5.1 Bessel functions 49

paper [145] by Narcowich and Ward. To give the proof for the second property would go

beyond the scope of this book; hence, we refer the reader to Watson [187], p. 199 or, alter-

natively, to Lebedev [102], p. 122. In the case = 0, the ¬rst property is a weaker version

of the second property. For higher derivatives we use repeatedly the recurrence relation

2Jν (z) = Jν’1 (z) ’ Jν+1 (z), which is a consequence of the formulas given in Proposition

5.4, to derive the desired asymptotic behavior.

We now turn to the Laplace transforms of some speci¬c functions involving Bessel

functions.

Lemma 5.7 For ν > ’1 and every r > 0 it is true that

∞

2ν+1 (ν + 3/2)r

Jν (t)t ν+1 e’r t dt = √ .

π(r 2 + 1)ν+3/2

0

Proof Let us start by looking at the binomial series. For 0 ¤ r < 1 and μ > 0 we have

∞

(’1)m (μ + m) m

’μ

(1 + r ) = r.

m! (μ)

m=0

Hence, if we replace r by 1/r 2 this gives

∞

(’1)m (μ + m) ’2m

r 2μ (1 + r 2 )’μ = r

m! (μ)

m=0

for r > 1. Moreover, Legendre™s duplication formula with z = ν + m + 1 > 0 yields

√

π (2ν + 2m + 2)

(ν + m + 3/2) = 2ν+2m+1 .

(ν + m + 1)

2

Thus, setting μ = ν + 3/2 > 1/2 shows that

∞

2ν+1 (ν + 3/2)r (’1)m (2ν + 2m + 2) ’2m’2ν’2

= .

√ r

22m+ν m! (ν + m + 1)

π(r 2 + 1)ν+3/2 m=0

Now we will have a look at the integral. Using the de¬nition of the Bessel function, and

interchanging summation and integration, allows us to make the following derivation:

∞

∞ ∞

(’1)m

ν+1 ’r t

t 2m+2ν+1 e’r t dt

dt =

Jν (t)t e

22m+ν m! (ν + m + 1)

0 0

m=0

∞

(’1) (2m + 2ν + 2) ’2m’2ν’2

m

= .

r

22m+ν m! (m + ν + 1)

m=0

In the last step we used that the integral representation of the -function can be expressed

as

∞

t 2m+2ν+1 e’r t dt = r ’2m’2ν’2 (2m + 2ν + 2).

0

Moreover, the interchange of summation and integration can be justi¬ed as follows. First

note that Stirling™s formula allows the bound (ν + m + 1) ≥ (1/cν )m!, where the constant

50 Auxiliary tools from analysis and measure theory

cν depends only on ν > ’1. Then

∞ ∞

(t/2)ν+2m t 2m

ν+1 ’r t

e’r t

¤ cν t 2ν+1

te

m! (ν + m + 1) 22m (m!)2

m=0 m=0

∞

t 2m

2ν+1 ’r t

¤ cν t e

(2m)!

m=0

¤ cν t 2ν+1 e’r t et ,

which is clearly in L 1 [0, ∞) provided that r > 1. Hence, Lebesgue™s convergence theorem

justi¬es the interchange.

Up to now we have shown that the stated equality holds for all r > 1. But, since both

sides are analytic functions in (r ) > 0 and | (r )| < 1, the equality extends to all r > 0 by

analytic continuation.

The following result is in the same spirit.

Lemma 5.8 For r > 0 it is true that

∞

1

J0 (t)e’r t dt = .

(1 + r 2 )1/2

0

Proof From the duplication formula of the -function it follows that

√

(2m)! π

(m + 1/2) = .

22m m!

Hence, as in the proof of Lemma 5.7 we get the representation

∞ ∞

(’1)m (m + 1/2) ’2m’1 (’1)m (2m)! ’2m’1

2 ’1/2

(1 + r ) = = .

r r

22m (m!)2

m! (1/2)

m=0 m=0

Thus, by interchanging summation and integration we derive

∞

∞ ∞

(’1)m

J0 (t)e’r t dt = t 2m e’r t dt

22m (m!)2

0 0

m=0

∞

(’1)m (2m + 1) ’2m’1

= r

22m (m!)2

m=0

= (1 + r 2 )’1/2

for r > 1. The interchange of summation and integration can be justi¬ed as in Lemma 5.7.

The equality extends to r > 0 by analytic continuation as before.

Our ¬nal result on Bessel functions of the ¬rst kind deals with J0 again.

Lemma 5.9 For the Bessel function J0 the following two properties are satis¬ed:

r

J0 (t)dt > 0 for all r > 0;

(1)

0

∞

J0 (t)dt = 1.

(2)

0