Find the zeros and singularities of the function in
. Also find the residue at the poles.
The given function is: [ f(z) = \frac{z}{4\cos^2 z - 1}. ] --- Step 1: Find the zeros of (f(z)) Zeros occur when the numerator is zero: [ z = 0. ] Since (|z| \le 1), the only zero is: [ z = 0. ] --- Step __________ __________ ________ ___ _________ _________ ______ _____ _________ __________ ______.
__________ _____ ________ _____ ____ _________ _____.
_____ ______ _______ __________ ______ ________.
_____ _________ _______ _______ ____ ______ __________ _______ _______ ___ ______ _____.
_________ _________ ________ ____ _______ ___ __________ _____ __________ ___.
__________ __________ __________ _____ ________ ____ _______ _________ _____ _____ _______.
_____ ____ _________ _________ _______.
_________ _________ __________ ____ __________ ____ ______ __________ ________.
______ ______ ___ _________ ________ ___.
_____ _________ __________ __________ _____.
_____ ______ ______ __________ _________ ___.
_________ _____ _________ _______ _____ ____ ___ __________ ___ _____ _______ _________.
_____ ____ ______ __________ ____ _______ _________.
___ ______ ____ ___ _____ ____ _____.
_____ ___ _________ ______ ____ _____ _____ __________ _______ ___ __________.
______ _____ ____ _______ _______ _______ ______.
____ ____ ________ ______ _______ _________ ______.
____ ________ ______ ____ ______ ________ ________ ______.
_____ ________ ____ __________ ______ ___.
______ _________ __________ _____ ______ _____ ___ _______ _________.
__________ ________ _________ ______ __________ _________ _____ _________ ________.
__________ __________ _______ ___ ________ _________ __________ ___ ____ ____ _________.
____ ________ ___ __________ _____ ___ ____.
Get Full Answer on WhatsApp
Find the image of the semi-infinite strip x > 0, 0 < y < 1 when . Sketch the strip and its image.
Prove that the linear fractional transformation maps the circle
into itself. Also prove that f(z) is conformal in
.
Find the image of the circle under the mapping
. What happens when
?
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.
a) If is entire such that
in
then show that f has the form
where
are constants with
.
Find the radius of convergence of the following series.
i)
ii)
Evaluate where c is the circle
.