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A symbolic calculus for some algebras of Mellin operators on the finite interval      J ≡  [ 0 , 1 ] J \equiv \left [ {0,1} \right ]       is developed. The algebras are ample enough to include singular integral operators and analytic double layer potentials and their adjoints on      C 1 {C^1}       curves and piecewise      C 1 {C^1}       curves with corners. Fredholmness and the index of the operators on      L p ( J ) {L^p}\left ( J \right )       are completely determined by the principal symbol on      L p ( J ) , Smb l 1 / p {L^p}\left ( J \right ),{\text {Smb}}{{\text {l}}^{1/p}}      .

A symbolic calculus for layer potentials on 𝐶¹ curves and 𝐶¹ curvilinear polygons