aboutsummaryrefslogtreecommitdiff
path: root/src/algebra/expexpan.spad.pamphlet
blob: 346bf3eb99e21bdff8c5186eb9e9624f59e6eb63 (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
\documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/src/algebra expexpan.spad}
\author{Clifton J. Williamson}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{domain EXPUPXS ExponentialOfUnivariatePuiseuxSeries}
<<domain EXPUPXS ExponentialOfUnivariatePuiseuxSeries>>=
)abbrev domain EXPUPXS ExponentialOfUnivariatePuiseuxSeries
++ Author: Clifton J. Williamson
++ Date Created: 4 August 1992
++ Date Last Updated: 27 August 1992
++ Basic Operations:
++ Related Domains: UnivariatePuiseuxSeries(FE,var,cen)
++ Also See:
++ AMS Classifications:
++ Keywords: limit, functional expression, power series, essential singularity
++ Examples:
++ References:
++ Description:
++   ExponentialOfUnivariatePuiseuxSeries is a domain used to represent
++   essential singularities of functions.  An object in this domain is a
++   function of the form \spad{exp(f(x))}, where \spad{f(x)} is a Puiseux
++   series with no terms of non-negative degree.  Objects are ordered
++   according to order of singularity, with functions which tend more
++   rapidly to zero or infinity considered to be larger.  Thus, if
++   \spad{order(f(x)) < order(g(x))}, i.e. the first non-zero term of
++   \spad{f(x)} has lower degree than the first non-zero term of \spad{g(x)},
++   then \spad{exp(f(x)) > exp(g(x))}.  If \spad{order(f(x)) = order(g(x))},
++   then the ordering is essentially random.  This domain is used
++   in computing limits involving functions with essential singularities.
ExponentialOfUnivariatePuiseuxSeries(FE,var,cen):_
      Exports == Implementation where
  FE  : Field
  var : Symbol
  cen : FE
  UPXS ==> UnivariatePuiseuxSeries(FE,var,cen)

  Exports ==> Join(UnivariatePuiseuxSeriesCategory(FE),OrderedAbelianMonoid) _
        with
    exponential : UPXS -> %
      ++ exponential(f(x)) returns \spad{exp(f(x))}.
      ++ Note: the function does NOT check that \spad{f(x)} has no
      ++ non-negative terms.
    exponent : % -> UPXS
      ++ exponent(exp(f(x))) returns \spad{f(x)}
    exponentialOrder: % -> Fraction Integer
      ++ exponentialOrder(exp(c * x **(-n) + ...)) returns \spad{-n}.
      ++ exponentialOrder(0) returns \spad{0}.

  Implementation ==> UPXS add

    Rep := UPXS

    exponential f == complete f
    exponent f == f pretend UPXS
    exponentialOrder f == order(exponent f,0)

    zero? f == empty? entries complete terms f

    f = g ==
    -- we redefine equality because we know that we are dealing with
    -- a FINITE series, so there is no danger in computing all terms
      (entries complete terms f) = (entries complete terms g)

    f < g ==
      zero? f => not zero? g
      zero? g => false
      (ordf := exponentialOrder f) > (ordg := exponentialOrder g) => true
      ordf < ordg => false
      (fCoef := coefficient(f,ordf)) = (gCoef := coefficient(g,ordg)) =>
        reductum(f) < reductum(g)
      before?(fCoef,gCoef)  -- this is "random" if FE is EXPR INT

    coerce(f:%):OutputForm ==
      ("%e" :: OutputForm) ** ((coerce$Rep)(complete f)@OutputForm)

