Another Euler Constant: The Value of an Euler Integral

Abstract We show many occurrences in various contexts of the constant <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mrow> <m:msub> <m:mrow> <m:mi>E</m:mi> </m:mrow> <m:mi>u</m:mi> </m:msub> <m:mo>:</m:mo> <m:mo>=</m:mo> <m:mfrac> <m:mrow> <m:msup> <m:mrow> <m:mi>π</m:mi> </m:mrow> <m:mn>2</m:mn> </m:msup> </m:mrow> <m:mn>4</m:mn> </m:mfrac> <m:mo>log</m:mo> <m:mn>2</m:mn> <m:mo>-</m:mo> <m:mfrac> <m:mn>7</m:mn> <m:mn>8</m:mn> </m:mfrac> <m:mi>ζ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mn>3</m:mn> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mn>2.507907542</m:mn> </m:mrow> </m:math> {E_u}: = {{{\pi ^2}} \over 4}\log 2 - {7 \over 8}\zeta \left( 3 \right) = 2.507907542 ... Since this constant is related to an integral studied by Euler, we propose to name it after Euler.