

− 2 xt +

x −

tw = 0 . (A.25)

∂