The dilogarithm function is defined as follows: Li2(x)
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The dilogarithm function is defined as follows: Li2(x)
The dilogarithm function is defined as follows: Li2(x) = â« x. 1 ln t. 1 - t dt. Identities: Li2(1 - x) - Li2(0) = - Li2(x) - ln(x) ln(1 - x),. Li2(x) + Li2(1/x) = -. 1. 2 ln2 x. (1).
The dilogarithm function is defined as follows: ∫ x ln t Li2 (x) = dt. 1 −t 1 Identities: Li2 (1 − x) − Li2 (0) = − Li2 (x) − ln(x) ln(1 − x), 1 Li2 (x) + Li2 (1/x) = − ln2 x. 2 (1) ∫ b ∫ b ∫ 0 ∫ b b ln(b) ln(1 − x/b) ln t ln(b − x) dx = dx+ dx = ln(b) ln( )− dt x x x a a a 1−a/b 1 − t a b a b a a a b a = ln(b) ln( )+Li2 (1− )−Li2 (0) = ln(b) ln( )−Li2 ( )−ln( ) ln(1− ) = ln(b−a) ln( )−Li2 ( ). a b a b b b a b (2) ∫ b ∫ b ∫ b ∫ 1−b/a ln(x − a) ln(−a) ln(1 − x/a) b ln t dx = dx+ dx = ln(−a) ln( )− dt x x x a 1 −t a a a 0 b b b b b b b b = ln(−a) ln( )−Li2 (1− )+Li2 (0) = ln(−a) ln( )−Li2 ( )+ln( ) ln(1− ) = ln(b−a) ln( )+Li2 ( ). a a a a a a a a (3) ∫ b a b 1 a 1 b−x ln dx = − Li2 ( ) − Li2 ( ) = ln2 ( ). x−a b a 2 b a x
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