a) If is entire such that
in
then show that f has the form
where
are constants with
.
Let ( f ) be an entire function such that ( f'(z) = f(z) ) for all ( z \in \mathbb{C} ). Since ( f ) is entire, it is holomorphic everywhere on the complex plane. The differential equation ( ______ _____ _______ ________ _______ ____ __________ ____ ____ _______.
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___ ________ ______ ________ ________ __________.
___ __________ _____ _________ _____ ______ ____ ____ ________ ___ ________ ____.
__________ ________ _______ __________ ___ ____ _____ _________ ______ ______ ____.
_______ _________ ___ _______ __________ ___ ________ _______ ____ __________ ____.
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_____ _________ __________ _____ ______ ________.
____ ____ _______ ___ ________ ____ ____ _______ _______ ___.
___ ______ ________ __________ _______.
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___ ____ __________ ______ _____ __________ ______ __________ ______ ____ ___ _________.
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If , then show that there exists a real R > 0 such that
for
.
Show that there is only one linear fractional transformation that maps three given distinct points z1, z2 and z3 in the extended z plane onto three specified distinct points w1, w2 and w3 in the extended w plane.
Find the zeros and singularities of the function in
. Also find the residue at the poles.
b) Consider and the closed circular region
. Find points in R where |f(z)| has its maximum and minimum values.
Find all solutions to the equation .
Find the constant c such that can be extended to be analytic at
, when
is fixed.
Prove that the linear fractional transformation maps the circle
into itself. Also prove that f(z) is conformal in
.