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I'm trying to understand a paper in which the authors use a number of test functions (are they the same as convergence factors) to make integrate unintegrable functions. Now here is my ignorant question: why is this acceptable? The product of the original function and the test function or convergence factor surely is an entirely new function. Why is the integral of this new function the same as the integral of the old function which I recall was unintegrable? In layperson terms, can we be sure that the integral of the new function is valid and or correct?
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