Past exam of the mathematics course of the University of Cambridge / 2003 / ib / Paper 2 / 16B / c

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- Solution c

Solution

Multiplication by

x

i d / d λ

. Write

p_{n} (x) = \sum_{j = 0}^{n} a_{j} x^{j}

and use

g = c g

for the Gaussian

g = e^{- x^{2} /2}

. Then

f_{n} (λ) = c \sum_{j = 0}^{n} a_{j} i^{j} \frac{d ^{j}}{d λ ^{j}} e^{- λ^{2} /2} = q_{n} (λ) e^{- λ^{2} /2} .

Successive differentiation of the Gaussian gives a polynomial of degree

j

with leading term

(- 1)^{j} λ^{j}

multiplying that same Gaussian. Thus

q_{n}

has leading coefficient

c a_{n} (- i)^{n} \neq = 0

. This proves $q_{n}$ is a polynomial of degree exactly $n$ , rather than merely at most

n

.

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