Bellman, Kalaba, and Lockett recently proposed a numerical method for inverting the Laplace transform. The method consists in first reducing the infinite interval of integration to a finite one by a preliminary substitution of variables, and then employing an
n
n
-point Gauss-Legendre quadrature formula to reduce the inversion problem (approximately) to that of solving a system of
n
n
linear algebraic equations. Luke suggests the possibility of using Gauss-Jacobi quadrature (with parameters
α
\alpha
and
β
\beta
) in place of Gauss-Legendre quadrature, and in particular raises the question whether a judicious choice of the parameters
α
\alpha
,
β
\beta
may have a beneficial influence on the condition of the linear system of equations. The object of this note is to investigate the condition number cond
(
n
,
α
,
β
)
(n,\alpha ,\beta )
of this system as a function of
n
n
,
α
\alpha
, and
β
\beta
. It is found that cond
(
n
,
α
,
β
)
(n,\alpha ,\beta )
is usually larger than cond
(
n
,
β
,
α
)
(n,\beta ,\alpha )
if
β
>
α
\beta > \alpha
, at least asymptotically as
n
→
∞
n \to \infty
. Lower bounds for cond
(
n
,
α
,
β
)
(n,\alpha ,\beta )
are obtained together with their asymptotic behavior as
n
→
∞
n \to \infty
. Sharper bounds are derived in the special cases
n
n
,
n
n
odd, and
α
=
β
=
±
1
2
\alpha = \beta = \pm \frac {1} {2}
,
n
n
arbitrary. There is also a short table of cond
(
n
,
α
,
β
)
(n,\alpha ,\beta )
for
α
\alpha
,
β
=
−
.8
(
.2
)
0
,
.5
,
1
,
2
,
4
,
8
,
16
,
β
≦
α
\beta = - .8(.2)0,.5,1,2,4,8,16,\beta \leqq \alpha
, and
n
=
5
,
10
,
20
,
40
n = 5,10,20,40
. The general conclusion is that cond
(
n
,
α
,
β
)
(n,\alpha ,\beta )
grows at a rate which is something like a constant times
(
3
+
√
8
)
n
{(3 + \surd 8)^n}
, where the constant depends on
α
\alpha
and
β
\beta
, varies relatively slowly as a function of
α
\alpha
,
β
\beta
, and appears to be smallest near
α
=
β
=
−
1
\alpha = \beta = - 1
. For quadrature rules with equidistant points the condition grows like
(
2
√
2
/
3
π
)
8
n
(2\surd 2/3\pi ){8^n}
.