130
P. RAZELOS
A New Integral
Representation
of the
Bessel Coefficients
By P. Razelos
The modified Bessel coefficients /„ are defined by the series [1]
,tyq+n
(1)
I(t)
with the integral representation
(2)
t
w
«-QSKS+ n)\
[1]
In(t) = - Í
e °°BXcos nxdx.
ir Jo
An integral representation of the coefficients In(t) is presented here where the path
of integration is extended to infinity.
Theorem:
The integral
An(l)
= - /
IT Jo
e<cosa: cos nx-dx
x
is the modiñed coefficient In(t) for any 0 < t < 1. There are many ways by which
the theorem can be proved, but we give here the following proof which consists of a
straightforward evaluation of the integral An . The function e'c°" Xis expanded in a
power series (which is absolutely convergent for all t) and we then integrate term
by term.
/o\
(.tCOSX)
e(coax = V"1
2---j-•
(3)
r=o
r\
Let us define
ia\
r\ r
(4)
2
f"
Qn = - I
r
sin ex ,
cos x cosnx-dx.
7TJn
x
Then
Qn tr
An(t)
Z^rtr¿InVJ =
= Z^
(5)
r=o
r-0
Introducing
(6)
r\
the expansion
1 <<■/*),-'
/r\
cosrz = -—:
2r~1
y,
k-o
( , ) cos(r — 2k)x
\k/
(where 5 = 1 or § for r even or odd, respectively)
(7)
Qn = \r 'g
Q
into (4), we obtain
lg(r - 2k + n) + g(r -2k-
Received March 31, 1964. Revised July 6, 1964.
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n)],
INTEGRAL REPRESENTATION
OF BESSEL
COEFFICIENTS
131
where
(8)
g(y) = - i
7T J0
cos yx ^-^
X
dx = 0,1,
for | y/t | greater than, less than, or equal to one, respectively [2]. Therefore, the
only term which is nonzero in (7) is the term g(0), if it exists.
Then
r
!
r!
(9)
^B = 2r (r - «\ '- -L--\
(10)
Q„r = 0
'
r - n and r' n both even 0r odd'
&)<;-¥)'
for r < n
or
r, n one even, one odd.
We can now write
r — n = 2q,
(11)
r + n = 2(g + n).
Thus,
'
r!
Qn' = br
2rç!(ç + n)!
(12)
Substituting (12) into (5), we obtain
(13)
A.(0-¿
.)2¿_ „-i-
Q-E-D.
,=oql(q + n)l
A similar expression can be readily obtained for the coefficients Jn(t). The value of
e = § gives the following interesting result. Let us define
(ia\
(14)
d d\
% fn
<1/2) r"
Bn(t) = - /
/
xJn-(i/2)
Jo
e
tC08x
2 f" ¿
■KJo
rotmax
dp /
= lim?
2e
Jo
f
dv f
¡CO rJo
= 2e t cos y sin
Consider
2/
,
i
now the integral
v sin(a;/2)
cos(ra)--—
x
,
dx
2etcOBXcos(vx)cos(vy) SÍn(x/2) dx
Ji
j//2
j
a;
It can be easily shown that Bn(t) = In(t).
= - /
sin(x/2)
cosca; ———- dvdx.
X
= e
by Fourier's theorem. Clearly B(t) = 2Z-«,ß„(i).
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132
C. D. SUTHERLAND
Acknowledgment. The author wishes to thank Professor H. G. Elrod, Jr. for
suggesting the problem.
Columbia University
New York, New York
1. G. N. Watson,
A Treatise on the Theory of Bessel Functions,
2nd ed., Cambridge
Univ.
Press, New York, 1944. MR 6, 64.
2. R. S. Burington,
Handbook of Mathematical
Tables and Formulas,
3rd ed., Handbook
Publishers Inc., Sandusky, Ohio, 1949.MR 22 #2514.
Footnote
to the Evaluation
of Certain
Complex
Elliptic Integrals
By C. D. Sutherland
The formulas for evaluating the elliptic integral of the third kind wi h a complex
parameter as given by Byrd and Friedman [1] have been corrected and simplified
)by Lang and Stevens [2]. There is, however, a further correction necessary in these
latter results.
The integral to be evaluated is
/ = (ai + *i)
where a2 is complex and A = \/(l
there appears the quantity
fPi
T2= I
/
T-
.
,
Jo (1 — a2 sin2 0)A
— k2 sin2 6). In the formulas for evaluating /
m2dx
m2 ,
-i,
,, N
T+h^2 = VF2tan WW'
where
sin tj>cos <t>
V2 = (1 + m2 sin2 </>)A'
We will consider the case where m2 ^ — 1. If this occurs we see that as </>goes
to ir/2, either p2 —> oo (m2 = —1) and [tan-1 (p2 \/h2)] —*ir/2, or p2 —>0 through
negative values (m2 < —1) and [tan-1 (p2 \/h2)] —*ir (and not to zero). To avoid
overlooking this possibility the proper representation
—1
\/h2
72 = —FT cos
-i /
A cos <t>
\
f
( -JTÏ.—• i -i i A2-T^\
I Ior
\V(h2 sin2 4>+ A2 cos2 <j>)/
for r2 is
m2 =
1
~ 1'
m2
-i (
A(l + m2 sin2 <f>)
\
cos
I -771—7~in-2
^ _l A2^i _l-■
, ^n )
Vh2
\V(h2 sin2 (j>cos2 <t>+ A2(l + m2 sin2 0)2)/
72 = -yr
.
for
m2 y¿ -1.
It is to be noted, in particular, that the formulas for the real and imaginary
parts of the complete integral should contain a term involving r2 whenever m2 ^ —1.
Received
Commission.
March 16, 1964. Work performed
under the auspices
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of the U. S. Atomic Energy