Huh...
cos(pi/u^2/2), first expression of your problem, has not, indeed, an
explicit solution that sage is able to find.
but, on your following attempts, you reach for the integral of
cos(pi*x^2/2), a horse of a different color (which is the one racing in
Wikipedia pages on "Euler spiral") :
sage: var("t")
t
sage: foo(x)=integrate(cos(pi*t^2/2),t,0,x)
sage: foo
x |--> -1/2*sqrt(1/2)*((I - 1)*(1/4)^(1/4)*pi*erf((1/2*I +
1/2)*sqrt(2)*(1/4)^(1/4)
Since
sage: bool(foo(x).imag()==0)
True
and, by the way
sage: bar(x)=integrate(sin(pi*t^2/2),t,0,x)
sage: bool(bar(x).imag()==0)
True
all hopes are open.
But the problem is elsewhere :
sage: parametric_plot([foo(x),bar(x)],[x,-5,5],figsize=4)
[ tons of error messages ... ]
TypeError: unable to coerce to a real number
Hmm. conversion problem ?
def Foo(x): return foo(x).N()
sage: def Bar(x): return bar(x).N()
sage: parametric_plot([Foo(x),Bar(x)],[x,-5,5],figsize=4)
Same problem... only different :
TypeError: Cannot evaluate symbolic expression to a numeric value.
Aha !
parametric_plot([lambda x:foo(x).N(), lambda
x:bar(x).N()],[x,-5,5],figsize=4)
Yet anopther problem :
TypeError: Unable to convert -0.498688874572196 - 4.99775951821661e-17*I to
float; use abs() or real_part() as desired
Hmmm. This time, it's a precison problem. The *numerical* computation of
the values is problematic : the imaginary part, which should be zero, is
not numerically null erf evaluation is suspect...). Let's try :
parametric_plot([lambda x:foo(x).N().real(), lambda
x:bar(x).N().real()],[x,-5,5],figsize=4)
and .. lo ! No error message, and Sage displays the enclosed figure.
However, this analysis is *NOT* satisfying : trying :
sage: def Foo(x): return foo(x).N().real()
sage: def Bar(x): return bar(x).N().real()
sage: parametric_plot([Foo(x),Bar(x)],[x,-5,5],figsize=4)
does *NOT* work (yet another ton of error messages ending with " Cannot
evaluate symbolic expression to a numeric value"). I do not understand why.
Better minds than mine are required.
My guess is that, since is forced to use erf on imaginary values (a kind of
casuus irreductibilis ?), there is no expression of the *real* value f this
integral of a *real* function in terms of "elementary" that does not
involve imaginary quantities... There might well be some expression of
"special functions" that avoid this, but those "special functions" have
usually been created to avoid this kind of problems. Even trig functions
can be seen as "special functions" curtaining e^i*x, but their geometric
interpretation predates largely (by about two millenia) their analytic
expression ; this is why they are thought of as "elementary".
HTH,
--
Emmanuel Charpentier
Le mercredi 10 septembre 2014 00:17:14 UTC+2, Jotace a écrit :
>
> Hi all,
>
> I want (my students) to plot Cornu's spiral, givent in parametric form by
>
> x(t) = integral cos(pi/u^2/2), u going from 0 to t , and y(t) defined
> analogously using the sine function. The integral connot be evaluated
> symbolically, I guess.
>
> The first attempt would be
>
> parametric_plot([integrate(cos(pi*u^2/2),u,0,t),integrate(sin(pi*u^2/2),u,0,t)],(t,-pi,pi))
> which failw (coercion)
>
> The second attempt would be:
>
> parametric_plot([integral_numerical(cos(pi*u^2/2),0,t),integral_numerical(sin(pi*u^2/2),0,t)],(t,-pi,pi))
> which also fails.
>
> I finally did:
> def x(t):
> return integral_numerical(cos(pi*u^2/2),0,t)[0]
>
> def y(t):
> return integral_numerical(sin(pi*u^2/2),0,t)[0]
>
> Points = [(x(t),y(t)) for t in sxrange(-pi,pi,2*pi/200)]
> line(Points).show(figsize=[5, 5],aspect_ratio=1)
>
> This works, but it looks highly inelegant. Also, i cannot expect my
> students to come up with something like this in a first year undergrad
> course.
>
> Is there a way to fix one of the first two options?
>
> Regards,
> JC
>
>
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