@
\section{domain UPXSSING UnivariatePuiseuxSeriesWithExponentialSingularity}
<<domain UPXSSING UnivariatePuiseuxSeriesWithExponentialSingularity>>=
)abbrev domain UPXSSING UnivariatePuiseuxSeriesWithExponentialSingularity
++ Author: Clifton J. Williamson
++ Date Created: 4 August 1992
++ Date Last Updated: 27 August 1992
++ Basic Operations:
++ Related Domains: UnivariatePuiseuxSeries(FE,var,cen),
++                  ExponentialOfUnivariatePuiseuxSeries(FE,var,cen)
++                  ExponentialExpansion(R,FE,var,cen)
++ Also See:
++ AMS Classifications:
++ Keywords: limit, functional expression, power series
++ Examples:
++ References:
++ Description:
++   UnivariatePuiseuxSeriesWithExponentialSingularity is a domain used to
++   represent functions with essential singularities.  Objects in this
++   domain are sums, where each term in the sum is a univariate Puiseux
++   series times the exponential of a univariate Puiseux series.  Thus,
++   the elements of this domain are sums of expressions of the form
++   \spad{g(x) * exp(f(x))}, where g(x) is a univariate Puiseux series
++   and f(x) is a univariate Puiseux series with no terms of non-negative
++   degree.
UnivariatePuiseuxSeriesWithExponentialSingularity(R,FE,var,cen):_
  Exports == Implementation where
  R   : Join(RetractableTo Integer,_
             LinearlyExplicitRingOver Integer,GcdDomain)
  FE  : Join(AlgebraicallyClosedField,TranscendentalFunctionCategory,_
             FunctionSpace R)
  var : Symbol
  cen : FE
  B       ==> Boolean
  I       ==> Integer
  L       ==> List
  RN      ==> Fraction Integer
  UPXS    ==> UnivariatePuiseuxSeries(FE,var,cen)
  EXPUPXS ==> ExponentialOfUnivariatePuiseuxSeries(FE,var,cen)
  OFE     ==> OrderedCompletion FE
  Result  ==> Union(OFE,"failed")
  PxRec   ==> Record(k: Fraction Integer,c:FE)
  Term    ==> Record(%coef:UPXS,%expon:EXPUPXS,%expTerms:List PxRec)
    -- the %expTerms field is used to record the list of the terms (a 'term'
    -- records an exponent and a coefficient) in the exponent %expon
  TypedTerm ==> Record(%term:Term,%type:String)
    -- a term together with a String which tells whether it has an infinite,
    -- zero, or unknown limit as var -> cen+
  TRec    ==> Record(%zeroTerms: List Term,_
                     %infiniteTerms: List Term,_
                     %failedTerms: List Term,_
                     %puiseuxSeries: UPXS)
  SIGNEF  ==> ElementaryFunctionSign(R,FE)

  Exports ==> Join(FiniteAbelianMonoidRing(UPXS,EXPUPXS),IntegralDomain) with
    limitPlus : % -> Union(OFE,"failed")
      ++ limitPlus(f(var)) returns \spad{limit(var -> cen+,f(var))}.
    dominantTerm : % -> Union(TypedTerm,"failed")
      ++ dominantTerm(f(var)) returns the term that dominates the limiting
      ++ behavior of \spad{f(var)} as \spad{var -> cen+} together with a
      ++ \spadtype{String} which briefly describes that behavior.  The
      ++ value of the \spadtype{String} will be \spad{"zero"} (resp.
      ++ \spad{"infinity"}) if the term tends to zero (resp. infinity)
      ++ exponentially and will \spad{"series"} if the term is a
      ++ Puiseux series.

  Implementation ==> PolynomialRing(UPXS,EXPUPXS) add
    makeTerm : (UPXS,EXPUPXS) -> Term
    coeff : Term -> UPXS
    exponent : Term -> EXPUPXS
    exponentTerms : Term -> List PxRec
    setExponentTerms_! : (Term,List PxRec) -> List PxRec
    computeExponentTerms_! : Term -> List PxRec
    terms : % -> List Term
    sortAndDiscardTerms: List Term -> TRec
    termsWithExtremeLeadingCoef : (L Term,RN,I) -> Union(L Term,"failed")
    filterByOrder: (L Term,(RN,RN) -> B) -> Record(%list:L Term,%order:RN)
    dominantTermOnList : (L Term,RN,I) -> Union(Term,"failed")
    iDominantTerm : L Term -> Union(Record(%term:Term,%type:String),"failed")

    retractIfCan(f: %): Union(UPXS,"failed") ==
      (numberOfMonomials f = 1) and (zero? degree f) => leadingCoefficient f
      "failed"

    recip f ==
      numberOfMonomials f = 1 =>
        monomial(inv leadingCoefficient f,- degree f)
      "failed"

