Starting/Ending Encoding
\def\UUU{\quad {\int_{u}^{n}}E(x,\tsp k)
{dx\over\sqrt{(\sin^{2} x-\sin^{2} u)
(\sin^{2} v-\sin^{2} x)}}}
\def\UU{\hphantom{\UUU}}\displaylines{\UUU
={}{1\over2\cos u\sin v}\mbi{E}(k)\mbi{K}
\left(\sqrt{1-{tg^{2}\tsp u\over
tg^{2}\tsp v}}\right)+\hfill\cr
\UU{}\hphantom{{}={}}{}+{k^{2}\sin v\over2\cos u}
\mbi{K}\left(\sqrt{1-{\sin^{2}
2u\over\sin^{2} 2v}}\right)\hfill\cr
\hfill[k^{2}=1-\ctg ^{2}\tsp u\ctg^{2}\tsp v].
(stmt () (== (integrate (* (userfunc E x k) (/ 1 (power
(* (- (power (sin x) 2) (power (sin u) 2)) (- (power
(sin v) 2) (power (sin x) 2))) (/ 1 2)))) (x u n )) (+
(* (* (/ 1 (* (* 2 (cos u)) (sin v))) (userfunc bold_E
k)) (userfunc bold_K (power (- 1 (/ (* (* t (power g
2)) u) (* (* t (power g 2)) v))) (/ 1 2)))) (* (/ (*
(power k 2) (sin v)) (* 2 (cos u))) (userfunc bold_K
(power (- 1 (/ (power (sin (* 2 u)) 2) (power (sin (* 2
v)) 2))) (/ 1 2)))))) (== (power k 2) (- 1 (* (power
(ctg u) 2) (power (ctg v) 2)))))
Although this looks pretty good at first sight, in fact there are at least two errors in G&R in this input: tg mean tangent, and thus tg^{2}v should not be (* t (power g 2) v) but (power (tan v) 2). Failing to use \tg instead of tg seems like a simple typo, but it makes the formula nonsensical in a purely mechanical form. A human probably wouldn’t even notice. Oh, the upper limit should be v not n. But the printed table is so smudged in my printed copy that you’d have trouble identifying that character except by context.