Useful

Table of Useful Operations acting On the Taylor Series part of Various Types

Taylor (t)

Complex Taylor (ct)

Real Polymorph (rp)

Complex Polymorph (cp)

:f: (f)

F×Ñ (F)

Damap/Gmap (M)

₁

Derivative dt/dx_i

t=t.d.i

ct=ct.d.i

Extract an order

t=t.sub.i

ct=ct.sub.i

rp=rp.sub.i

cp=cp.sub.i

f=f.sub.i

F.sub.i

M.sub.i

Truncate order i and above

t=t.cut.i

ct=ct.cut.i

rp=rp.cut.i

cp=cp.cut.i

f=f.cut.i

F.cut.i

M.cut.i

₄

Create Monomial

r x₁^j(1)...x_nv^j(nv)

r x_i

t=r.mono.j(nv)

t=r.mono.'j₁...j_nv'

t=r.mono.i

ct=c.mono.j(nv)

ct=c.mono.'j₁...j_nv'

₅

Peek coefficient r of monomial r x₁^j(1)...x_nv^j(nv)

r=t.sub.j(nv)

r=t.sub.'j₁...j_nv'

c=ct.sub.j(nv)

c=ct.sub.'j₁...j_nv'

r=rp.sub.j(nv)

r=rp.sub.'j₁...j_nv'

c=cp.sub.j(nv)

c=cp.sub.'j₁...j_nv'

₆

Peek coefficient of x₁^j(1)...x_n^j(n)_{as a Taylor series whenre n<nv}

t=t.par.j(n)

t=t.par.'j₁...j_n'

ct=ct.par.j(n)

ct=ct.par.'j₁...j_n'

rp=rp.par.j(n)

rp=rp.par.'j₁...j_n'

cp=cp.par.j(n)

cp=cp.par.'j₁...j_n'

₇

Generalization of .par. (item 6)

t=t.part.info

Shift exponents downwards by k

t=t<=k

ct=ct<=k

Peek and shift (6+8 combined): t=(t.par.j(n))<=n

t= t<=j(n)

t= t<='j₁...j_n'

ct= ct<=j(n)

ct= ct<='j₁...j_n'

Tiny Real Polynomials

r+ x_i

r(1)+r(2) x_i

t= r.var.i

t= r(2).var.i

Tiny Complex Polynomials

Re(c) x_j(1)+Im(c) x_j(2)

c(1)+c(2) (x_j(1)+i x_j(2))

ct= c.var.j(2)

ct= c(2).var.j(2)

Pseudo-derivative

dxⁿ/dx_i=x_i^n-1

t=t.k.i

ct=ct.k.i

₁₃

Poisson Bracket

t= t.pb.t