Take 15 minutes to prepare this exercise.
Then, if you lack ideas to begin, look at the given clue and start searching for the solution.
A detailed solution is then proposed to you.
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Question
a) Using the Taylor-expansion of application
find the Taylor-expansion of order
around
of applications
b) Find the Taylor-expansion of order
around
of applications
c) Find the Taylor-expansion of order
around
of applications
.
a) Write the Taylor-expansion
then set
Then set
then replace
by
b) and c) Replace
by
in the Taylor-expansion of
then integrate it while checking that the integration constant is zero.
a) We have:
With
we obtain:
Multiplying the numerator and the denominator by
we obtain:
In conclusion:
With
we obtain:
Multiplying the numerator and the denominator by
we obtain:
In conclusion:
Substituing
to
we obtain:
b) Substituing
to
in the previous Taylor-expansion, we get to:
c) Integrating the previous Taylor-expansion, we have:
