OK, now that I have that off my chest, I have attached a worksheet that one of the Community members helped me develop. Do not pay attention to why I want to evaluate the particular integral, since I was going to post this to the Physics section. It is the real mutual attraction model that has never been solved....the classical point-mass integral we are all taught is a heuristic.

Consequently, I set out to prove my point. The issue became that the mutual attraction between sphere's is so close to the point-mass model that the solutions are really perturbations on the point-mass approach....which is why no issues have ever been identified. If the model is extended to cylinders and internal masses...a test mass within a distributed mass such as a sphere or disk, the results are smooth and totally unlike the point-mass results...the solutions are not simply perturbations. My goal has been to understand the detail of the spherical solutions. However, the perturbations are on the same order as the variances in the numerical results when I switch between coordinate-system descriptions of the spheres.

That being said, the worksheet models have been set up to explicitly evaluate at one point and I was attempting to generalize the approach to allow a parametric evaluation over the separation distance. It is actually easy to do and I have voluminous plots showing the result. However, when using the worksheet as a template and getting ready to post this material in the physics community, I got the behavior that I have been grumbling over and I do not understand what is going on. So far, I still don't understand the issue. To may simplistic way of looking at things, the integrals should not have evaluated at all until I defined the variable parameter...which it did in my original clunky methodology. I have no doubts about the validity of the results, but I wanted to create a more sophisticated approach and ran into these issues.

I am a bear about details, which is why I noticed that the Newtonian gravity model...or static electric interaction models as we use them...are heuristics. Now the issue is that I simply cannot understand why the numerical models evaluate at all when they are incomplete.

Regards and thanks for your forbearance.

Luther

Attachments:

I am not comfortable with working in the complex domain when the arguments to integrals are real and finite. I went through all the advanced mathematics and complex variable analysis coursework in my youth but in the pre-Mathematica days I just used Gratshteyn and Ryzhik and kept my fingers crossed. If that did not work out I resorted to writing numerical integral code using the trapezoidal or Simpson rules and stayed away from the complex plane. I simply do not want to have to understand the complex solutions that arise when what are finite real integrals throw off complex solutions.

There are two issues. The first is that if there is nothing to execute, then why show any results? I have input the function NIntegrate[x^a, {a, 1., 2., .1}], it should not execute, and when it is first executed, in fact I simply get back NIntegrate[x^a, {a, 1., 2., .1}]. If I re-execute an evaluation on the same function, it evaluates. Why?

My goal is to set fff=NIntegrate[x^a, {a, 1., 2., .1}] and then to use fff over variable x to do a parametric evaluation. But if, when setting fff, the function evaluates to something, what is it evaluating to and what is the function fff that I am using? I don't really have to do it this way but I did and I got an output that is ambiguous...at least to me. To my simple way of looking at things, if it should not evaluate, and so, then, it should not give back something I don't understand.

I simply want to understand what is going on and what the "false" evaluation is leaving hidden in or associated with the function fff.

If I set a variable, call it d and evaluate d without giving it a value, I simply get d back....no matter how many times I evaluate d.

In the simple function I used as an example, when I evaluate that function without defining certain variables, I get the function back with no comment. If I evaluate this function a second time or more, I get the comment back that it evaluates with indeterminate values.

If a more complicated function with the accuracy and precision and working precision limits set but no variables defined, again, the first evaluation gives back the function with no comments. The second and later evaluations with the variable still not defined gives back the statement that it tried to evaluate the function and that the function's precision was less than the working precision. What can this possibly mean, since the function should not evaluate at all? And, I do not want some spurious evaluation to "contaminate" the function.