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Index




of highest level, 134
k-summability, 105
of level µ, 134
at in¬nity, 123
blocked
in a direction, 100
diagonally, 214
admissible, 165 triangularly, 214
multidirection, 161 blocks
algebra, 215 direct sum of, 214
commutative, 215 Jordan, 211
di¬erential, 216 Borel operator, 80
with unit element, 215 formal, 80
anti-Stokes™ direction, 137
closed
associated function, 146
sector, 60
asymptotic
w. r. to continuation, 57, 132
equality, 65
complex plane
expansion, 65
compacti¬ed, 15
remainders of, 65
con¬‚uent hypergeometric
of Gevrey order, 70
equation, 24, 25
space, 75
function, 23
Banach space, 216 generalized, 24
Bessel™s connection
equation, 24 matrix, 193
function, 23 problem, 193
Beta Integral, 228 convergence
block structure compact, 243
290 Index

locally uniform of highest level, 55
Laplace operator, 79
of an integral, 223
Laurent series, 72, 124
convolution, 178
power series, 64
formal, 178
solution, 42
of kernel functions, 160
system, 40
curve, 62
of Gevrey order s, 40
data pairs of FFS, 132 formula
ordered, 134 Hankel™s, 228
derivation, 216 Stirling™s, 229
di¬erentiability Fuchsian
at the origin, 62 system, 188
di¬erential function
¬eld, 73 associated, 146
di¬erential algebra, 216 Bessel™s, 23
direction bounded at the origin, 61
anti-Stokes™, 137 continuous at the origin, 61
Stokes™, 137 entire, 233
distinct eigenvalue case, 44 Gamma, 227
hypergeometric, 26
entire function, 233 con¬‚uent, 23
equation generalized, 26
Bessel™s, 24 kernel, 85, 89
hypergeometric, 26 Kummer™s, 23
con¬‚uent, 24, 25 matrix-valued, 2
generalized, 27 Mittag-Le¬„er™s, 233
indicial, 36 aymptotic of, 84
Legendre™s, 5 of a matrix, 237
equivalence order of a, 232
analytic, 28, 153 root of a, 230
meromorphic, 153 single-valued, 227
exponential type of a, 233
growth, 62 fundamental solution, 5
in a direction, 63 computation of, 6
order, 232 existence of, 6
shift, 39 formal, 42

factorial series, 110 general summability method, 97
¬nite type, 233 Gevrey order
Floquet asymptotic, 70
exponent, 20 of a system, 40
solution, 20 of a transformation, 39
for scalar ODE, 34 good spectrum, 18
formal
fundamental solution, 131 Hankel™s formula, 228
Index 291

highest level Legendre
equation, 5
normal solution, 138
polynomials, 5
HLFFS
level
data pairs of a, 56
highest, 134
holomorphic
set, 134
continuation, 224
logarithm
matrix-valued f., 2
Riemann surface of the, 226
holomorphy
in Banach spaces, 219
M¨bius transformation, 15
o
weak, 219
matrix
homotopic paths, 225
canonical form of a, 211
hypergeometric
companion, 4, 52
equation, 26
connection, 193
con¬‚uent, 24
Jordan, 211
generalized, 27
method, 97
function, 26
monodromy, 9
con¬‚uent, 23
nilpotent
generalized, 26
order of a, 212
series
permutation, 143
generalized, 107
power series, 3
system, 25
spectrum of a, 211
con¬‚uent, 21
unit, 211
moderate growth, 55
indicial equation, 36
moment
integrability
condition, 86
at the origin, 86
function, 86
invariants, 153
moment methods, 107
moment sequence, 119
Jordan
moments, 107
block, 211
monodromy
matrix, 211
factor, 9
kernel function, 85 formal, 143
kernel functions matrix, 9
convolution of, 160 formal, 143
kernel of small order, 89 multi-index, 198
multidirection, 161
Laplace singular, 167
operator, 78 of level j, 168
Laplace operator multisummability, 173
¬nite, 74 type of, 165
formal, 79
Laurent series nilpotent matrices
formal, 124 superiority, 45
leading term, 37 norm, 216
292 Index

normal covering, 118 almost, 55
for scalar equations, 34
operator residue, 224
acceleration, 176 Riemann surface
formal, 176 of the logarithm, 226
Borel, 80 ring, 8
formal, 80 root
deceleration, 177 of a function, 230
Laplace, 78 order of a, 230
¬nite, 74
sector, 60
formal, 79
bisecting direction of a, 60
order
closed, 60
exponential, 232
opening of a, 60
of a root, 230
radius of a, 60
of kernel functions, 86
sectorial region, 60
of moments, 86
at in¬nity, 123
series
paths, 62
convergent, 64
homotopic, 225
formal, 64
Pochhammer symbol, 22
formal Laurent, 72
Poincar´ rank, 14
e
Laurent, 224
power series
sum of a, 64
matrix, 3
similarity, 211
vector, 3
singular
power series regularity, 183
multidirection, 167
principal value
singular set, 168
of log z, 9
singularity
problem

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