Special Functions - Heat Conduction: Mathematical Models and Analytical Solutions - page 434

3. Generating Function and Integral Formula

The generating function of Bessel functions is the function of two variables whose
power series expansion has the Bessel functions as its coefficients. The generating
function of Bessel functions of the first kind is

2 t t

x 1

−

w ( x , t )= e .

t − 1 is Its power series expansion in t and −

2 t t

x 1

−

( , )=

w x t e

⎡ ⎤

x

2 2

x

2 3

x
2 t

t 2 t 3

⎣ ⎦

= 1 +

1! +

+ + ···

2! 3!

⎡
⎣

⎤

x
2

2 x

2

3

x
2 t − 1

t − 2 t − 3

⎦

· 1 −

1! +

− + ···

2! 3!

t − 2 J − 2 t − 1 J − 1 t 2 J 2

= ··· + ( )+ ( )+ ( )+ ( )+ ( )+ ···

x x J 0 x tJ 1 x x

+ ∞
∑

t n = J n ( x ) .

n = − ∞

To obtain an integral formula of Bessel functions, consider a variable transformation

e iθ = t − 1 t = cos θ + isin θ or = cos θ − isin θ .

N J N (

Note also that J − N ( x )=( − 1 ) x ) . Thus

2 t − t

x 1

e i x sinθ = e = cos ( x sin θ )+ isin ( x sin θ ) .

Also

t

1
t

t − 1 J 1 ( x ) t + J − 1 ( x ) = J 1 ( x ) − = 2i J 1 ( x ) sin θ ,

t 2

1
t 2

t 2 t − 2 J 2 ( x ) + J − 2 ( x ) = J 2 ( x ) + = 2 J 2 ( x ) cos2 θ ,

······ .

Therefore

2 t t

x 1

−

e = J 0 ( x )+ 2 [ J 2 ( x ) cos2 θ + J 4 ( x ) cos4 θ + ··· ]

+ 2i [ J 1 ( x ) sin θ + J 3 ( x ) sin3 θ + ··· ] .