Abstract
We derive a power series expansion for the function \(\,x\mapsto x \log {(\tan x)} -x \log {x} +x - \mathrm{Ti_2} (\tan x)\,\), and then use it to obtain the expansion of the function \(\,x\mapsto \mathrm{Ti_2} (\tan x)\); as usual, \(\mathrm{Ti_2}\) denotes the inverse tangent integral. Two similar expansions involving Legendre’s chi-function follow from the above. These representations are used to obtain new rapidly convergent numerical series involving zeta values.
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Markov, L. (2019). A Functional Expansion and a New Set of Rapidly Convergent Series Involving Zeta Values. In: Georgiev, K., Todorov, M., Georgiev, I. (eds) Advanced Computing in Industrial Mathematics. BGSIAM 2017. Studies in Computational Intelligence, vol 793. Springer, Cham. https://doi.org/10.1007/978-3-319-97277-0_22
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DOI: https://doi.org/10.1007/978-3-319-97277-0_22
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