Sage:
def h(x): return 1/((x + 1)^(1/3) + 1)
hint = integral(h(x), (x, -6, -2))
hint.n()
Exception ValueError: ValueError('negative number to a fractional power not real',) in 'sage.gsl.integration.c_ff'
Maple:
int(1/((x+1)^(1/3) + 1), x = -6..-2):
evalf(%);
1.562667835 - 1.105160054 I
If you consider that you work on the real field, Maple answers a complex number, which is just weird. Sage's Symbolic Ring does the same artificial choice since:
sage: h(-3.3)
0.408623955144076 - 0.281398327127290*I
So, i agree that there is a kind of bug here, see also this question.
If you consider that you work on the complex field, your function is not well defined since there are 3 different choices for the cubical root, leading to different integrals.
If you want the cube root to be the real one, your integral will go to -Infinity as x approaches -2. And Sage can see this. Let us specify the real cube root as follows for the negative numbers, so that you will get the real cube root (not a complex one):
sage: cuberoot(x) = -((abs(x))^(1/3))
Then,
sage: h(x) = 1/(cuberoot((x + 1)) + 1)
sage: integral_numerical(h, -6, -2)
(-inf, nan)
Not sure if Maple's answer is weird. Mathematica gives the same answer: http://www.wolframalpha.com/input/?i=int%281%2F%28%28x%2B1%29%5E%281%2F3%29+%2B+1%29%2C+x+%3D+-6..-2%29+
tmonteil, thanks for your comments! (1) Working on the complex field I assume that the principal root is chosen by default; Maple and Mathematica seem to do this also. Do I have to declare this in Sage explicitly? (2) When choosing the real-valued root the Cauchy principal value is -oo, as you say. How do I assure that Sage uses the real-valued root in the above integral? (3) Now, what is the meaning of the Maple and Mathematica solutions? I think they compute an indefinite integral (antiderivative) and than take the difference at the end points. This leads to the value 1.562667835 - 1.105160054*I which both programs give. Let's try to do this with Sage: def h(x): return 1/((x + 1)^(1/3) + 1) hint = integral(h(x), x) (hint(x=-6) - hint(x=-2)).n() sage: 2.76637207714311 - 1.394212302715
For your question (2), you can use the function `cuberoot()` i defined in my answer. For your question (1), you should know that the other cube roots come from the real one by multiplying by `j` or `j^2` as explained in [this answer](http://ask.sagemath.org/question/2518/get-variants-of-complex-cube-root?answer=3451#3451). Concerning your question (3), i have no idea on how does Maple and Mathematica work here (i do not have them). Concerning the choice of a "principal root" in general, you have to know that there are some issues on chosing one root globally, if a curve winds around zero, chosing the "principal root" let you jump from one branch of the cube root to another one on the way... This may explain the different results, and why i do not like how the `Symbolic Ring` handles this.
Asked: 2013-09-02 09:20:49 +0100
Seen: 492 times
Last updated: Sep 02 '13
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