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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.

The shortest path between two truths in the real domain passes through the complex domain. -Jacques Hadamard

Luther, you're probably not making a mistake. This kind of problem is inherent in mathematics. Integrate uses general methods to obtain an antiderivative of the given function. Those methods generally use complex multivalued functions. When the intended result is real, it often involves canceling imaginary parts of partial results. That, in turn, requires the appropriate choice of the values to use for the multivalued functions. There is no general way to compute this that is right (for some definition of "right") in all circumstances. Furthermore, numerical evaluation of an expression that symbolically represents a real number may result in a complex number with a small (or ~zero but imprecise) imaginary part because the cancellation isn't perfect using approximate numbers.

Do you have examples available? Without that it's impossible to tell if you are running into numerical issues or getting legitimate but surprising results.

You sound like a lawyer: it depends! LOL

Is there a way of either eliminating the component or deciding that it can be ignored? Is this even the correct question?

In responding to Daniel Lichtblau's requests, I discovered that I have eliminated this issue by my choice of doing NIntegrate and avoiding Integrate. I cannot find the worksheet that was giving me the imaginary results.

I may have wasted the group's time on this topic, despite what I have learned, since the issue now is whether my NIntegrate and NumericQ parametric plots are leading me astray.

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}]

Well, at least there are some functions which behave like I expected. Thanks

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.

Thank you, David. I must give credit where credit is due, to Leonhard Euler!

I was wondering whether someone might suggest a parametric plot for representing real and imaginary components. It is Pi Day after all! An alternate, continuous representation that includes the subset you've explored above:

ParametricPlot[{Cos[u], Sin[u]}, {u, 0, 2 Pi}, 
 PlotLabel -> "Cos \[Phi] + \[ImaginaryI] Sin \[Phi]", 
 AxesLabel -> {"Re", "Im"}]

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.

This is curious:

Plot[Re[x^(1/3)], {x, -10, 10}]