The dubious function is defined as follows : and
Evaluate the sum
Let . Evaluate the integral:
We have successively:
Let . Evaluate the integral
The key substitution is . Applying it we see that
Thus , .
Let . Prove that .
The characteristic polynomial of is . This in return means and . Thus,
Converting the product to a sum and using duplication formula for the gamma function and telescoping,
Using Stirling formula
we get that