    makeTerm(coef,expon) == [coef,expon,empty()]
    coeff term == term.%coef
    exponent term == term.%expon
    exponentTerms term == term.%expTerms
    setExponentTerms_!(term,list) == term.%expTerms := list
    computeExponentTerms_! term ==
      setExponentTerms_!(term,entries complete terms exponent term)

    terms f ==
      -- terms with a higher order singularity will appear closer to the
      -- beginning of the list because of the ordering in EXPPUPXS;
      -- no "expnonent terms" are computed by this function
      zero? f => empty()
      concat(makeTerm(leadingCoefficient f,degree f),terms reductum f)

    sortAndDiscardTerms termList ==
      -- 'termList' is the list of terms of some function f(var), ordered
      -- so that terms with a higher order singularity occur at the
      -- beginning of the list.
      -- This function returns lists of candidates for the "dominant
      -- term" in 'termList', i.e. the term which describes the
      -- asymptotic behavior of f(var) as var -> cen+.
      -- 'zeroTerms' will contain terms which tend to zero exponentially
      -- and contains only those terms with the lowest order singularity.
      -- 'zeroTerms' will be non-empty only when there are no terms of
      -- infinite or series type.
      -- 'infiniteTerms' will contain terms which tend to infinity
      -- exponentially and contains only those terms with the highest
      -- order singularity.
      -- 'failedTerms' will contain terms which have an exponential
      -- singularity, where we cannot say whether the limiting value
      -- is zero or infinity. Only terms with a higher order sigularity
      -- than the terms on 'infiniteList' are included.
      -- 'pSeries' will be a Puiseux series representing a term without an
      -- exponential singularity.  'pSeries' will be non-zero only when no
      -- other terms are known to tend to infinity exponentially
      zeroTerms : List Term := empty()
      infiniteTerms : List Term := empty()
      failedTerms : List Term := empty()
      -- we keep track of whether or not we've found an infinite term
      -- if so, 'infTermOrd' will be set to a negative value
      infTermOrd : RN := 0
      -- we keep track of whether or not we've found a zero term
      -- if so, 'zeroTermOrd' will be set to a negative value
      zeroTermOrd : RN := 0
      ord : RN := 0; pSeries : UPXS := 0  -- dummy values
      while not empty? termList repeat
        -- 'expon' is a Puiseux series
        expon := exponent(term := first termList)
        -- quit if there is an infinite term with a higher order singularity
        (ord := order(expon,0)) > infTermOrd => leave "infinite term dominates"
        -- if ord = 0, we've hit the end of the list
        (ord = 0) =>
          -- since we have a series term, don't bother with zero terms
          leave(pSeries := coeff(term); zeroTerms := empty())
        coef := coefficient(expon,ord)
        -- if we can't tell if the lowest order coefficient is positive or
        -- negative, we have a "failed term"
        (signum := sign(coef)$SIGNEF) case "failed" =>
          failedTerms := concat(term,failedTerms)
          termList := rest termList
        -- if the lowest order coefficient is positive, we have an
        -- "infinite term"
        (sig := signum :: Integer) = 1 =>
          infTermOrd := ord
          infiniteTerms := concat(term,infiniteTerms)
          -- since we have an infinite term, don't bother with zero terms
          zeroTerms := empty()
          termList := rest termList
        -- if the lowest order coefficient is negative, we have a
        -- "zero term" if there are no infinite terms and no failed
        -- terms, add the term to 'zeroTerms'
        if empty? infiniteTerms then
          zeroTerms :=
            ord = zeroTermOrd => concat(term,zeroTerms)
            zeroTermOrd := ord
            list term
        termList := rest termList
      -- reverse "failed terms" so that higher order singularities
      -- appear at the beginning of the list
      [zeroTerms,infiniteTerms,reverse_! failedTerms,pSeries]

