arcsin vs. Arcsin??
riverman
and 'arcsin'?
He claims that 'Arcsin' (equivalent to 'Sin ^-1' ) is the function with
the restricted range of -pi/2 -> pi/2, while 'arcsin' (equivalent to
'sin^-1' ) has no restrictions.
None of my reference texts show this distinction, but he asserts that it
is so, merely falling out of style.
Which is is? Anyone know?
--
riverman
.........................
I think, therefore I thwim;
Carpe ropum.
rbp #2
Jake Wildstrom
>He claims that 'Arcsin' (equivalent to 'Sin ^-1' ) is the function with
>the restricted range of -pi/2 -> pi/2, while 'arcsin' (equivalent to
>'sin^-1' ) has no restrictions.
This is true. A standard procedure for multivalued functions is to use
a lowercased version to indicate the multivalued variant and an uppercase
version to indicate the 'principal branch', which is usually defined along
with the function. Different functions have different principal branches. Here
are a few straightforward ones for result in the form x + iy:
Arcsin(z): -pi/2 <= x <= pi/2
Arccos(z): 0 <= x <= pi
Arctan(z): -pi/2 < x < pi/2
Log(z): 0 <= y < 2pi
>None of my reference texts show this distinction, but he asserts that it
>is so, merely falling out of style.
Check a book on complex analysis, where distinctions between branches becomes
important. I know that Brown & Churchhill and Alfohrs both make this
distinction.
+--First Church of Briantology--Order of the Holy Quaternion--+
| A mathematician is a device for turning coffee into |
| theorems. -Paul Erdos |
+-------------------------------------------------------------+
| Jake Wildstrom |
+-------------------------------------------------------------+
david_w_...@my-deja.com
wil...@mit.edu (Jake Wildstrom) wrote:
> In article <38906B6C...@the.machine>, riverman
<ra...@the.machine> wrote:
> >He claims that 'Arcsin' (equivalent to 'Sin ^-1' ) is the function
with
> >the restricted range of -pi/2 -> pi/2, while 'arcsin' (equivalent to
> >'sin^-1' ) has no restrictions.
>
> This is true. A standard procedure for multivalued functions is to use
> a lowercased version to indicate the multivalued variant and an
uppercase
> version to indicate the 'principal branch', which is usually defined
along
> with the function.
I wish that what you have said were true! I think the convention
you have described is very reasonable. But alas, there is no consistency
in usage. Quite a few authors, I hate to say, distinguish between a
multivalued relation and the corresponding "principal value" function in
precisely the *opposite* manner, that is, lower case for the function
and upper case for the multivalued relation. For one example, see An
Atlas of Functions, Spanier and Oldham, p. 340. Confusing, right?
Perhaps even worse nowadays, some references have obliterated any
distinction whatsoever between the multivalued relation and the
function. For example, the curious statement
sin^(-1)z = (-1)^k sin^(-1)z + k pi appears in the newest edition of the
CRC Standard Math. Tables and Formulae, p. 466. Previous editions of
that work would have given something like
arcsin z = (-1)^k Arcsin z + k pi, which is quite proper in my opinion.
(By the way, I have already mentioned to the current editor of the CRC
SMTF that some distinction should be made between the multivalued
relation and the function, but I never got a response from him
concerning the matter.)
Cheers,
David C.
> Different functions have different principal
branches. Here
> are a few straightforward ones for result in the form x + iy:
>
> Arcsin(z): -pi/2 <= x <= pi/2
> Arccos(z): 0 <= x <= pi
> Arctan(z): -pi/2 < x < pi/2
> Log(z): 0 <= y < 2pi
>
> >None of my reference texts show this distinction, but he asserts that
it
> >is so, merely falling out of style.
>
> Check a book on complex analysis, where distinctions between branches
becomes
> important. I know that Brown & Churchhill and Alfohrs both make this
> distinction.
Sent via Deja.com http://www.deja.com/
Before you buy.
G. A. Edgar
wrote:
> A colleague sprung this on me: What's the difference between 'Arcsin'
> and 'arcsin'?
>
> He claims that 'Arcsin' (equivalent to 'Sin ^-1' ) is the function with
> the restricted range of -pi/2 -> pi/2, while 'arcsin' (equivalent to
> 'sin^-1' ) has no restrictions.
