Legendre's Identity

A nice proof of this follows by (i) differentiating both sides - one gets zero- and then taking the limit at $k /to 0$. We illustrate this in [Maple]. It is interesting to note that LegendreUs faith in and knowledge of limits precluded his developing this proof. Instead he evaluated all quantities at $(3^(1/2)-1)/2$ (the third [singular value]) a considerable tour de force!

`diff/LegendreKc` := proc(k,x)
diff(k,x)*(-1/k*LegendreKc(k)+1/k/(1-k^2)*LegendreEc(k));
end:

`diff/LegendreKc1` := proc(k,x)
diff(k,x)*(k/(1-k^2)*LegendreKc1(k)-1/k/(1-k^2)*LegendreEc1(k));
end:

`diff/LegendreEc` := proc(k,x)
diff(k,x)*(-1/k*LegendreKc(k)+1/k*LegendreEc(k));
end:

`diff/LegendreEc1` := proc(k,x)
diff(k,x)*(k/(1-k^2)*LegendreKc1(k)-k/(1-k^2)*LegendreEc1(k));
end:

alias (Kc=LegendreKc,Ec= LegendreEc,Kc1= LegendreKc1, Ec1=LegendreEc1);

LL:=Kc*Ec1+Ec*Kc1-Kc*Kc1;

diff(LL(t),t); simplify(");

LL((sqrt(3)-1)/2); evalf(");