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It is getting too many follow-up questions that deserve their own discussions.

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I agree with John. Mathematics is generally wider and deeper than the problems we present to it. In physics, it is amazing to me how many times a model has been stretched, and the unexpected output actually found in nature, Like Dirac finding the positron on paper.

But for many problems, we just have to be aware of what it's saying. When Mathematica delivers imaginary parts so small compared to the magnitude of the result that Chop clears them, I generally ascribe them to numerical issues. On the other hand, if you're solving for leg "b" of a right triangle, given the hypotenuse "h" and the other leg "a", it's very possible to ask for Sqrt(h^2-a^2) with a > h. And when Mathematica gives you a complex number for the length of leg "b", it's telling you something important.

Kind regards, David

I am sure my reply is off topic, but I have puzzled over a similar issue of reals and imaginary solutions.

In one problem, I have a messy integral whose integrand has no poles and is regular. Yet when I perform a numerical integration over some limits it will often throw off an imaginary solution. I have tried to suppress this behavior, because I don't quite know what to do with the imaginary part of the solutions.

In the past, some respondents who answered my pleas for help said that if the imaginary part is small, they ignore it! Whoa. In some of my limits of integration, the imaginary part is as big as or larger than the real part. I am not familiar with Surd and will look into that, but otherwise, is this behavior in these integrals just user error on my part?

I am not quite a newby but lack depth in complex algebra and mathematical analysis.

What you say is true, to a point. Theoretical physicists often do let the math do the physics, which is a terrible mistake. The whole high energy physics domain is replete with various predictions built on mathematical symmetries and manipulations. Hence the predictions. It only took 50 years to find a Higgs to match one such theory. I personally believe the Higgs is bogus. What good is a theory about mass-energy when you cannot predict either in the model?

When I set up a physical problem, I build my understanding of the dynamics into that problem. When Mathematica gives me back results that do not match my expectations, then I have to make a choice based on my understanding of the dynamics or I have to see if perhaps it is telling me something that is not well understood. Even in taking data, many old timers simply culled the data to collect that which supported their theories or models. Occasionally, they missed a gem...like the lambda point in low temperature physics. That "pole" was real.

Enough said. I think I am zeroing in on what may be real and what may be idiosyncratic in my models. I am an experimentalist, and my models meet the needs of setting up measurement procedures and in analyzing the resulting data and determining what I measured and what it means. I have a lot more confidence in my data when I have confidence in my mathematics, and Mathematica is so powerful that I am loath to a priori cull the model's solution sets without being sure I have not missed my lambda point. On the other hand, I am not keen on predicting another Higgs-like phenomenon that can only be stumbled upon after several generations have passed...and me along with it. That is too much like the old "blind pig finding an acorn."

Thanks for your help.

David, I think in your second example there is a type mistake : you used X instead of x as plot variable:

psurd = Plot[Surd[x, 3], {x, -10, 10}]

I think it's on-topic. It still falls short of Stefano's desire to have a general specification that keeps calculations in the Real branch. But CubeRoot does work for this. Surd however seems to generate the real roots when used directly, but not with Plot.

pcr = Plot[CubeRoot[x], {x, -10, 10}]

In[28]:= Surd[-10, 3]

Out[28]= -10^(1/3)
psurd = Plot[Surd[x, 3], {X, -10, 10}]

Actually Surd works fine in a plot. You used x as the variable in your plotted expression, but X in the plot range.

Thanks, that sounds interesting. I will look into it

Nice observation.

Hi,

What about

x /. Solve[x^3 == -10 && x \[Element] Reals, x] // N

You can use a similar idea to get your plot.

f := x /. Solve[x^3 == # && x \[Element] Reals, x] &

Then

Plot[f[y], {y, -10, 10}]

gives

![enter image description here][1]

I am not quite sure whether that is what you want though.

Cheers,

M.

All, I think that the "Re[]" takes just the real part of the solution(s). Of course, negative real numbers have complex roots. If you want to see the real part of (each of) the solutions :

in:= Re[x /. Solve[x^3 == -10, x]] // N
out:= {1.07722, -2.15443, 1.07722}

As I recall, graphically, these are going to be where your curve "crosses" the axis.

A nice way to visualize all three:

in:= rePart = Re[x /. Solve[x^3 == -10, x]] // N
in: = imPart = Im[x /. Solve[x^3 == -10, x]] // N
in:= Partition[Riffle[rePart, imPart], 2]
in:= ListPlot[%]

Note, if you are looking for a way to "filter" and choose only those solutions that are Real (no imaginary part) you can add something that keeps only those solutions where the imaginary part is equal to zero.