Evaluate the following integrals:
i) .
ii)
Assume that (f(z)) is analytic on and inside the unit circle and has the power series expansion (f(z)=\sum_{n=0}^{\infty} a_n z^n). Then (f(e^{i\theta})=\sum_{n=0}^{\infty} a_n e^{in\theta}). Using [ \cos^2\frac{\theta}{2}=\frac{1+\cos\theta}{2}, \qquad \sin^2\frac{\theta}{2}=\frac{1-\cos\theta}{2}, ] we __________ ___ __________ ____ _____ ________ ______ __________ ____ _______.
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Evaluate the following integrals
a) .
b) .
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 points where the function is not analytic.
Find the image of the semi-infinite strip x > 0, 0 < y < 1 when . Sketch the strip and its image.
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.
Evaluate where c is the circle
.
a) If is entire such that
in
then show that f has the form
where
are constants with
.