Parameter-dependent integrals and sums

When we have a function defined by an integral depending on a parameter, where the parameter plays the role of variable, we may be led to look for an equivalent of this function when the variable tends to + ∞; similarly, when we have a function defined by the sum of a series of functions, we may be led to look for an analogous equivalent.

Three examples come under this heading: the Laplace transform of a function, rotating integrals and the sum of an integer series. They lead respectively to the three sub-paragraphs that follow.