    termsWithExtremeLeadingCoef(termList,ord,signum) ==
      -- 'termList' consists of terms of the form [g(x),exp(f(x)),...];
      -- when 'signum' is +1 (resp. -1), this function filters 'termList'
      -- leaving only those terms such that coefficient(f(x),ord) is
      -- maximal (resp. minimal)
      while (coefficient(exponent first termList,ord) = 0) repeat
        termList := rest termList
      empty? termList => error "UPXSSING: can't happen"
      coefExtreme := coefficient(exponent first termList,ord)
      outList := list first termList; termList := rest termList
      for term in termList repeat
        (coefDiff := coefficient(exponent term,ord) - coefExtreme) = 0 =>
          outList := concat(term,outList)
        (sig := sign(coefDiff)$SIGNEF) case "failed" => return "failed"
        (sig :: Integer) = signum => outList := list term
      outList

    filterByOrder(termList,predicate) ==
      -- 'termList' consists of terms of the form [g(x),exp(f(x)),expTerms],
      -- where 'expTerms' is a list containing some of the terms in the
      -- series f(x).
      -- The function filters 'termList' and, when 'predicate' is < (resp. >),
      -- leaves only those terms with the lowest (resp. highest) order term
      -- in 'expTerms'
      while empty? exponentTerms first termList repeat
        termList := rest termList
        empty? termList => error "UPXSING: can't happen"
      ordExtreme := (first exponentTerms first termList).k
      outList := list first termList
      for term in rest termList repeat
        not empty? exponentTerms term =>
          (ord := (first exponentTerms term).k) = ordExtreme =>
            outList := concat(term,outList)
          predicate(ord,ordExtreme) =>
            ordExtreme := ord
            outList := list term
      -- advance pointers on "exponent terms" on terms on 'outList'
      for term in outList repeat
        setExponentTerms_!(term,rest exponentTerms term)
      [outList,ordExtreme]

    dominantTermOnList(termList,ord0,signum) ==
      -- finds dominant term on 'termList'
      -- it is known that "exponent terms" of order < 'ord0' are
      -- the same for all terms on 'termList'
      newList := termsWithExtremeLeadingCoef(termList,ord0,signum)
      newList case "failed" => "failed"
      termList := newList :: List Term
      empty? rest termList => first termList
      filtered :=
        signum = 1 => filterByOrder(termList,#1 < #2)
        filterByOrder(termList,#1 > #2)
      termList := filtered.%list
      empty? rest termList => first termList
      dominantTermOnList(termList,filtered.%order,signum)

    iDominantTerm termList ==
      termRecord := sortAndDiscardTerms termList
      zeroTerms := termRecord.%zeroTerms
      infiniteTerms := termRecord.%infiniteTerms
      failedTerms := termRecord.%failedTerms
      pSeries := termRecord.%puiseuxSeries
      -- in future versions, we will deal with "failed terms"
      -- at present, if any occur, we cannot determine the limit
      not empty? failedTerms => "failed"
      not zero? pSeries => [makeTerm(pSeries,0),"series"]
      not empty? infiniteTerms =>
        empty? rest infiniteTerms => [first infiniteTerms,"infinity"]
        for term in infiniteTerms repeat computeExponentTerms_! term
        ord0 := order exponent first infiniteTerms
        (dTerm := dominantTermOnList(infiniteTerms,ord0,1)) case "failed" =>
          return "failed"
        [dTerm :: Term,"infinity"]
      empty? rest zeroTerms => [first zeroTerms,"zero"]
      for term in zeroTerms repeat computeExponentTerms_! term
      ord0 := order exponent first zeroTerms
      (dTerm := dominantTermOnList(zeroTerms,ord0,-1)) case "failed" =>
        return "failed"
      [dTerm :: Term,"zero"]

    dominantTerm f == iDominantTerm terms f

    limitPlus f ==
      -- list the terms occurring in 'f'; if there are none, then f = 0
      empty?(termList := terms f) => 0
      -- compute dominant term
      (tInfo := iDominantTerm termList) case "failed" => "failed"
      termInfo := tInfo :: Record(%term:Term,%type:String)
      domTerm := termInfo.%term
      (type := termInfo.%type) = "series" =>
        -- find limit of series term
        (ord := order(pSeries := coeff domTerm,1)) > 0 => 0
        coef := coefficient(pSeries,ord)
        member?(var,variables coef) => "failed"
        ord = 0 => coef :: OFE
        -- in the case of an infinite limit, we need to know the sign
        -- of the first non-zero coefficient
        (signum := sign(coef)$SIGNEF) case "failed" => "failed"
        (signum :: Integer) = 1 => plusInfinity()
        minusInfinity()
      type = "zero" => 0
      -- examine lowest order coefficient in series part of 'domTerm'
      ord := order(pSeries := coeff domTerm)
      coef := coefficient(pSeries,ord)
      member?(var,variables coef) => "failed"
      (signum := sign(coef)$SIGNEF) case "failed" => "failed"
      (signum :: Integer) = 1 => plusInfinity()
      minusInfinity()

