363:
Every element of a Hardy field is eventually either strictly positive, strictly negative, or zero. This follows fairly immediately from the facts that the elements in a Hardy field are eventually differentiable and hence
425:
529:. But they both tend to infinity. In this sense the ordering tells us how quickly all the unbounded functions diverge to infinity. Even finite limits being equal is not enough: consider
315:
Another example is the field of functions that can be expressed using the standard arithmetic operations, exponents, and logarithms, and are well-defined on some interval of the form
345:
368:
and eventually either have a multiplicative inverse or are zero. This means periodic functions such as the sine and cosine functions cannot exist in Hardy fields.
371:
This avoidance of periodic functions also means that every element in a Hardy field has a (possibly infinite) limit at infinity, so if
52:
Initially at least, Hardy fields were defined in terms of germs of real functions at infinity. Specifically we consider a collection
151:
modulo this equivalence relation under the induced addition and multiplication operations. Moreover, if every function in
557:
The modern theory of Hardy fields doesn't restrict to real functions but to those defined in certain structures expanding
688:
385:
178:. However, in practice the elements are normally just denoted by the representatives themselves, so instead of
285:
749:
37:
744:
734:
600:
77:
318:
581:). The properties of Hardy fields in the real setting still hold in this more general setting.
351:. Much bigger Hardy fields (that contain Hardy L-functions as a subfield) can be defined using
739:
140:
is eventually zero. The equivalence classes of this relation are called germs at infinity.
117:
33:
598:
Boshernitzan, Michael (1986), "Hardy fields and existence of transexponential functions",
174:
to denote the class of functions that are eventually equal to the representative function
8:
365:
25:
711:
670:
617:
29:
621:
558:
241:
300:. Adding and multiplying rational functions gives more rational functions, and the
147:
forms a field under the usual addition and multiplication of functions then so will
715:
703:
660:
609:
566:
233:
is, and since the derivative of every function in this field is 0 which must be in
170:
Elements of a Hardy field are thus equivalence classes and should be denoted, say,
573:
that are defined for all sufficiently large elements forms a Hardy field denoted
461:
is eventually strictly positive. Note that this is not the same as stating that
199:
304:
shows that the derivative of rational function is again a rational function, so
707:
728:
301:
56:
of functions that are defined for all large real numbers, that is functions
438:
352:
76:. Here and in the rest of the article we say a function has a property "
41:
17:
206:
then we can consider it as a Hardy field by considering the elements of
674:
613:
288:, such a rational function will be defined for all sufficiently large
665:
569:
expansion of a field, then the set of unary definable functions in
155:
is eventually differentiable and the derivative of any function in
477:. For example, if we consider the germs of the identity function
167:
modulo the above equivalence relation is called a Hardy field.
280:
are polynomials with real coefficients. Since the polynomial
210:
as constant functions, that is by considering the number α in
651:
Rosenlicht, Maxwell (1983), "The Rank of a Hardy Field",
240:
A less trivial example of a Hardy field is the field of
388:
321:
80:" if it has the property for all sufficiently large
419:
339:
40:. They are named after the English mathematician
653:Transactions of the American Mathematical Society
726:
689:"Valuation theory of exponential Hardy fields I"
687:Kuhlmann, Franz-Viktor; Kuhlmann, Salma (2003),
686:
635:Properties of Logarithmico-Exponential Functions
420:{\displaystyle \lim _{x\rightarrow \infty }f(x)}
390:
256:). This is the set of functions of the form
597:
650:
664:
284:can have only finitely many zeros by the
727:
637:, Proc. London Math. Soc. (2), 54–90,
347:. Such functions are sometimes called
296:larger than the largest real root of
84:, so for example we say a function
13:
552:
400:
331:
36:at infinity that are closed under
14:
761:
680:
644:
627:
591:
437:It also means we can place an
414:
408:
397:
334:
322:
286:fundamental theorem of algebra
1:
584:
489:and the exponential function
358:
229:to α. This is a field since
96:if there is some real number
47:
434: ∪ {−∞,+∞}.
7:
517:) > 0 for all
340:{\displaystyle (x,\infty )}
189:
10:
766:
473:is less than the limit of
708:10.1007/s00209-002-0460-4
696:Mathematische Zeitschrift
214:as the constant function
601:Aequationes Mathematicae
108:) = 0 for all
64:,∞) to the real numbers
312:) forms a Hardy field.
292:, specifically for all
68:, for some real number
421:
341:
422:
342:
237:it is a Hardy field.
182:one would just write
34:real-valued functions
750:Algebraic structures
509:) −
386:
319:
118:equivalence relation
745:Field (mathematics)
735:Asymptotic analysis
457: −
136: −
614:10.1007/BF02189932
559:real closed fields
417:
404:
337:
242:rational functions
116:. We can form an
549:) = 0.
