Find the constant c such that can be extended to be analytic at
, when
is fixed.
Consider ( f(z)=\dfrac{1}{z^{n}+z^{,n-1}+\cdots+z-n}+\dfrac{c}{z-1} ). Let ( g(z)=z^{n}+z^{,n-1}+\cdots+z=\sum_{k=1}^{n} z^{k} ), so the denominator of the first term is ( g(z)-n ). _____ _____ _____ ________ _______ __________ ________ _______ ___ ______ _________ _____.
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Find all the singularities of the function
Find the image of the semi-infinite strip x > 0, 0 < y < 1 when . Sketch the strip and its image.
Find the zeros and singularities of the function in
. Also find the residue at the poles.
Evaluate the following integrals
a) .
b) .
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.
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.