Re: InverseFunction[]
- To: mathgroup at smc.vnet.net
- Subject: [mg42010] Re: InverseFunction[]
- From: Murray Eisenberg <murray at math.umass.edu>
- Date: Mon, 16 Jun 2003 03:57:51 -0400 (EDT)
- Organization: Mathematics & Statistics, Univ. of Mass./Amherst
- References: <bbt23g$nj3$1@smc.vnet.net> <bc1jh1$bjp$1@smc.vnet.net> <200306110749.DAA02507@smc.vnet.net> <bc7pet$6f0$1@smc.vnet.net> <paul-000944.15584413062003@news.uwa.edu.au>
- Reply-to: murray at math.umass.edu
- Sender: owner-wri-mathgroup at wolfram.com
Well, to take the example of the function you pose for comparison, Sin:
The section "Some Notes on Internal Implementation" in the Mathematica
Book's "Mathematica Reference Guide", says:
Exponential and trigonometric functions use Taylor series,
stable recursion by argument doubling, and functional relations.
But I'm guilty of not scrolling down far enough on that same page to
have discovered:
PolyLog uses Euler-Maclaurin summation, expansions in terms
of incomplete gamma functions and numerical quadrature.
That's the sort of thing I was looking for and somehow had previously
overlooked!
Paul Abbott wrote:
> In article <bc7pet$6f0$1 at smc.vnet.net>,
> Murray Eisenberg <murray at math.umass.edu> wrote:
>
>
>>What does "knows about" mean in your answer? Evidently Mathematica
>>knows a NAME (ProductLog) for the inverse function and can evaluate it
>>numerically.
>
>
> And it knows (some) algebraic properties of ProductLog. For example, it
> knows how to differentiate and integrate it.
>
>
>>But what is the definition of the function as Mathematica knows it?
>
>
> I do not understand your question. What is the definition of _any_
> function in Mathematica? E.g., can you tell me what is the definition of
> Sin (in Mathematica)? Further, InverseFunction[Sin] yields ArcSin and
> Mathematica "knows about" ArcSin in the sense that it can evaluate it
> numerically _and_ algebraically.
>
>
>>Reply to "REPLY TO" address and NOT to the "FROM" address!!
>>Otherwise I will never see your reply!!!!!!!!!!!!!!!!!!!!!!
>
>
> I don't understand this: According to the message headers I get, your
> "REPLY TO" and "FROM" addresses are identical?
>
> Cheers,
> Paul
>
>
>>Paul Abbott wrote:
>>
>>>In article <bc1jh1$bjp$1 at smc.vnet.net>,
>>> wself at msubillings.edu (Will Self) wrote:
>>>
>>>
>>>
>>>>You can easily write down a function which has an inverse, but the
>>>>inverse cannot be expressed in closed form, or in symbols that anyone
>>>>has invented. For example, f[x_]= x*E^x. This function is increasing
>>>>on (for example) the interval [1,3], therefore an inverse exists, at
>>>>least for 1 <= x <= 3. But (as far as I know) this inverse function
>>>>has no name.
>>>
>>>
>>>Your point is valid. However, try Solve[x*E^x == y, x] and you'll see
>>>Mathematica knows about the inverse function for your example.
>>>
>>>Cheers,
>>>Paul
>>>
>>
>>--
>>
>>Murray Eisenberg murray at math.umass.edu
>>Mathematics & Statistics Dept.
>>Lederle Graduate Research Tower phone 413 549-1020 (H)
>>University of Massachusetts 413 545-2859 (W)
>>710 North Pleasant Street fax 413 545-1801
>>Amherst, MA 01003-9305
>
>
--
Reply to "REPLY TO" address and NOT to the "FROM" address!!
Otherwise I will never see your reply!!!!!!!!!!!!!!!!!!!!!!
Murray Eisenberg murray at math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower phone 413 549-1020 (H)
University of Massachusetts 413 545-2859 (W)
710 North Pleasant Street fax 413 545-1801
Amherst, MA 01003-9305
- References:
- Re: InverseFunction[]
- From: Paul Abbott <paul@physics.uwa.edu.au>
- Re: InverseFunction[]