>
> None of my reference texts show this distinction, but he asserts that it
> is so, merely falling out of style.
>
> Which is is? Anyone know?
>
I would say that research mathematicians have normally not made
such distinctions, but calculus (and pre-calculus) textbooks have
sometimes done this.
--
Gerald A. Edgar ed...@math.ohio-state.edu
Virgil
wrote:
>A colleague sprung this on me: What's the difference between 'Arcsin'
>and 'arcsin'?
>
>He claims that 'Arcsin' (equivalent to 'Sin ^-1' ) is the function with
>the restricted range of -pi/2 -> pi/2, while 'arcsin' (equivalent to
>'sin^-1' ) has no restrictions.
>
>None of my reference texts show this distinction, but he asserts that it
>is so, merely falling out of style.
>
>Which is is? Anyone know?
In some contexts, this distinction is useful, and is occasionally used.
When used, it should be stated clearly, as there is no universal
agreement favoring your colleague's interpretation, and there are many
contexts in which 'arcsin' is used to represent the function.
In contexts where you do not need both interpretations, you can use
either to mean whichever is needed.
--
Virgil
vm...@frii.com
John Prussing
>In article <38906B6C...@the.machine>, riverman <ra...@the.machine>
>wrote:
>> A colleague sprung this on me: What's the difference between 'Arcsin'
>> and 'arcsin'?
>>
>> He claims that 'Arcsin' (equivalent to 'Sin ^-1' ) is the function with
>> the restricted range of -pi/2 -> pi/2, while 'arcsin' (equivalent to
>> 'sin^-1' ) has no restrictions.
>>
>> None of my reference texts show this distinction, but he asserts that it
>> is so, merely falling out of style.
>>
>> Which is is? Anyone know?
>>
>I would say that research mathematicians have normally not made
>such distinctions, but calculus (and pre-calculus) textbooks have
>sometimes done this.
>--
>Gerald A. Edgar ed...@math.ohio-state.edu
I typically give engineering students (especially freshmen) a problem
(pointing an antenna or whatever) for which the solution is a value of
an inverse trig function that is not the principal value. They have to
face the enlightening prospect that the answer their calculator gives them
is wrong and they have to use their brains.
--
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
John E. Prussing
Dept. of Aeronautical & Astronautical Engineering
University of Illinois at Urbana-Champaign
http://www.uiuc.edu/~prussing
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
David W. Cantrell
writes:
>In article <38906B6C...@the.machine>, riverman <ra...@the.machine>
>wrote:
>
>>A colleague sprung this on me: What's the difference between 'Arcsin'
>>and 'arcsin'?
>>
>>He claims that 'Arcsin' (equivalent to 'Sin ^-1' ) is the function with
>>the restricted range of -pi/2 -> pi/2, while 'arcsin' (equivalent to
>>'sin^-1' ) has no restrictions.
>>
>>None of my reference texts show this distinction, but he asserts that it
>>is so, merely falling out of style.
>>
>>Which is is? Anyone know?
>
>In some contexts, this distinction is useful, and is occasionally used.
>When used, it should be stated clearly, as there is no universal
>agreement favoring your colleague's interpretation, and there are many
>contexts in which 'arcsin' is used to represent the function.
Yes, but surely such notational ambiguity is not good. Why can't there simply
be some "universal agreement" to distinguish between a multivalued relation and
a corresponding principal-valued function? Although I would prefer that the
relation be lowercase and the function uppercase, I would also be happy to
accept it vice versa if that were chosen instead. But there should be
*some*single*standard*. Of course (as I have pointed out previously in
sci.math), this is hardly the only situation where there is a regrettable lack
of standardization in mathematics! What's the problem? Why don't we
mathematicians standardize?
>In contexts where you do not need both interpretations, you can use
>either to mean whichever is needed.
Would a statement such as
arcsin z = (-1)^k arcsin z + k pi
be acceptable to you? Seeing such a statement, I would assume that the
principal-valued function was intended on the right. Using my preferred
notation, I would have written instead that
arcsin z = (-1)^k Arcsin z + k pi
Surely this is a context which calls for "both interpretations", each with its
distinct notation.
Cheers,
David C.