Question
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.
Answer :
Word Count : 795
Let's analyze each statement one by one: ### (i) \[ \text{If } z = a + ib, \text{ where } a \text{ and } b \text{ are integers, then } |1+z+z^2+\cdots+z^n| \geq |z|^n \text{ if } a > 0. \] Verdict: False Counterexample: Let \( z = 2 \) (i.e., \( a = 2, b = 0 \)). Then, \[ S_n = 1 + 2 + 4 + \dots + 2^n = \frac{2^{n+1} - 1}{2 - 1} = 2^{n+1} - 1. \] Clearly, \( |S_n| = 2^{n+1} - 1 \) and \( |z|^n = 2^n \). However, for \( n = 1 \), \[ |1 + 2| = 3 < 2^1 = 2. \] Thus, the inequality does not always hold, proving the statement false. --- ### (ii) \[ \text{If } f(z) \text{ and } \overline{f(z)} \text{ are analytic functions in a domain, then } f \text{ is necessarily a constant.} \] Verdict: True Proof: If \( f(z) \) ___ ____ ____ _____ _____ ___ ______ _____.
________ ______ ________ _____ _________ _______.
___ _____ _______ ____ _____ ________.
__________ ______ _____ ___ _______ _______ ________ ___ _________ ______ ________ _____.
_____ _____ ______ ___ _____ ___ _______ ________ _________ ________ ________ _________.
_______ _____ ____ _____ ___ _______ _____ __________ _________.
____ ____ _______ _______ _____ __________ ________ ____ _________.
__________ ______ ____ __________ ______.
________ ______ ________ ________ ____ _____.
________ ___ ___ _____ ______ _______ ____ __________ ____.
_________ ________ ___ ________ _________ _______ _______ _________ ____ ______ ___ __________.
_________ ___ ____ __________ ______ ______ _________.
_______ ______ ___ _______ __________ ___ __________ ______ ___ _______.
_____ ______ _______ ______ __________.
_______ _______ __________ ________ ___ ____ _____ ___ _______ ____.
______ _______ _______ _____ _____ _________ ____.
______ __________ _____ ______ _________ _____ ________.
___ __________ ________ __________ _____ ____ ___ _______.
___ __________ ________ _________ __________.
_______ ___ ____ ______ _________.
________ _________ ________ ______ ____ __________ _________ _________ __________ ________ _________.
________ ______ ________ ______ ______ _____ _______ ________ ___ ________.
________ ___ ____ ________ __________ __________ _____ ____ ______ _______ ________ _______.
___ ________ _______ __________ ____ _________ ___ _______ __________.
______ ___ _________ _____ ___ _________ ________ ______ ____ ____.
____ ___ ____ _____ ________ __________ _______ ______ ______ ________ ______ __________.
_______ _________ _______ ______ __________ ______.
__________ ______ ___ ___ ____.
___ ______ ______ _____ _________ ________ ____ ____ _____ __________.
_______ ___ ___ _____ ____ _______.
________ ______ ___ __________ __________ ________ __________ _________ ______ __________ __________.
______ __________ _____ __________ _________ _________ ________ _____ ________.
______ __________ __________ ______ ___ _______.
______ ___ ________ ___ ____ ____ __________.
________ __________ ___ __________ ____ _________ _______ ___ _________.
__________ _____ ______ ____ _______ ______ __________ _________ __________ _____ _______ ______.
______ _________ _______ _______ __________.
_____ _____ ____ ________ ___ ________ ________ ________ __________ _____ __________ ______.
