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 points where the function is not analytic.
Find all solutions to the equation sin z=5.
5.a) