Stuttgart, den 15. Februar 2005
Hallo Herr Kuska,
danke fuer Ihre ausfuehrlichen Infos zum Thema Distributionen,
dargestellt an DiracDelta[] !
Meine Fragen entstanden aus der Beschaeftigung mit
ET-Uebertragungsfunktionen wie im angehaengten Notebook.
[ mail_05/DiracDeltaFun050215.nb ]
Da gibt es eben Ausdruecke wie
DiracDelta[ 2 Pi s + Pi ]
und aehnlich.
Wenn man versucht, das mit FullSimplify zu vereinfachen, kommt man
nicht ohne weiteres weiter. Also versuchte ich, das ueber einen
Booleschen Test ( nennen wir es hier einmal so ), abzuklaeren.
Warum sollte FullSimplify nicht auch Distributionen richtig
behandeln koennen ?
So entstand diese Anfrage, die zu einigen interessanten Antworten
fuehrte.
Dafuer noch einmal vielen Dank in diese Runde !
Mit freundlichen Gruessen,
Gunter Woysch
------------- Begin Forwarded Message -------------
From: "Jens-Peer Kuska" <kuska@XXXXXXX.de>
To: "Thomas Hahn" <hahn@XXXXXXX.de>, "Andre El-Ama"
<Andre@XXXXXXX.de>
Cc: "DMUG" <demug@XXXXXXX.ch>
Subject: Re: Re(Woysch): Frage zur Aequivalenz von modifizierten
DiracDelta-Funktionen
Date: Tue, 15 Feb 2005 12:38:24 +0100
Hallo,
das ist ja HAARSTRÄUBEND !
DiracDelta[] ist eine *Distribution*, das heißt sie ist nur in
Ausdrücken
Integrate[f[x]*DiracDelta[x],{x,-Infinity,Infinity}]
erlaubt und sinnvoll. Und auch nicht für beliebige Funktionen f[]
sondern nur für eine bestimmte Klasse z.B den Raum der
(quadratisch) integrablen Funktionen.
Eine nackte DiracDelta[] Funktion kann es nicht geben. Es gibt
immer nur Integrale in denen eine solche Funktion steht. Um genau
zu sein, ist DiracDelta[] diejenige Distribution, die
Integrate[f[x]*DiracDelta[x],{x,-Infinity,Infinity}] -> f[0]
zuordnet. Es ist also höchst sinnlos darüber zu spekulieren
ob DiracDelta[s]==DiracDelta[5 s] ist oder nicht, denn es ist nur
ein Ausdruck
Integrate[f[x]*DiracDelta[x],{x,-Infinity,Infinity}]==
Integrate[f[x]*DiracDelta[ 5 x],{x,-Infinity,Infinity}]
überhaupt definiert, und der ist natürlich
f[0]==f[0]/s
ob das True oder False ist kann man so nicht sagen, TrueQ[] sollte
aber auf jeden Fall False ergeben ...
Natürlich ist es schlimm, wenn so etwas überhaupt ausgewertet
wird, denn der Ausdruck DiracDelta[s]==DiracDelta[5 s] ist
sinnlos, und das FullSimplify[] das auch noch zu True vereinfacht
ist bedauerlich, weil eigentlich Indeterminate heraus kommen
sollte.
Aber vermutlich ist es kein Bug, wenn bei der Eingabe von Unsinn
auch Unsinn rauskommt ...
Es ist eine beliebte Ungenauigkeit einfach das Integral und die
Funktion f[] wegzulassen und die Daumen zu drücken, das alle
Integrale konvergieren, das hat aber nix mit Mathematik zu tun,
sonder mit (Schreib-)Faulheit, genauso, wie man in
Differentialgleichungen gern die Funktionsargumente weg läßt.
Aus einer gewissen Laxheit/Faulheit heraus aber zu schlußfolgern,
das
DiracDelta[s]==DiracDelta[5 s]
überhaupt etwas sinnvolles ergibt ist recht vermessen. Selbst
mit der üblichen "auch ganz doofe Studenten sollen das
verstehen"-Erklärung:
"Eine Diracsche Delta-Funktion ist überall null außer dort wo das
Argument verschwindet, dort ist sie unendlich." ist klar das man
"Unendlich" und "Unendlich/5" wohl eher nicht vergleichen kann.
Jedenfalls hat es die Anzahl der Postings in der DMUG dramatisch
erhöht
Gruß
Jens
------------- End Forwarded Message -------------
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