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apparent, 16
connection, 193
essential, 224
central, 193
essentially irregular, 55
lateral, 193
of ¬rst kind, 14
projection, 226
of second kind, 14
punctured
of the ¬rst kind
plane, 226
for scalar equations, 34
punctured disc, 8
pole, 224
rank regular, 33
Poincar´, 14
e almost, 55
reduced system, 12 for scalar equations, 34
type of, 12 removable, 224
region, 219 solution
sectorial, 60 Floquet, 20
at in¬nity, 123 for scalar ODE, 34
simply connected, 1 fundamental, 5
regular-singular point, 33 computation of, 6
Index 293

existence of, 6 q-analytic, 39
q-meromorphic, 39
formal, 42, 131
analytic, 27
of highest level, 55
at in¬nity, 38
logarithmic, 35
of Gevrey order s, 39
normal
constant, 27
of highest level, 138
exponential shift, 39
space
formal analytic
asymptotic, 75
at in¬nity, 38
spectrum, 211
M¨bius, 15
o
good, 18
meromorphic, 39
Stirling™s formula, 229
formal, 39
Stokes™ direction, 137
shearing, 39
Stokes™ multipliers
unrami¬ed, 39
of highest level, 140
terminating, 38
sum
transformed system, 28, 39
of a series, 64
type, 233
summability
¬nite, 233
domain, 97
method
unit element, 215
general, 97
matrix, 97
variation of constants, 11
of series in roots, 124
vector
summability factor, 109
cyclic, 5
system
power series, 3
elementary, 14
unit, 211
formal, 40
of Gevrey order s, 40
Wronski™s identity, 6
Fuchsian, 188
hypergeometric, 25
con¬‚uent, 21
meromorphic, 14
nonlinear, 198
homogeneous, 198
normalized, 145
w. nilpotent l. t., 47
rank-reduced, 187
reduced, 12
type of, 12
w. nilpotent l. t., 47
transformed, 28, 39

transform
Cauchy-Heine, 116
formal, 117
transformation
List of Symbols




Here we list the symbols and abbreviations used in the book, giving a short
description of their meaning and, whenever necessary, the number of the
page where they are introduced.


∼ The function on the left has the power series on the right
=
as its asymptotic expansion (p. 65)
∼s The function on the left has the power series on the right
=
as its Gevrey expansion of order s (p. 70)

≺ We write m ≺ n whenever, after a rotation of C d making
the cuts point upward, the nth cut is to the right of the
mth one (p. 148)
∞(„ )
An integral from a to in¬nity along the ray arg(u ’ a) = „
a
(pp. 78, 79)

duk Short for k uk’1 du (p. 78)

δ Short for z(d/dz) (p. 24)

γk („ ) The path of integration following the negatively oriented
boundary of a sector of ¬nite radius, opening larger than
π/k and bisecting direction „ (p. 80)

(±)n Pochhammer™s symbol (p. 22)
296 Symbols

Ak,k Ecalle™s acceleration operator (p. 176)
˜

ˆ˜
Ak,k Ecalle™s formal acceleration operator (p. 176)

A(k) (S, E ) The space of all E -valued functions that are holomorphic,
bounded at the origin and of exponential growth at most
k in a sector S of in¬nite radius (p. 62)

A(G, E ) The space of all E -valued functions that are holomorphic
in a sectorial region G and have an asymptotic expansion
at the origin (p. 67)

As (G, E ) The space of all E -valued functions that are holomorphic
in a sectorial region G and have an asymptotic expansion
of Gevrey order s (p. 71)

As,m (G, E ) The space of all E -valued functions that are meromorphic
in a sectorial region G and have a Laurent series as asymp-
totic expansion of Gevrey order s (p. 73)

A(k) (S, E ) The intersection of As (G, E ) and A(k) (S, E ) (p. 79)
s

As,0 (G, E ) The set of ψ ∈ As (G, E ) with J(ψ) = ˆ
0 (p. 116)

Bk The Borel operator of order k (p. 80)
ˆ
Bk The formal Borel operator of order k (p. 80)

C The ¬eld of complex numbers

Cν The Banach space of column vectors of length ν with com-
plex entries (p. 2)

C ν—ν The Banach algebra of ν — ν matrices with complex entries
(p. 3)

Cd A complex plane with ¬nitely many cuts along rays arg(u’
um ) = ’r d (p. 145)

C± (z) The kernel of Ecalle™s acceleration operator (p. 175)

CHa The Cauchy-Heine operator (p. 116)

CHa The formal Cauchy-Heine operator (p. 117)

D(z0 , ρ) The disc with midpoint z0 and radius ρ (p. 2)

Degree or valuation of a matrix power series in z ’1 (p. 40)
ˆ
deg T (z)

E,F Banach spaces, resp. Banach algebras (p. 219)
Symbols 297

E— The set of continuous linear maps from E into C (p. 219)
E [[z]] The space of formal power series whose coe¬cients are
in E (p. 64)
E [[z]]s The space of formal power series with coe¬cients in E and
Gevrey order s (p. 64)
E {z} The space of convergent power series whose coe¬cients are
in E (p. 64)
E {z}k,d The space of power series with coe¬cients in E that are
k-summable in direction d (p. 102)
E {z}k The space of power series with coe¬cients in E that are
k-summable in all but ¬nitely many directions (p. 105)
E {z}T,d The space of power series with coe¬cients in E that are
T -summable in direction d (p. 108)
E {z}T ,d The space of power series with coe¬cients in E that are
T -summable in the multidirection d (p. 161)
e Euler™s constant (= exp[1])
e, ˆ
ˆ0 The formal power series whose constant term is e, resp. 0,
while the other coe¬cients are equal to 0 (pp. 64, 70)
e1 — e2 The convolution of kernel functions (p. 160)
E± (z) Mittag-Le¬„er™s function (p. 233)
F (±; β; z) Con¬‚uent hypergeometric function (p. 22)
F (±, β; γ; z) Hypergeometric function (p. 26)
Similar notation is used for the generalized con¬‚uent hy-
pergeometric function (p. 23)
resp. generalized hypergeometric function (p. 26)
resp. generalized hypergeometric series (p. 107)
FFS Short for formal fundamental solution (p. 131)
f —k g Convolution of functions f and g (p. 178)
ˆˆ ˆ
f —k g Convolution of formal power series f and g
ˆ (p. 178)
G A region in the complex domain, resp, on the Riemann
surface of the logarithm (p. 2)
G(d, ±) A sectorial region with bisecting direction d and opening
± (p. 61)
298 Symbols

H(G, E ) The space of functions, holomorphic in G, with values in
E (p. 221)

HLFFS Short for highest-level formal fundamental solution (p. 55)

HLNS Short for highest-level normal solution (p. 138)

J The linear map that maps functions to their asymptotic
expansion (p. 67)

Jµ (z) Bessel™s function (p. 23)

j0 Number of singular directions in a half-open interval of
length 2π (p. 137)

j1 Number of singular directions in a half-open interval of
length µπ/(qr ’ p) (p. 137)

Lk The Laplace operator of order k (p. 78)
ˆ
Lk The formal Laplace operator of order k (p. 79)

L(E , F) The Banach algebra of bounded linear maps from E into
F (p. 219)

N The set of natural numbers; observe that we here assume
0∈N

N0 Is equal to N ∪ {0}

ODE Short for ordinary di¬erential equation

p. Short for page

PDE Short for partial di¬erential equation

resp. Short for respectively

R The ¬eld of real numbers

The set of z with 0 < |z| < ρ

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. 60
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