I’ve discovered the channel “blackpenredpen” and it is delightful.

It is a nice escape into mathematics that, while far from research level, is “fun” and beyond mere fluff.

And that got me to thinking about . Yes, this can be done by residues

But I’ll look at this with Laplace Transforms.

We know that

But note that the antiderivative of with respect to is That might not seem like much help, but then notice (assuming

So why not:

Now since the left hand side is just a double integral over the first quadrant (an infinite rectangle) the order of integration can be interchanged:

and that is equal to .

Note: is sometimes called the function

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