Abstract
We present a method using contour integration to evaluate the definite integral of arctangent reciprocal logarithmic integrals in terms of infinite sums. In a similar manner, we evaluate the definite integral involving the polylogarithmic function L i k ( y ) in terms of special functions. In various cases, these generalizations give the value of known mathematical constants such as Catalan’s constant G, Aprey’s constant ζ ( 3 ) , the Glaisher–Kinkelin constant A, l o g ( 2 ) , and π .
Highlights
We use our new method to evaluate definite integrals in the form of a series [1]. This is a novel approach to these problems in mathematics and has not been used before to our knowledge
This method involves using a form of the Cauchy integral formula
Both the definite integral and infinite sum can be written in terms of the same contour integral, and we can equate the two
Summary
We use our new method to evaluate definite integrals in the form of a series [1]. This is a novel approach to these problems in mathematics and has not been used before to our knowledge. This method involves using a form of the Cauchy integral formula. Lim (cy) over y y ∈ [0, ∞) in the form of known functions. The parameters in these integrals are complex in general
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