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 Jun 10, 2005 - A Generalization of Adamchik's Formulas   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

Theorem.  where coefficients are determined by  Proof. Since    We get  Therefore,  Comment. The first few are        The proof for Adamchik's Proposition 4 can also be simpler:

Theorem.  Proof.    ## References

The first reference is Victor Adamchik's paper. The other
papers also discuss expression of definite integrals
in terms of derivatives of products of and functions.

Geddes, K. O. and Scott, T. C. (1989) "Recipes for Classes of Definite Integrals Involving Exponentials and Logarithms", Computers and Mathematics, Kaltofen, E. and Watt, S. M. (eds.), 192-201, Springer-Verlag.

Williams, Kenneth S. and Zhang, Nan Yue (1993) "Special Values of the Lerch Zeta Function and the Evaluation of Certain Integrals", Proc. Amer. Math. Soc., 119(1), 35-49.

Zhang, Nan Yue and Williams, Kenneth S. (1993) "Applications of the Hurwitz Zeta Function to the Evaluation of Certain Integrals", Canad. Math. Bull., 36(3), 373-384.