389:
375:is an element of
349:Hardy L-functions
128:is equivalent to
757:
719:
718:
693:
684:
678:
677:
668:
648:
642:
631:
625:
624:
595:
537:) = 1/
525: >
469:if the limit of
465: <
449: <
426:
424:
423:
418:
403:
346:
344:
343:
338:
221:that maps every
765:
764:
760:
759:
758:
756:
755:
754:
725:
724:
723:
722:
691:
685:
681:
666:10.2307/1999639
649:
645:
632:
628:
596:
592:
587:
555:
553:In model theory
393:
387:
384:
383:
361:
320:
317:
316:
220:
192:
181:
173:
132:if and only if
94:eventually zero
50:
38:differentiation
12:
11:
5:
763:
753:
752:
747:
742:
737:
721:
720:
702:(4): 671–688,
679:
659:(2): 659–671,
643:
626:
608:(1): 258–280,
589:
588:
586:
583:
561:. Indeed, if
554:
551:
497:) =
485:) =
428:
427:
416:
413:
410:
407:
402:
399:
396:
392:
360:
357:
336:
333:
330:
327:
324:
218:
191:
188:
179:
171:
49:
46:
28:consisting of
9:
6:
4:
3:
2:
762:
751:
748:
746:
743:
741:
738:
736:
733:
732:
730:
717:
713:
709:
705:
701:
697:
690:
683:
676:
672:
667:
662:
658:
654:
647:
640:
636:
633:G. H. Hardy,
630:
623:
619:
615:
611:
607:
603:
602:
594:
590:
582:
580:
576:
572:
568:
564:
560:
550:
548:
544:
540:
536:
532:
528:
524:
521:we have that
520:
516:
512:
508:
504:
500:
496:
492:
488:
484:
480:
476:
472:
468:
464:
460:
456:
452:
448:
444:
440:
435:
433:
411:
405:
394:
382:
381:
380:
378:
374:
369:
367:
356:
354:
350:
328:
325:
313:
311:
307:
303:
302:quotient rule
299:
295:
291:
287:
283:
279:
275:
271:
267:
263:
259:
255:
251:
247:
243:
238:
236:
232:
228:
224:
217:
213:
209:
205:
201:
197:
187:
185:
177:
168:
166:
162:
158:
154:
150:
146:
141:
139:
135:
131:
127:
123:
119:
115:
112: ≥
111:
107:
103:
99:
95:
91:
87:
83:
79:
75:
72:depending on
71:
67:
63:
59:
55:
45:
43:
39:
35:
31:
27:
23:
19:
740:Model theory
699:
695:
682:
656:
652:
646:
638:
634:
629:
605:
599:
593:
578:
574:
570:
562:
556:
546:
542:
538:
534:
530:
526:
522:
518:
514:
510:
506:
502:
498:
494:
490:
486:
482:
478:
474:
470:
466:
462:
458:
454:
450:
446:
442:
436:
431:
429:
376:
372:
370:
362:
348:
314:
309:
305:
297:
293:
289:
281:
277:
273:
269:
265:
261:
257:
253:
249:
245:
239:
234:
230:
226:
222:
215:
211:
207:
203:
195:
193:
183:
175:
169:
164:
160:
156:
152:
148:
144:
142:
137:
133:
129:
125:
121:
113:
109:
105:
101:
97:
93:
89:
85:
81:
73:
69:
65:
61:
57:
53:
51:
21:
15:
501:then since
353:transseries
159:is also in
42:G. H. Hardy
22:Hardy field
18:mathematics
729:Categories
585:References
445:by saying
430:exists in
366:continuous
359:Properties
248:, denoted
124:by saying
100:such that
78:eventually
60:that map (
48:Definition
622:121021048
567:o-minimal
401:∞
398:→
332:∞
439:ordering
272:) where
200:subfield
190:Examples
716:6679449
675:1999639
641:, 1911
379:, then
714:
673:
620:
565:is an
712:S2CID
692:(PDF)
671:JSTOR
618:S2CID
198:is a
163:then
30:germs
26:field
24:is a
541:and
276:and
20:, a
704:doi
700:243
661:doi
657:280
610:doi
453:if
441:on
391:lim
244:on
225:in
202:of
194:If
143:If
120:on
92:is
88:in
32:of
16:In
731::
710:,
698:,
694:,
669:,
655:,
639:10
616:,
606:30
604:,
355:.
264:)/
186:.
44:.
706::
663::
612::
579:R
577:(
575:H
571:R
563:R
547:x
545:(
543:g
539:x
535:x
533:(
531:f
527:f
523:g
519:x
515:x
513:(
511:f
507:x
505:(
503:g
499:e
495:x
493:(
491:g
487:x
483:x
481:(
479:f
475:g
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467:g
463:f
459:f
455:g
451:g
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443:H
432:R
415:)
412:x
409:(
406:f
395:x
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373:f
335:)
329:,
326:x
323:(
310:x
308:(
306:R
298:Q
294:x
290:x
282:Q
278:Q
274:P
270:x
268:(
266:Q
262:x
260:(
258:P
254:x
252:(
250:R
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235:F
231:F
227:R
223:x
219:α
216:f
212:F
208:F
204:R
196:F
184:f
180:∞
176:f
172:∞
165:H
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104:(
102:f
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74:f
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66:R
62:u
58:f
54:H
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