7/27/2019 Some asymptotics on Bessel functions

1/2

A note about Bessel functions of the first kind

Jack DAurizio

July 11, 2013

The starting point of our study is the integral:

In() =1

0

cos(nx) e cosxdx.

By considering the Taylor series of ey and the Fourier series of (cos x)m, we have that:

In() =+=0

(/2)n+2

!(n + )!.

Moreover, integrating by parts and using Briggs formulas we have:

In =

2n(In1 In+1) ,

d

dIn =

1

2(In1 + In+1) .

From the second identity, we have that the ratio between In and In1 is the continued fraction:

InIn1

= 12n + 12

(n+1)+...

,

that obviously gives:

In (/2)n

n!I0.

By defining:

Jn =n!

(/2)nIn,

so J0 = I0, we have that Jn is decreasing as n goes from 0 to +, and since:

Jn =+

=0

(/2)2

!2 n+

exp 24(n + 1)

we have that limn+ Jn = 1.

Jn = Jn1 2

4n(n + 1)Jn+1.

Jn 2

4I0 + (I0 1)n

2

4 + (I0 1)n,

1

Jn 1

1

I0 1+

4n

2.

M=0

1

n+

=

n

n 1

1

1

M+nM+1

1n+

= 1 nn + 1

1k=0

1n+1+k

k

1

7/27/2019 Some asymptotics on Bessel functions

2/2

M=0

1

!2(M )!2=

(2M)!

M!4

M=0

1

!2(M )!2n+

= n!(2M+ n)!(M+ n)!2M!2M=0

1

!2(M )!2n+n

n+M

n

= n!2(2M+ 2n)!(M+ 2n)!(M+ n)!2M!

+=0

(/2)2

!2 n+

= +=0

(/2)2

!2

(/2)2+2

(+ 1)!2

n

n 1

1

1n++1

Jn [1;4n

2, (I0 1)]

JnJn1 = [0; 1,

4

2 n(n + 1),n + 2

n ,4

2 n(n + 3), . . .]

Jn+1 = n + 1

/2d

dJn

Conosciamo lespressione della derivata logaritmica rispetto ad di Jn e vogliamo provare che

J0 Jn =

nk=1

(/2)2

k(k + 1)Jk+1

4n

2(Jn 1)(J0 1).

O semplicemente provare la convessita (rispetto ad n) della funzione 1Jn1

. Abbiamo anche:

J0 Jn = nn + 1

+=1

(/2)2

!2

1k=0

1n+1+kk

= nn + 1+k=1

1n+kk1

+=k

(/2)2

!2

che sembra estremamente promettente. In particolare si ha:

(Jn 1) 1

n+11

(J0 1) ,(Jn 1)

1n+22

J0 1 + n2

(/2)2,

(Jn 1) 1

n+33 (J0 1) +

1

n+11 1

n+33

(/2)2 +

1

n+22 1

n+33

(/2)4

2!2

Jn 1 1

n+KK

(J0 1) + K1=1

1

n+

1n+KK

(/2)2

!2

Inoltre, al crescere di K le disuguaglianze ottenute sono progressivamente piu strette,

k k1 =n(k 1)!

(n + 1) . . . (n + k)

+=k

(/2)2

!2

Pagina 2 di 2