for z ∈ C.

It is a meromorphic function, well investigated in classical analysis. Some of its relevant

properties are collected in the next proposition:

Proposition 5.2 The -function has the following properties:

(1) 1/ (z) is an entire function;

√

(2) (1) = 1, (1/2) = π ;

∞

(3) (z) = 0 e’t t z’1 dt for (z) > 0 (Euler™s representation);

(4) (z + 1) = z (z) (recurrence relation);

46

5.1 Bessel functions 47

(z) (1 ’ z) = π/ sin(π z) (re¬‚ection formula);

(5)

(x + 1)

(6) 1 ¤ √ ¤ e1/(12x) , x > 0 (Stirling™s formula);

x e’x

2π x x

22z’1

(7) (2z) = √ (z) (z + 1/2) (Legendre™s duplication formula).

π

The -function and its properties are well known. Proofs of the formulas just stated can

be found in any book on special functions. A particular choice would be the book [102] by

Lebedev. Now, we continue by introducing Bessel functions.

De¬nition 5.3 The Bessel function of the ¬rst kind of order ν ∈ C is de¬ned by

∞

(’1)m (z/2)2m+ν

Jν (z) :=

m! (ν + m + 1)

m=0

for z ∈ C \ {0}.

The power z ν in this de¬nition is de¬ned by exp[ν log(z)], where log is the principal branch

of the logarithm, i.e. ’π < arg(z) ¤ π .

The Bessel function can be seen as a function of z and also as a function of ν and

the following remarks are easily veri¬ed. Obviously, Jν (z) is holomorphic in C \ [0, ∞)

as a function of z for every ν ∈ C. Moreover, the expansion converges pointwise also

for z < 0. If ν ∈ N then Jν has an analytic extension to C. If (ν) ≥ 0 then we have

a continuous extension of Jν (z) to z = 0. Finally, if z ∈ C \ {0} is ¬xed then Jν (z) is a

holomorphic function in C as a function of ν. We state further elementary properties in the

next proposition.

Proposition 5.4 The Bessel function of the ¬rst kind has the following properties:

(1) J’n = (’1)n Jn if n ∈ N;

dν

{z Jν (z)} = z ν Jν’1 (z);

(2)

dz

d ’ν

{z Jν (z)} = ’z ’ν Jν’1 (z);

(3)

dz

2 2

(4) J1/2 (z) = J’1/2 (z) =

sin z, cos z.

πz πz

Proof For the ¬rst property we simply use that 1/ (n) = 0 for n = 0, ’1, ’2, . . .. The

second and the third property follow by differentiation under the sum using also the recur-

√

rence formula for the -function. The latter together with (1/2) = π demonstrates the

last item.

There exist many integral representations for Bessel functions. The one that matters here

is given in the next proposition.

48 Auxiliary tools from analysis and measure theory

Proposition 5.5 If we denote for d ≥ 2 the unit sphere by S d’1 = {x ∈ Rd : x = 1}

2

then we have, for x ∈ Rd ,

’(d’2)/2

T

ei x y d S(y) = (2π)d/2 x J(d’2)/2 ( x 2 ). (5.1)

2

S d’1

Proof Obviously, both sides of (5.1) are radially symmetric. Hence with spherical coordi-

nates and r = x 2 we can derive

π

2π (d’1)/2

eir cos θ sind’2 θ dθ

ixT y ir y1

d S(y) = d S(y) =

e e

((d ’ 1)/2)

S d’1 S d’1 0

using that the surface area of S d’2 is given by

2π (d’1)/2

ωd’2 = .

((d ’ 1)/2)

The initial and the last integral are obviously restrictions of entire functions to r > 0. Hence,

we can calculate the last integral by expanding the exponent in the integral and integrating

term by term, which gives

∞

π

i kr k

ir cos θ

θ dθ =

d’2

e sin ak

k!

0 k=0

π

cosk θ sind’2 θ dθ. By induction it is possible to show a2k+1 = 0 and

with ak := 0

(2k)! ((d ’ 1)/2) (1/2)

a2k = .

22k k! ((k + d)/2)

Collecting everything together gives the stated representation. The exchange of integration

and summation can easily be justi¬ed. Since we will give similar arguments in later proofs,

this time we will leave the details to the reader.

Our next result is concerned with the asymptotic behavior of the Bessel functions of the

¬rst kind.

Proposition 5.6 The Bessel function has the following asymptotic behavior:

√

(1) Jν( ) (r ) = O(1/ r ) for r ’ ∞ and ν ∈ R, ∈ N0 ;

νπ π

2

+ O(r ’3/2 )

(2) Jν (r ) = cos r ’ ’

πr 2 4

for r ’ ∞ and ν ∈ R;

2d+2

(3) Jd/2 (r ) ¤ for r > 0 and d ∈ N;

2

rπ

1

(4) limr ’0 r ’d Jd/2 (r ) = d 2 for d ∈ N.

2

(d/2 + 1)

2

Proof The last property is an immediate consequence of the de¬nition of the Bessel func-

tions. The penultimate property is obviously true in the case d = 1, since in this case

√

J1/2 (r ) = 2/(πr ) sin r . The case d ≥ 2 is more complicated. It is based mainly on Weber™s

“crude” inequality for Hankel functions (see Watson [187], p. 211). The complete proof