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Notes | Introducing a parameter into an integral

Posted at 2024-07-09 Notes Integrals Methods 

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Some integrals are awkward to compute directly; introducing a parameter can make them easier. In my original note I called this an “integrating-factor method,” but what I actually use here is differentiation of a parameter-dependent integral, not the integrating factor from differential equations.

Consider

Insert into the integrand, taking for now, and write

One cannot move the derivative inside the integral merely by saying that differentiation and integration commute; the statement of a theorem in the original note was left unfinished. For , if we restrict to a closed interval away from zero, the exponential provides enough decay to justify differentiation. Differentiating introduces , which cancels the in the denominator:

Thus . As , both the integral and tend to zero, so . Letting gives

This last step takes a limit; it is not a direct substitution of into a differentiation formula proved only for . The convergence of the oscillatory integral also needs to be justified. My original note passed over this too quickly.

The same idea brought to mind Euler’s integral for the factorial. For , we start from

Differentiate both sides times with respect to . Each derivative of the integrand introduces another , so

or

Putting yields . The original note had in the denominator; I have corrected it to . Here too, differentiating an improper integral requires an integrable bound in a neighbourhood of ; uniform continuity on an interval alone is not enough.