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Re: Re: New Analytical Functions - Mathematica Verified
*To*: mathgroup at smc.vnet.net
*Subject*: [mg66871] Re: [mg66853] Re: New Analytical Functions - Mathematica Verified
*From*: Daniel Lichtblau <danl at wolfram.com>
*Date*: Fri, 2 Jun 2006 04:08:47 -0400 (EDT)
*References*: <200605280104.VAA23436@smc.vnet.net><e5jss1$e73$1@smc.vnet.net> <200606011055.GAA20733@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
Mohamed Al-Dabbagh wrote:
> Daniel Lichtblau wrote:
>
>> [...]
>> I'll point out some things with regard to your reformulation of a
>> sawtooth in terms of elementary functions. It is not hard to define the
>> derivative of FractionalPart as 1, and this is a.e. correct, that is,
>> correct except at jumps, where the derivative, strictly speaking, is not
>> defined. So a question to ask is what do you want the derivative to do
>> at such points? Your gPaper, shown below, will give an Indeterminate at
>> x = 12 (which one might regard as a good thing). So if that is the
>> behavior you want, well and good.
>>
>> f[x_] := x^5
>>
>> Fr[x_] := 1/2 - ArcTan[Cot[Pi*x]]/Pi
>>
>> gPaper[x_] := Fr[f[x]]
>>
>> If we simply do
>>
>> Derivative[1][FractionalPart][x_] := 1 + DiracDelta[x-IntegerPart[x]]
>> Derivative[n_Integer /; n>1][FractionalPart][x_] := 0
>>
>
>
> Will the above two definitions change the behaviour of FractionalPart
> function?
They will change the behavior of its derivatives. Actually I think I did
not want to use the DiracDelta component in that particular definition
(or else to use further derivatives thereof in the definition for higher
derivatives of FractionalPart).
>> gMath[x_] := FractionalPart[f[x]]
>>
>> we'll get an exact result for gMath''[61/5] that agrees with yours
>> numerically but is in vastly simpler form. We'll also get a rational
>> result for gMath''[12] that might or might not be what you'd want.
>
>
>
> Is that really simpler??!!
Did you try it?
In[11]:= InputForm[gPaper''[61/5]]
Out[11]//InputForm=
(907924*Csc[(574*Pi)/3125]^2)/(25*(1 + Cot[(574*Pi)/3125]^2)) +
(383414625994562*Pi*Cot[(574*Pi)/3125]*Csc[(574*Pi)/3125]^2)/
(15625*(1 + Cot[(574*Pi)/3125]^2)) -
(383414625994562*Pi*Cot[(574*Pi)/3125]*Csc[(574*Pi)/3125]^4)/
(15625*(1 + Cot[(574*Pi)/3125]^2)^2)
In[12]:= InputForm[gMath''[61/5]]
Out[12]//InputForm= 907924/25
> Moreover, you rather used an exact rational
> form and you didn't use the approximate 12.2`400 (12.2 correct to 400
> digits) to see what my experiment will be?
It occurs to me that by taking the code I wrote anyone interested can
quite easily check the approximate case. Anyway, here they are. You will
notice gMath'' gets around 14-15 more digits than gPaper''.
In[9]:= InputForm[gMath''[N[61/5,400]]]
Out[9]//InputForm=
36316.95999999999999999999999999999999999999999999999999999999999999999999999\
99999999999999999999999999999999999999999999999999999999999999999999999999999\
99999999999999999999999999999999999999999999999999999999999999999999999999999\
99999999999999999999999999999999999999999999999999999999999999999999999999999\
99999999999999999999999999999999999999999999999999999999999999999999999999999\
99999999999999999999999999999998868`399.52287874528037
In[10]:= InputForm[gPaper''[N[61/5,400]]]
Out[10]//InputForm=
36316.96000000000000000000000000000000000000000000000000000000000000000000000\
00000000000000000000000000000000000000000000000000000000000000000000000000000\
00000000000000000000000000000000000000000000000000000000000000000000000000000\
00000000000000000000000000000000000000000000000000000000000000000000000000000\
00000000000000000000000000000000000000000000000000000000000000000000000000000\
00000000000002`385.50064348271536
> Again, you should use my
> definition for FractionalPart function to get precise results.
>
> Now why you used [x-IntegerPart[x]]? Why didn't you simply use
> FractionalPart?
FractionalPart could have been used in the delta function, now that you
mention it.
> I will do experiments again, and will post results.. I
> hope moderator wont take long to publish them.
Messages posted to the group usually appear on the next day.
> Mohamed Al-Dabbagh
Daniel Lichtblau
Wolfram Research
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