__________ _____ __________ _______ _______ _____ ________.
__________ _______ _______ ________ ___.
__________ ______ _____ _____ __________ __________ _____ ________ _______ _____.
______ _____ ______ _______ ______.
___ ______ _______ ______ _________ _______ _______ ________ ______ ______.
____ __________ ___ ________ ______.
_______ __________ ________ ______ ___ _______ _________ ________ _________.
____ ____ ___ _____ ___ ______ ___ ___ _________ ______ ________ ___.
_____ _______ _______ ________ ___ ____ __________ _____.
_______ ___ ________ ________ _______.
_________ ______ ____ _________ _____ __________ ____.
________ _______ ______ _____ ______ ________ ________ ______ _____.
______ _______ ___ __________ ________ __________ _________ ______ ____ ________ __________ ________.
____ _____ ____ ___ __________ ________.
________ __________ __________ ________ ___ ___ ___ ________ _____ _____.
__________ ________ __________ _____ _________.
_________ ________ ________ ______ __________ _________ _______ __________ _______ ________.
______ ______ _______ _______ ______ _____ ____ _________.
________ _________ ____ _____ __________ ____ _________.
________ _____ _________ _____ ______ _________.
_____ _________ ________ ________ ________ _________ ____ ____ _________.
___ _______ __________ ________ _______ _________ ______ _____ ________ _____.
_______ ___ ________ ______ ___ ________ ___ ______ __________ _________ __________.
_____ ______ ____ ________ __________ _______ ____.
__________ _________ ___ ________ _____ _________ _____ _____.
__________ ____ _____ ___ _______ ______.
_________ ___ _______ _________ ________ ____ ___ ________ _____ ___ _________.
______ __________ _______ ________ ______ __________.
___ __________ __________ __________ ___ ___.
_________ ____ ________ __________ ____.
_______ ____ ____ ___ _________ _________ __________ ___.
____ ______ _______ _______ ___ __________ _________ _____ __________ __________ ______.
____ _________ _______ _______ __________ ____ _______.
______ ________ ______ _____ ______ ____ __________ _____ __________ _________.
_____ _________ ___ ________ ___ _____ ______.
________ _______ _____ ____ __________ ______ ___ ___ _________ _________ ___ _______.
________ ______ ________ _________ __________ _____ ________ __________ _________ __________ ____.
_______ __________ _______ _________ ________ _________ ___ ____ _____ _____ _______ ______.
___ _______ __________.
Get Full Answer on WhatsApp
Let's analyze each statement one by one: ### (i) \[ \text{If } z = a + ib, \text{ where } a \text{ and } b \text{ are integers, then } |1+z+z^2+\cdots+z^n| \geq |z|^n \text{ if } a > 0. \] Verdict: False Counterexample: Let \( z = 2 \) (i.e., \( a = 2, b = 0 \)). Then, \[ S_n = 1 + 2 + 4 + \dots + 2^n = \frac{2^{n+1} - 1}{2 - 1} = 2^{n+1} - 1. \] Clearly, \( |S_n| = 2^{n+1} - 1 \) and \( |z|^n = 2^n \). However, for \( n = 1 \), \[ |1 + 2| = 3 < 2^1 = 2. \] Thus, the inequality does not always hold, proving the statement false. --- ### (ii) \[ \text{If } f(z) \text{ and } \overline{f(z)} \text{ are analytic functions in a domain, then } f \text{ is necessarily a constant.} \] Verdict: True Proof: If \( f(z) \) ___ ____ ____ _____ _____ ___ ______ _____.
