Find all the singularities of the function
The singularities of the function ( f(z) = \exp!\left(\dfrac{z}{\sin z}\right) ) arise from the points where the denominator (\sin z) vanishes. The zeros of (\sin z) occur at (z = n\pi), where (n ____ ___ ____ ____ __________.
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a) If is entire such that
in
then show that f has the form
where
are constants with
.
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 .
Determine whether each of the following statement is true or false. Justify your answer with a short proof or a counter example.
i) If , where a and b are integers, then
if a > 0.
ii) If f(z) and are analytic functions in a domain, then f is necessarily a constant.
iii) A real-valued function u(x, y) is harmonic in D iff u(x, -y) is harmonic in D.
iv) .
v) The inequality holds for
.
vi) If has the property that
converges, then f is necessarily an entire function.
vii) If a power series converges for |z| < 1 and if
is such that |bn| < n2 |an| for all
, then
converges for |z| < 1.
viii) If f is entire and for all z, then there exists an entire function g such that
for all
.
ix) A mobius transformation which maps the upper half plane onto itself and fixing
and no other points, must be of the form
for some
and
.
x) If f is entire and is bounded as
, then f is constant.
Find the points where the function is not analytic.
Find the image of the circle under the mapping
. What happens when
?