106
VIII – Cauchy Theory
f (x) = a 1 x
u1 + . . . + a n x
un + O (x
un+1 )
(13.6’)
in the neighbourhood of 0, with real exponents u 1 < . . . < u n < u n+1 . As
1
0
x
s d
∗ x = 1/s if Re(s) > 0 ,
integral (5’) converges for Re(s) > −u 1 and, is equal to
1≤k≤n
a k
s + u k
+
1
0
O (x
un+1 ) x
s d
∗ x
(13.7)
on this strip. The
is a rational function whose poles and residues are
prominently displayed, though the integral in the last term converges and is
holomorphic for Re(s) > −u n+1 . (5’) can, therefore, be defined by analytic
extension on this half-plane. If f admits an unbounded asymptotic expansion
62 of the form
f (x) ≈
a n x
un , x −→ 0 with u n < u n+1 and lim u n = +∞ ,
(13.8’)
function (5’) can be analytically extended to all of C, with simple poles
and residues equal to a n at all points −u n . If for example f (x) is C
∞ on
R + , with the origin included, then by MacLaurin’s formula [Chap. V, n
◦ 18,
eq. (18.11)], there is an unbounded asymptotic expansion
f (x) ≈
f
(n) (0)x
n /n! .
Function (5’), a priori defined for Re(s) > 0, can, therefore, be extended to C
with the points 0, −1, −2, . . . removed, points where it has simple poles and
residues equal to the corresponding numbers f
(n) (0)/n!. This has already
been seen for f (x) = e
−x in relation to Euler’s Γ function; in this case, the
MacLaurin series can even be integrated term by term over ]0, 1], which gives
an expansion
Γ
−
f (s) =
f
(n) (0)/n!(s + n)
with a convergent series for all s. The same holds for any analytic function
f in the neighbourhood of 0 provided the radius of convergence R of its
MacLaurin series is > 1. If R ≤ 1, decompose R
∗
+ at (0, a) and (a, +∞) with
62 As a general rule, formula f (x) ≈
un(x) means that (i) un+1(x) = o [un(x)]
in the neighbourhood of the point considered, (ii) f (x) = u1(x) + . . . + un(x) +
O [un+1(x)] for any n. This does not in any way imply that the series
un(x)
converges to f (x); it can in fact diverge and that is often the case in practice,
in particular for the Taylor series of a non-analytic C
∞ function. See Chap. VI,
n
◦ 10.
VIII – Cauchy Theory
f (x) = a 1 x
u1 + . . . + a n x
un + O (x
un+1 )
(13.6’)
in the neighbourhood of 0, with real exponents u 1 < . . . < u n < u n+1 . As
1
0
x
s d
∗ x = 1/s if Re(s) > 0 ,
integral (5’) converges for Re(s) > −u 1 and, is equal to
1≤k≤n
a k
s + u k
+
1
0
O (x
un+1 ) x
s d
∗ x
(13.7)
on this strip. The
is a rational function whose poles and residues are
prominently displayed, though the integral in the last term converges and is
holomorphic for Re(s) > −u n+1 . (5’) can, therefore, be defined by analytic
extension on this half-plane. If f admits an unbounded asymptotic expansion
62 of the form
f (x) ≈
a n x
un , x −→ 0 with u n < u n+1 and lim u n = +∞ ,
(13.8’)
function (5’) can be analytically extended to all of C, with simple poles
and residues equal to a n at all points −u n . If for example f (x) is C
∞ on
R + , with the origin included, then by MacLaurin’s formula [Chap. V, n
◦ 18,
eq. (18.11)], there is an unbounded asymptotic expansion
f (x) ≈
f
(n) (0)x
n /n! .
Function (5’), a priori defined for Re(s) > 0, can, therefore, be extended to C
with the points 0, −1, −2, . . . removed, points where it has simple poles and
residues equal to the corresponding numbers f
(n) (0)/n!. This has already
been seen for f (x) = e
−x in relation to Euler’s Γ function; in this case, the
MacLaurin series can even be integrated term by term over ]0, 1], which gives
an expansion
Γ
−
f (s) =
f
(n) (0)/n!(s + n)
with a convergent series for all s. The same holds for any analytic function
f in the neighbourhood of 0 provided the radius of convergence R of its
MacLaurin series is > 1. If R ≤ 1, decompose R
∗
+ at (0, a) and (a, +∞) with
62 As a general rule, formula f (x) ≈
un(x) means that (i) un+1(x) = o [un(x)]
in the neighbourhood of the point considered, (ii) f (x) = u1(x) + . . . + un(x) +
O [un+1(x)] for any n. This does not in any way imply that the series
un(x)
converges to f (x); it can in fact diverge and that is often the case in practice,
in particular for the Taylor series of a non-analytic C
∞ function. See Chap. VI,
n
◦ 10.