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.
i) True. If (a > 0) and (a x + b = 0), then (x = -\frac{b}{a}) is a unique solution. ii) False. If (f(z)) and (g(z)) are analytic in a domain, (f(z) = g(z)) does not imply (f) is constant; for example, (f(z) = z) and (g(z) = z) are analytic but not constant. iii) True. If (u(x, y)) is harmonic ((\nabla^2 _________ ______ _________ _____ _________.
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Find the image of the circle under the mapping
. What happens when
?
Find the points where the function is not analytic.
b) Consider and the closed circular region
. Find points in R where |f(z)| has its maximum and minimum values.
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 radius of convergence of the following series.
i)
ii)
Evaluate , where c is the eight like figure shown in Fig. 1.
a) If is entire such that
in
then show that f has the form
where
are constants with
.