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
Proof. If we let
we have for each
This can be further simplified to
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.
Simplification leads to the stated result: