Show that there is only one linear fractional transformation that maps three given distinct points, and
in the extended
plane onto three specified distinct points
, and w,3 in the extended w plane.
To show that there is only one linear fractional transformation (also known as a Möbius transformation) that maps three distinct points \( z_1, z_2, z_3 \) in the extended complex plane (Riemann sphere) to three specified distinct points \( w_1, w_2, w_3 \) in the extended \( w \)-plane, we proceed as follows: ### Step 1: General form of the linear fractional transformation A linear fractional transformation has the general form: ___ ______ ______ ______ ______ _________ __________.
______ ______ ____ ___ ____ _________ __________ ____ _________ __________.
_________ _______ _________ ___ __________.
__________ _____ ____ _____ _______ ________ ____ _________.
__________ ____ ___ ______ ____ ___ ____.
_____ _______ _______ ___ ____ ________.
________ ________ ___ _____ _________ _________ _______ __________ _____.
____ ________ ___ ____ ______ _______ _________ ___ ____ _________.
_______ __________ __________ ___ _______ _______ ______ _________ ___ __________ ______.
_____ ___ ______ ______ ___ __________ _______ ____ _____.
________ ___ ___ __________ ________ _____ ____ _______ ________.
__________ ______ ___ _____ __________.
__________ __________ ________ ________ ___ ____.
_______ ________ ____ _______ ______ ____ ____ ____ _________.
________ _______ _______ _________ ____.
____ ______ ________ ______ _____ ____ ________ ________ ____ _____ _______.
________ ________ ________ _____ ________ ____.
___ ___ _________ ________ __________ ___ _____ ___ ________ ________.
__________ ___ ________ ________ ____ _________.
_______ ___ _______ ________ _______ __________ _______ ________ __________ ___ ____ ______.
_______ ______ _________ _____ ___ ______ _________ ___ ___.
____ ______ _______ ____ ______ _________ ______ _____.
________ ____ _________ ____ ___ ____ _____.
______ _________ _____ ______ _________ ____ _____ _______ _____ _________ _____ ___.
___ ____ ____ _________ _______ _________ ________ ________ __________ ______ __________ ______.
_______ ______ __________ ____ ____ __________ _______ _____ ___ __________ ________.
________ ____ ___ ______ _____ _______ _______ __________ ________ _______.
____ ________ _____ ___ ____ _______.
______ ______ ____ ________ ________ ___ ______ ____ ___.
_________ _______ _____ ______ ______ ________ ________ _____ ______ _________.
_____ _______ _______ ______ ___ _______ ______ ___ ___ ________ _____ _________.
_________ ____ _______ ____ ____ ____ ____.
__________ ________ ___ _____ ____ _________ __________ _______ ______ ____ ___.
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22-1 Prove that the linear fractional transformation maps the circle
into itself. Also prove that
is conformal in
Determine whether each of the following statement is true or false. Justify your answer with a short proof or a counter example.
i) where a and b are integers, the
ii) 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 w(x,y) is harmonic in D.
iv)
v) The inequality holds for
vii) If a power series converges for
and if
is such that
then converges for r | z |
1.
viii) If is entire and
for all, then there exists an entire function g such that
ix) A mobius transformation which maps the upper half plane onto itself and fixing 0,
and no other points, must be of the form
for some
and
x) If is entire and Re
is bounded as
then
is constant.
Consider and the closed circular region
Find points in R where |
| has its maximum and minimum values.
Expand in a Laurent series valid for
i) ii)
Evaluate the following integrals
a)
b)
is entire such that
in C then show that
has the form
where
, b are constants with Re
=0.
Find the radius of convergence of the following series.
i) ii)