Find the radius of convergence of the following series
i)
ii)
i) \sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k!} (z-1-i)^k
This is a Taylor series expansion of the function e^{-(z-1-i)} centered at 1+i.
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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 and
are analytic functions in
domain, then
is necessarily a constant.
iii) A real-valued function u(x, y) is harmonic in iff u(x, − y)
is harmonic in .
iv)
v) The inequality holds for
.
vi) If has the property that
converges, then
is necessarily an entire function.
vii) If a power series converges for
and if
is such that
for all
, then
converges for
.
viii) If 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 is entire and Re f(z) is bounded as
, then
is constant.
If f = u + iv is entire such that ux + vy = 0 in C then show that f has the form f (z) = az + b where a, b are constants with Re a = 0
Find all solutions to the equation sin z = 5
Find the image of the circle under the mapping
. What happens when
?
Evaluate the following integrals
a)
b)
Find the maximum modulus of on the closed circular region defined by
Find the constant c such that can be extended to be analytic at
, when
is fixed.