Find all solutions to the equation .
(\sin z=\dfrac{e^{iz}-e^{-iz}}{2i}=5). Let (w=e^{iz}). Then [ \frac{w-\frac1w}{2i}=5 ;\Rightarrow; w-\frac1w=10i ;\Rightarrow; w^2-10iw-1=0. ] __________ _______ ________ ______ ___ __________.
________ _______ _____ _________ _________.
________ _____ _____ _________ ____ ________ __________ _______.
___ __________ ____ ___ ___ ____ ________ _______ ______.
___ __________ ___ __________ ______ ___ __________ __________.
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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 maximum modulus of on the closed circular region defined by
.
If , then show that there exists a real R > 0 such that
for
.
Find the constant c such that can be extended to be analytic at
, when
is fixed.
Find the zeros and singularities of the function in
. Also find the residue at the poles.
Evaluate the following integrals
a) .
b) .
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.