I want to plot power series with symbolic functions

Question

var('x')
f = sin
def P(n,x):
    return sum([(-1)^k*x^(2*k+1)/factorial(2*k+1) for k in range(n)]) 

sinplot = plot(f(x),(x,-2*pi,2*pi),color='red')

@interact
def _(n=(1..10)):
    seriesplot = plot(P(n,x),(x,-2*pi,2*pi),color='blue')
    html('$P(%s,x) = %s$'%(latex(n),latex(P(n,x))))
    show(sinplot+seriesplot,ymin=-4,ymax=4)

Notice that they have to define a Python function in order for this to work. I could not get this to work with a Sage callable function like P(n,x) = sum([...]) for the life of me. I tried lambdas, everything.

Now, likely either

I've already answered this question somewhere else on the Internet, or
It's not possible.

But I'd like confirmation of this. It's really annoying that one has to use a Python function to do this.

Answer 1

This code with a lambda function works. As you mentioned in the comment, it is still not what you want.

var('x')
n=var('n')
assume(n,'integer')
f = sin
P = lambda n,x:sum([(-1)^k*x^(2*k+1)/factorial(2*k+1) for k in range(n)]) 

sinplot = plot(f(x),(x,-2*pi,2*pi),color='red')

@interact
def _(n=(1..10)):
    seriesplot = plot(P(n,x),(x,-2*pi,2*pi),color='blue')
    html('$P(%s,x) = %s$'%(latex(n),latex(P(n,x))))
    show(sinplot+seriesplot,ymin=-4,ymax=4)

Answer 2

What's about assigning a python function to a sage callable function?

var('x')
f = sin
def P(n,x):
    return sum([(-1)^k*x^(2*k+1)/factorial(2*k+1) for k in range(n)]) 

print 'type(P)',type(P)

sinplot = plot(f(x),(x,-2*pi,2*pi),color='red')

@interact
def _(n=(1..10)):
    p(x) = P(n,x)
    print 'type(p)',type(p)
    print  'p.parent()', p.parent()
    seriesplot = plot(p,(-2*pi,2*pi),color='blue')
    html('$P(%s,x) = %s$'%(latex(n),latex(p(x))))
    show(sinplot+seriesplot,ymin=-4,ymax=4)

Hmm, good point. Of course, then I might as well do what the original worksheet I found did anyway... but still worth having as an option.
kcrisman (Apr 30 '12)

Answer 3

The best I could come up with was

var('x')
f = sin
# note the different, symbolic convention for calling sum,
# *not* the standard python give-it-a-list way.
P(n,x)=sum((-1)^k*x^(2*k+1)/factorial(2*k+1), k, 0, n-1)

sinplot = plot(f(x),(x,-2*pi,2*pi),color='red')

@interact
def _(n=(1..10)):
    seriesplot = plot(P(n,x),(x,-2*pi,2*pi),color='blue')
    html('$P(%s,x) = %s$'%(latex(n),latex(P(n,x))))
    show(sinplot+seriesplot,ymin=-4,ymax=4)

However, I was still getting errors. It's odd that this doesn't work:

sage: var('i,x,k,n')
(i, x, k, n)
sage: a=(-1)^k*x^(2*k+1)/factorial(2*k+1)
sage: b=a.sum(k,0,n-1)
sage: b.subs(n=3,x=5)
sum((-1)^k*5^(2*k + 1)/factorial(2*k + 1), k, 0, 2)
sage: (b.subs(n=3,x=5)).n()
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
/Users/grout/<ipython-input-14-3dd82cd01b78> in <module>()
----> 1 (b.subs(n=3,x=5)).n(
      2 )

/Users/grout/sage-trees/sage-5.0.beta12/local/lib/python2.7/site-packages/sage/symbolic/expression.so in sage.symbolic.expression.Expression._numerical_approx (sage/symbolic/expression.cpp:18440)()

TypeError: cannot evaluate symbolic expression numerically

I was hoping to avoid using the symbolic sum like this, because it will likely invoke Maxima, but I guess this is more or less what I was looking for. But you are right that I can't get it to work.
kcrisman (Apr 25 '12)

I want to plot power series with symbolic functions

3 Answers:

Your answer

Question tools

Tags:

Stats:

Related questions