In this post I present my argument which proves integral no. 5 under the Radical Integrals section of The Integral Corner. The result in question reads as follows.

Radical Integral 5. Let and real satisfy . Consider the function

for where is the incomplete gamma function with integral representation

Then,

with

and

Proof. If we let

we have for each

This can be further simplified to

after establishing

for . To see this, note that the “lower” incomplete gamma function is a holomorphic function with singularities at points where or is a non-positive integer (check out the Incomplete gamma function Wikipedia page). Moreover, it admits the representation

Setting and , we find for each

Rearranging, we get

But notice that, formally,

As such, we arrive at the desired identity for the infinite sum, justifying the identity for . Writing out as

we see that is differentiable over with derivative given by

Consequently, is monotone decreasing over . Therefore, our proposed integral for given can be evaluated as follows.

Glad to see you’ve been keeping busy in quarantine 🙂

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