@
\section{domain EXPEXPAN ExponentialExpansion}
<<domain EXPEXPAN ExponentialExpansion>>=
)abbrev domain EXPEXPAN ExponentialExpansion
++ Author: Clifton J. Williamson
++ Date Created: 13 August 1992
++ Date Last Updated: 27 August 1992
++ Basic Operations:
++ Related Domains: UnivariatePuiseuxSeries(FE,var,cen),
++                  ExponentialOfUnivariatePuiseuxSeries(FE,var,cen)
++ Also See:
++ AMS Classifications:
++ Keywords: limit, functional expression, power series
++ Examples:
++ References:
++ Description:
++   UnivariatePuiseuxSeriesWithExponentialSingularity is a domain used to
++   represent essential singularities of functions.  Objects in this domain
++   are quotients of sums, where each term in the sum is a univariate Puiseux
++   series times the exponential of a univariate Puiseux series.
ExponentialExpansion(R,FE,var,cen): Exports == Implementation where
  R   : Join(RetractableTo Integer,_
             LinearlyExplicitRingOver Integer,GcdDomain)
  FE  : Join(AlgebraicallyClosedField,TranscendentalFunctionCategory,_
             FunctionSpace R)
  var : Symbol
  cen : FE
  RN       ==> Fraction Integer
  UPXS     ==> UnivariatePuiseuxSeries(FE,var,cen)
  EXPUPXS  ==> ExponentialOfUnivariatePuiseuxSeries(FE,var,cen)
  UPXSSING ==> UnivariatePuiseuxSeriesWithExponentialSingularity(R,FE,var,cen)
  OFE      ==> OrderedCompletion FE
  Result   ==> Union(OFE,"failed")
  PxRec    ==> Record(k: Fraction Integer,c:FE)
  Term     ==> Record(%coef:UPXS,%expon:EXPUPXS,%expTerms:List PxRec)
  TypedTerm ==> Record(%term:Term,%type:String)
  SIGNEF   ==> ElementaryFunctionSign(R,FE)

  Exports ==> Join(QuotientFieldCategory UPXSSING,RetractableTo UPXS) with
    limitPlus : % -> Union(OFE,"failed")
      ++ limitPlus(f(var)) returns \spad{limit(var -> a+,f(var))}.
    coerce: UPXS -> %
      ++ coerce(f) converts a \spadtype{UnivariatePuiseuxSeries} to
      ++ an \spadtype{ExponentialExpansion}.

  Implementation ==> Fraction(UPXSSING) add
    coeff : Term -> UPXS
    exponent : Term -> EXPUPXS
    upxssingIfCan : % -> Union(UPXSSING,"failed")
    seriesQuotientLimit: (UPXS,UPXS) -> Union(OFE,"failed")
    seriesQuotientInfinity: (UPXS,UPXS) -> Union(OFE,"failed")

    Rep := Fraction UPXSSING

    ZEROCOUNT : RN := 1000/1

    coeff term == term.%coef
    exponent term == term.%expon

    --!! why is this necessary?
    --!! code can run forever in retractIfCan if original assignment
    --!! for 'ff' is used
    upxssingIfCan f ==
      one? denom f => numer f
      "failed"

    retractIfCan(f:%):Union(UPXS,"failed") ==
      --ff := (retractIfCan$Rep)(f)@Union(UPXSSING,"failed")
      --ff case "failed" => "failed"
      (ff := upxssingIfCan f) case "failed" => "failed"
      (fff := retractIfCan(ff::UPXSSING)@Union(UPXS,"failed")) case "failed" =>
        "failed"
      fff :: UPXS

