Find the image of the circle under the mapping
. What happens when
?
Let ( w = \dfrac{z - i}{z + i} ). Solving for ( z ) in terms of ( w ), [ w(z+i)=z-i ;\Rightarrow; wz+wi=z-i ;\Rightarrow; z(w-1)=-i(1+w), ] so [ z=\frac{-i(1+w)}{,w-1,}. ] The ____ _______ ___ ____ ________ ____ _________.
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Find the zeros and singularities of the function in
. Also find the residue at the poles.
Find the radius of convergence of the following series.
i)
ii)
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 image of the circle under the mapping
. What happens when
?
Prove that the linear fractional transformation maps the circle
into itself. Also prove that f(z) is conformal in
.
Evaluate where c is the circle
.
Find the constant c such that can be extended to be analytic at
, when
is fixed.