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Kunze Linear Algebra
· Problem 4.5.4
Polynomials
Mathematics — Methods · generalized-taylors-formula
Problem
Prove the following generalization of Taylor's formula. Let
f
f
f
,
g
g
g
, and
h
h
h
be polynomials over a subfield of the complex numbers, with
deg
f
≤
n
\deg f \leq n
de
g
f
≤
n
. Then
f
(
g
)
=
∑
k
=
0
n
1
k
!
f
(
k
)
(
h
)
(
g
−
h
)
k
.
f(g) = \sum_{k=0}^n \frac{1}{k!} f^{(k)}(h)(g-h)^k.
f
(
g
)
=
k
=
0
∑
n
k
!
1
f
(
k
)
(
h
)
(
g
−
h
)
k
.
(Here
f
(
g
)
f(g)
f
(
g
)
denotes '
f
f
f
of
g
g
g
.')
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