restart;
Define truncated power
pp:=(z,n)->piecewise(z>=0,z^n,0);
Define Peano Kernel (for Peano's Theorem)
PeanoKernel:=proc(L::procedure,d::nonnegint,t::name)
description "Compute Peano Kernel";
local K;
K:=1/d!*eval(L(x->pp(x-t,d)));
return K;
end proc;
Corrolary to Peano's theorem
PeanoCorr:=proc(L::procedure,d::nonnegint)
description "Peano error term";
local K;
K:=L(x->x^(d+1));
K:=1/(d+1)!*factor(simplify(K))*D[1$(d+1)](f)(xi);
return K;
end proc;
Estimation based on Peano's theorem
PeanoEstimation:=proc(L::procedure,d::nonnegint,a,b)
description "Peano estimation";
local K,t;
K:=PeanoKernel(L,d,t);
M[d+1]*int(abs(K),t=a..b);
end proc;
RG:=f->int(f(x),x=a..b)-(b-a)*f((a+b)/2);
KG:=simplify(PeanoKernel(RG,1,t) assuming a<b);
KG:=convert(%,piecewise,t) assuming a<b;
a:=-1: b:=1: plot(KG,t=-1..1);
unassign('a','b');
pe:=PeanoEstimation(RG,0,a,b);
assume(t>=a and t<=b);
simplify(pe);
save pp, PeanoKernel, PeanoCorr, PeanoEstimation, "Peanoth.m";
restart;
read "Peanoth.m";
print(PeanoKernel);
print(PeanoCorr);