________ ______ ________ _____ _________ _______.
___ _____ _______ ____ _____ ________.
__________ ______ _____ ___ _______ _______ ________ ___ _________ ______ ________ _____.
_____ _____ ______ ___ _____ ___ _______ ________ _________ ________ ________ _________.
_______ _____ ____ _____ ___ _______ _____ __________ _________.
____ ____ _______ _______ _____ __________ ________ ____ _________.
__________ ______ ____ __________ ______.
________ ______ ________ ________ ____ _____.
________ ___ ___ _____ ______ _______ ____ __________ ____.
_________ ________ ___ ________ _________ _______ _______ _________ ____ ______ ___ __________.
_________ ___ ____ __________ ______ ______ _________.
_______ ______ ___ _______ __________ ___ __________ ______ ___ _______.
_____ ______ _______ ______ __________.
_______ _______ __________ ________ ___ ____ _____ ___ _______ ____.
______ _______ _______ _____ _____ _________ ____.
______ __________ _____ ______ _________ _____ ________.
___ __________ ________ __________ _____ ____ ___ _______.
___ __________ ________ _________ __________.
_______ ___ ____ ______ _________.
________ _________ ________ ______ ____ __________ _________ _________ __________ ________ _________.
________ ______ ________ ______ ______ _____ _______ ________ ___ ________.
________ ___ ____ ________ __________ __________ _____ ____ ______ _______ ________ _______.
___ ________ _______ __________ ____ _________ ___ _______ __________.
______ ___ _________ _____ ___ _________ ________ ______ ____ ____.
____ ___ ____ _____ ________ __________ _______ ______ ______ ________ ______ __________.
_______ _________ _______ ______ __________ ______.
__________ ______ ___ ___ ____.
___ ______ ______ _____ _________ ________ ____ ____ _____ __________.
_______ ___ ___ _____ ____ _______.
________ ______ ___ __________ __________ ________ __________ _________ ______ __________ __________.
______ __________ _____ __________ _________ _________ ________ _____ ________.
______ __________ __________ ______ ___ _______.
______ ___ ________ ___ ____ ____ __________.
________ __________ ___ __________ ____ _________ _______ ___ _________.
__________ _____ ______ ____ _______ ______ __________ _________ __________ _____ _______ ______.
______ _________ _______ _______ __________.
_____ _____ ____ ________ ___ ________ ________ ________ __________ _____ __________ ______.
__________ _____ __________ _______ _______ _____ ________.
__________ _______ _______ ________ ___.
__________ ______ _____ _____ __________ __________ _____ ________ _______ _____.
______ _____ ______ _______ ______.
___ ______ _______ ______ _________ _______ _______ ________ ______ ______.
____ __________ ___ ________ ______.
_______ __________ ________ ______ ___ _______ _________ ________ _________.
____ ____ ___ _____ ___ ______ ___ ___ _________ ______ ________ ___.
_____ _______ _______ ________ ___ ____ __________ _____.
_______ ___ ________ ________ _______.
_________ ______ ____ _________ _____ __________ ____.
________ _______ ______ _____ ______ ________ ________ ______ _____.
______ _______ ___ __________ ________ __________ _________ ______ ____ ________ __________ ________.
____ _____ ____ ___ __________ ________.
________ __________ __________ ________ ___ ___ ___ ________ _____ _____.
__________ ________ __________ _____ _________.
_________ ________ ________ ______ __________ _________ _______ __________ _______ ________.
______ ______ _______ _______ ______ _____ ____ _________.
________ _________ ____ _____ __________ ____ _________.
________ _____ _________ _____ ______ _________.
_____ _________ ________ ________ ________ _________ ____ ____ _________.
___ _______ __________ ________ _______ _________ ______ _____ ________ _____.
_______ ___ ________ ______ ___ ________ ___ ______ __________ _________ __________.
_____ ______ ____ ________ __________ _______ ____.
__________ _________ ___ ________ _____ _________ _____ _____.
__________ ____ _____ ___ _______ ______.
_________ ___ _______ _________ ________ ____ ___ ________ _____ ___ _________.
______ __________ _______ ________ ______ __________.
___ __________ __________ __________ ___ ___.
_________ ____ ________ __________ ____.
_______ ____ ____ ___ _________ _________ __________ ___.
____ ______ _______ _______ ___ __________ _________ _____ __________ __________ ______.
____ _________ _______ _______ __________ ____ _______.
______ ________ ______ _____ ______ ____ __________ _____ __________ _________.
_____ _________ ___ ________ ___ _____ ______.
________ _______ _____ ____ __________ ______ ___ ___ _________ _________ ___ _______.
________ ______ ________ _________ __________ _____ ________ __________ _________ __________ ____.
_______ __________ _______ _________ ________ _________ ___ ____ _____ _____ _______ ______.
___ _______ __________.
Get Full Answer on WhatsApp
Related Questions
Find the image of the semi-infinite strip when
. Sketch the strip and its image.
Expand in a Laurent series valid for
i) ii)
then show that there exists a real R>0 such that
for
Find the points where the function is not analytic.
Evaluate where e is the eight like figure shown in Fig. 1.
22-1 Prove that the linear fractional transformation maps the circle
into itself. Also prove that
is conformal in
is entire such that
in C then show that
has the form
where
, b are constants with Re
=0.
IGNOU NEWS
Assignment Submission Last Date Extended Till 30 June 2026 Click Here★★★IGNOU June 2026 TEE Date Sheet Released Click Here★★★