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