I thought I mentioned here some months ago Steve Finch's strange Integrate[((2 + 4*v + 3*v^2)* Sqrt[-((1 + 2*v + 3*v^2)/(4 + 8*v + 3*v^2))])/(1 + 2*v)^2, {v, -2, -2/3}] that Mathematica can do INdefinitely, but cannot take the endpoint limits. The integral is the surface area of the convex hull of two edge-to-edge orthogonal unit disks. (http://arxiv.org/abs/1211.4514). Working a couple of days, Neil and I numerically conjectured the messy expression on p10, subsequently reduced, again numerically, to (1/(54*(-I + 2*Sqrt[2])))* (-243*I*EllipticE[(1/81)*(17 - 56*I*Sqrt[2])] + 27*Sqrt[-17 + 56*I*Sqrt[2]]* EllipticE[(1/81)*(17 + 56*I*Sqrt[2])] + 2*(81*Sqrt[2]*EllipticK[1/9] - 81*Sqrt[2]* EllipticK[(1/81)*(17 + 56*I*Sqrt[2])] + 3*(8*I - 7*Sqrt[2])* EllipticPi[(1/27)*(7 - 4*I*Sqrt[2]), (1/81)*(17 - 56*I*Sqrt[2])] + (104*I + 71*Sqrt[2])* EllipticPi[(1/3)*(7 - 4*I*Sqrt[2]), (1/81)*(17 - 56*I*Sqrt[2])])) A completely different problem: Limit[(1/( 2 s))(2 y + s (-1 + t - y + z) - t (1 + t - y + z) + 2 ContinuedFractionK[1/2 n^2 (-8 t - 2 t^2) + 1/2 n (-2 t^2 - 4 s y - 2 t (4 + s (-3 + z))) + 1/2 (-2 t + 3 s t - 5 t^2 - 2 s y + 6 t y - 2 y^2 + 3 t^2 z - t (s + 2 y - 2 z) z), s (2 + t) + n s (2 + t) + 2 y - t (1 + t - y + z), {n, 0, ∞}]), t -> 0, Direction -> -1] == -1 - y + z + (E^-y y^z)/Gamma[z, y] where s:=Sqrt[t (4 + t)]. I can't even do the special case z=1, y=2: Limit[(1/( 2 s))((-2 + s - t) (-2 + t) + 2 cfk[-4 + (1 + 2 n) s (-2 + t) - t (-4 + t + n (1 + n) (4 + t)), (2 + s + n s - t) (2 + t), {n, 0, \[Infinity]}]), t -> 0, Direction -> -1] == 0 I can get these as ratios of 2F1s with exploding parameters... It would help if Mma knew that Limit[Hypergeometric2F1[a, b/t, c, t*z], t -> 0] == HypergeometricPFQ[{a}, {c}, b*z] and didn't hang up for literally hours trying to express the ContinuedFractionK in closed form. --rwg