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I am currently writing my dissertation, and there are a couple of assumptions in my work that I need to be true in order for my proofs to be valid. Without venturing into too many details, I'm currently depending on using the fact that a handful of integrals are convergent. They are quite difficult integrals, so given that I am unable to prove their convergence directly at the moment, I've made use of Mathematica. This software quickly shows that they converge to a particular number in a matter of seconds, and with a very small margin of error.

Now, I'm fully aware that this does not constitute a proof. Indeed, computers can be wrong, and should be used as a guide rather than as a substitute for a rigorous argument. My question is a very general one: when is it appropriate to use numerical evidence in this way (or in this case, a computer package which evaluates integrals numerically) as justification for a particular assumption in a proof to hold? What are examples where this would be frowned upon, and instances where this would be encouraged? Would my work be an example where such an assumption should be heavily discouraged? (I appreciate that it is difficult to pass judgement in this way without knowing the details of what my assumptions are used to prove, but I can provide details if necessary.)

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Mathematica, MATLAB and other numerical tools are perhaps good if you're unsure about how to approach your problem, and you want to see the final solution to give yourself an idea on what to do next, how to start out the proof.

But if you're writing a thesis, it's incorrect to assume that a certain integral converges based on a finite number of iterations done by an application, and I don't think that your reviewers would react positively to such a "proof" in your paper.

There are many ways of proving that an integral converges without calculating it, so maybe you could read up on some of those theorems if you haven't quite yet gotten around to it.

That being said, your awareness of the fact that a numerical calculation is not a formal proof is commendable.

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