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forum.math.uoa.gr • Προβολή θέματος - Ρητή συνάρτηση

forum.math.uoa.gr

Forum του Τμήματος Μαθηματικών
Ημερομηνία 19 Νοέμ 2017, 06:35

Όλοι οι χρόνοι είναι UTC + 2 ώρες [ DST ]




Δημιουργία νέου θέματος Απάντηση στο θέμα  [ 4 δημοσιεύσεις ] 
Συγγραφέας Μήνυμα
 Θέμα δημοσίευσης: Ρητή συνάρτηση
ΔημοσίευσηΔημοσιεύτηκε: 23 Οκτ 2016, 11:57 
Χωρίς σύνδεση

Εγγραφή: 10 Μαρ 2016, 14:08
Δημοσ.: 33
Να αποδείξετε ότι η συνάρτηση [unparseable or potentially dangerous latex formula] δεν είναι ρητή.


Κορυφή
 Προφίλ  
 
 Θέμα δημοσίευσης: Re: Ρητή συνάρτηση
ΔημοσίευσηΔημοσιεύτηκε: 23 Οκτ 2016, 12:58 
Χωρίς σύνδεση

Εγγραφή: 11 Δεκ 2014, 19:45
Δημοσ.: 49
Η [unparseable or potentially dangerous latex formula] για x>0 δεν μπορεί σίγουρα να είναι πολυώνυμο!

Αν ήταν, ας πούμε ότι είχε την μορφή [unparseable or potentially dangerous latex formula], τότε από την ιδιότητα του λογαρίθμου [unparseable or potentially dangerous latex formula] θα είχαμε

[unparseable or potentially dangerous latex formula]

που θα σήμαινε ότι [unparseable or potentially dangerous latex formula] για κάθε x>0, πράγμα αδύνατο. Άρα, ο λογάριθμος δεν είναι πολυώνυμο.

_________________
"Problems worthy of attack prove their worth by fighting back" -Paul Erdős


Κορυφή
 Προφίλ  
 
 Θέμα δημοσίευσης: Re: Ρητή συνάρτηση
ΔημοσίευσηΔημοσιεύτηκε: 23 Οκτ 2016, 15:32 
Χωρίς σύνδεση

Εγγραφή: 10 Μαρ 2016, 14:08
Δημοσ.: 33
Βασίλη, ωραίο αυτό που έγραψες.

Μπορούσαμε και ως εξής :

Αν [unparseable or potentially dangerous latex formula] , όπου \displaystyle{f} είναι πολυωνυμική συνάρτηση

με βαθμό μεγαλύτερο ή ίσο της μονάδας, τότε με

παραγώγους παίρνουμε [unparseable or potentially dangerous latex formula] και

συγκρίνοντας βαθμούς, καταλήγουμε σε άτοπο.

Παραμένει ωστόσο το αρχικό ερώτημα : Υπάρχουν πολυωνυμικές συναρτήσεις [unparseable or potentially dangerous latex formula] με [unparseable or potentially dangerous latex formula]

παντού, έτσι ώστε [unparseable or potentially dangerous latex formula] ;


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 Θέμα δημοσίευσης: Re: Ρητή συνάρτηση
ΔημοσίευσηΔημοσιεύτηκε: 23 Οκτ 2016, 21:23 
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Regular Forumer

Εγγραφή: 05 Φεβ 2008, 03:03
Δημοσ.: 420
Μπορούμε να μελετήσουμε τη συμπεριφορά στο άπειρο: έστω ότι [unparseable or potentially dangerous latex formula], όπου τα f, g είναι πολυώνυμα. Αν [unparseable or potentially dangerous latex formula], τότε το όριο [unparseable or potentially dangerous latex formula] είναι πεπερασμένο, το οποίο είναι άτοπο. Επομένως [unparseable or potentially dangerous latex formula]. Αν [unparseable or potentially dangerous latex formula], τότε [unparseable or potentially dangerous latex formula], το οποίο είναι άτοπο. Αν τώρα [unparseable or potentially dangerous latex formula], τότε το όριο [unparseable or potentially dangerous latex formula] είναι πεπερασμένο και μη μηδενικό, το οποίο είναι ξανά άτοπο.

_________________
\emptyset\not=\{\emptyset\}


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