A treatise on the theory of Bessel functions, by G. N. Watson.

2-12]

THE BESSEL COEFFICIENTS

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By considering all the terms of the series for J, (z) except the first, it is
found that
(5) J~ ( 2)= — ) (1 + 0),
(- a l (\ ^ 12 ex ( I z 2) -
where 0 exp ( ) -1 (< _.
It should be observed that the series on the right in ~ 2'1 (1) converges uniformly in
any bounded domain of the variables z and t which does not contain the origin in the
t-plane. For if 8, A and R are positive constants and if
|.^I A, Iz l^R,
the terms in the expansion of exp (~zt) exp (iz/t) do not exceed in absolute value the corresponding terms of the product exp (-RA) exp (I-R/8), and the uniformity of the convergence
follows from the test of Weierstrass. Similar considerations apply to the series obtained
by term-by-term differentiations of the expansion 2tnJn (z), whether the differentiations be
performed with respect to z or t or both z and t.
2'12. The recurrence formulae.
The equations*
2n
(1) JJn-l (*) + Jn+l (Z) = - Jn (Z),
(2) Jn-1 (z)- J^,+ (z) = 2Jn (),
which connect three contiguous functions are useful in constructing Tables of
Bessel coefficients; they are known as recurrence formulae.
To prove the former, differentiate the fundamental expansion of ~ 2'1,
namely
e2 z(t-l/t)
= e tnJn(,
with respect to t; we get
Z (1 + 1/t2) e'z(t/t) = nt 1
( )2 e ntWl — Jn (z),
so that
IZ(1 + 1/t2) t Jn(z) ntn- Jn (z).
If the expression on the left is arranged in powers of t and coefficients of t"-l
are equated in the two Laurent series, which are identically equal, it is evident
that
z { Jn-i (z) + Jn-t (z)} = nJ (z),
which is the first of the formulaet.
* Throughout the work primes are used to denote the derivate of a function with respect to
its argument.
t Differentiations are permissible because (~ 2 11) the resulting series are uniformly convergent.
The equating of coefficients is permissible because Laurent expansions are unique.

W. B. F.

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