    f:UPXSSING / g:UPXSSING ==
      (rec := recip g) case "failed" => f /$Rep g
      f * (rec :: UPXSSING) :: %

    f:% / g:% ==
      (rec := recip numer g) case "failed" => f /$Rep g
      (rec :: UPXSSING) * (denom g) * f

    coerce(f:UPXS) == f :: UPXSSING :: %

    seriesQuotientLimit(num,den) ==
      -- limit of the quotient of two series
      series := num / den
      (ord := order(series,1)) > 0 => 0
      coef := coefficient(series,ord)
      member?(var,variables coef) => "failed"
      ord = 0 => coef :: OFE
      (sig := sign(coef)$SIGNEF) case "failed" => return "failed"
      (sig :: Integer) = 1 => plusInfinity()
      minusInfinity()

    seriesQuotientInfinity(num,den) ==
      -- infinite limit: plus or minus?
      -- look at leading coefficients of series to tell
      (numOrd := order(num,ZEROCOUNT)) = ZEROCOUNT => "failed"
      (denOrd := order(den,ZEROCOUNT)) = ZEROCOUNT => "failed"
      cc := coefficient(num,numOrd)/coefficient(den,denOrd)
      member?(var,variables cc) => "failed"
      (sig := sign(cc)$SIGNEF) case "failed" => return "failed"
      (sig :: Integer) = 1 => plusInfinity()
      minusInfinity()

    limitPlus f ==
      zero? f => 0
      (den := denom f) = 1 => limitPlus numer f
      (numerTerm := dominantTerm(num := numer f)) case "failed" => "failed"
      numType := (numTerm := numerTerm :: TypedTerm).%type
      (denomTerm := dominantTerm den) case "failed" => "failed"
      denType := (denTerm := denomTerm :: TypedTerm).%type
      numExpon := exponent numTerm.%term; denExpon := exponent denTerm.%term
      numCoef := coeff numTerm.%term; denCoef := coeff denTerm.%term
      -- numerator tends to zero exponentially
      (numType = "zero") =>
        -- denominator tends to zero exponentially
        (denType = "zero") =>
          (exponDiff := numExpon - denExpon) = 0 =>
            seriesQuotientLimit(numCoef,denCoef)
          expCoef := coefficient(exponDiff,order exponDiff)
          (sig := sign(expCoef)$SIGNEF) case "failed" => return "failed"
          (sig :: Integer) = -1 => 0
          seriesQuotientInfinity(numCoef,denCoef)
        0 -- otherwise limit is zero
      -- numerator is a Puiseux series
      (numType = "series") =>
        -- denominator tends to zero exponentially
        (denType = "zero") =>
          seriesQuotientInfinity(numCoef,denCoef)
        -- denominator is a series
        (denType = "series") => seriesQuotientLimit(numCoef,denCoef)
        0
      -- remaining case: numerator tends to infinity exponentially
      -- denominator tends to infinity exponentially
      (denType = "infinity") =>
        (exponDiff := numExpon - denExpon) = 0 =>
          seriesQuotientLimit(numCoef,denCoef)
        expCoef := coefficient(exponDiff,order exponDiff)
        (sig := sign(expCoef)$SIGNEF) case "failed" => return "failed"
        (sig :: Integer) = -1 => 0
        seriesQuotientInfinity(numCoef,denCoef)
      -- denominator tends to zero exponentially or is a series
      seriesQuotientInfinity(numCoef,denCoef)

@
\section{License}
<<license>>=
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--All rights reserved.
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
--    - Redistributions of source code must retain the above copyright
--      notice, this list of conditions and the following disclaimer.
--
--    - Redistributions in binary form must reproduce the above copyright
--      notice, this list of conditions and the following disclaimer in
--      the documentation and/or other materials provided with the
--      distribution.
--
--    - Neither the name of The Numerical ALgorithms Group Ltd. nor the
--      names of its contributors may be used to endorse or promote products
--      derived from this software without specific prior written permission.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
@
<<*>>=
<<license>>

<<domain EXPUPXS ExponentialOfUnivariatePuiseuxSeries>>
<<domain UPXSSING UnivariatePuiseuxSeriesWithExponentialSingularity>>
<<domain EXPEXPAN ExponentialExpansion>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}