This note (PDF version) discusses a generalization of some formulas in "A Class of Logarithmic Integrals" by Victor Adamchik. Adamchik's paper proves 6 different general propositions allowing Adamchik to solve integrals such as
Generally, I like Adamchik's paper. The most interesting proposition is Proposition 1 which Adamchik attributes to G. Almkvist and A. Meurman. The proofs for Adamchik's Proposition 3, Proposition 5, and Proposition 6 are more drawn out and tortured than necessary. My theorem below is proved in less space and generalizes half the propositions in Adamchik's paper.
Definition of Two Functions
Definition. For convenience, define
Function has properties
The following theorem generalizes Adamchik's Proposition 3, Proposition 5, and Proposition 6.
Generalization of Adamchik's Propositions 3, 5, and 6
where coefficients are determined by
Comment. The first few are
Adamchik's Proposition 4
The proof for Adamchik's Proposition 4 can also be simpler:
The first reference is Victor Adamchik's paper. The other
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