This was incredibly useful. Thanks. I've added those graphs plus others to the
latest draft of the hopefully-soon-to-be-latexed version of
Granville's calculus:
http://sage.math.washington.edu/home/wdj/teaching/granville-calculus/
(from the pdf, click on for example section 5.28 in the table of
contents).
On Dec 25, 2007 11:40 AM, William Stein <[EMAIL PROTECTED]> wrote:
>
> On Dec 25, 2007 9:18 AM, David Joyner <[EMAIL PROTECTED]> wrote:
> >
> > Hi:
> >
> > (a) I'm not sure if this is a bug or something missing, but it seems
> > to me it should be easy to plot y=arccsc(x) in SAGE, since it
> > is a basic function of trigonometry and calculus. Two problems:
> > (1) it seems arccsc is not defined,
>
> It is acsc, just like asin, etc. This works fine:
>
> sage: show(plot(acsc, 1,2))
>
> > (2) after defining it, it does not seem easy to plot it:
> >
> > sage: acsc = lambda x: CDF(x,0).arccsc()
> > sage: acsc(1.1)
> > 1.14109666064
> > sage: acsc(1.9)
> > 0.554261834452
> > sage: P = plot(RR(acsc(x)),1,2)
>
> RR(acsc(x)) makes no sense; you're pluggin a symbolic variable into
> a lambda function, then trying to convert the result to a real field element.
> You meant to do
>
> sage: acsc = lambda x: float(abs(CDF(x,0).arccsc()))
> sage: show(plot(acsc, 1,2))
>
> Sorry sage is so hard to use! What can we learn from the above?
> The main problem is acsc versus arccsc, which caused confusion.
> Should we change the names of the "arc" functions to arc* instead of a*?
>
> Maple: uses arcsin:
> sage: maple.eval('arcsin(1)')
> '1/2*Pi'
> sage: maple.eval('asin(1)')
> 'asin(1)'
>
> Mathematica: uses ArcSin:
> sage: mathematica.eval('ASin[1]')
> ASin[1]
> sage: mathematica.eval('ArcSin[1]')
>
> Pi
> --
> 2
>
> Maxima: Uses asin (which is why we currently do):
> sage: maxima.eval('arcsin(1)')
> 'arcsin(1)'
> sage: maxima.eval('asin(1)')
> '%pi/2'
>
>
> If nobody strongly objects in a day or two, I'll open a trac ticket
> to change a*'s to arc*'s. Better now than later. And if something
> like this is confusing David Joyner, then it's to be taken seriously.
>
> > ---------------------------------------------------------------------------
> > <type 'exceptions.TypeError'> Traceback (most recent call last)
> >
> > /home/wdj/sagestuff/sage-2.8.7.rc1/<ipython console> in <module>()
> >
> > /home/wdj/sagestuff/sage-2.8.7.rc1/<ipython console> in <lambda>(x)
> >
> > /home/wdj/sagestuff/sage-2.8.7.rc1/complex_double.pyx in
> > sage.rings.complex_double.ComplexDoubleField_class.__call__()
> >
> > /home/wdj/sagestuff/sage-2.8.7.rc1/complex_double.pyx in
> > sage.rings.complex_double.ComplexDoubleElement.__init__()
> >
> > <type 'exceptions.TypeError'>: a float is required
> >
> > (b) In fact, what I'd like to do is plot in SAGE what calculus teachers draw
> > frequently on the board: not just one branch of arccsc but rather
> > several of them: ie, the plot of y=csc(x) over say -2\pi to 2*\pi,
> > flipped about the 45^o line. Is this easy to do?
>
> This will do it. I hope it isn't too ugly:
>
> sage: v = [(csc(x),x) for x in srange(-2*float(pi),2*float(pi),0.1) if x]
> sage: show(line(v), xmin=-20, xmax=20)
>
> The tricks above:
> (1) use float(pi) so the iteration through the range of inputs is very fast
> (2) don't evaluate csc at 0.
> (3) use a line and flip the order of the points in the graph.
> (4) use xmin, xmax, since otherwise one large value will through
> off the whole graph.
>
> William
>
> >
> > - David Joyner
> >
> > Can anyone see what I'm doing wrong here?
> >
> > - David Joyner
> >
> > >
> >
>
>
>
> --
> William Stein
> Associate Professor of Mathematics
> University of Washington
> http://wstein.org
>
> >
>
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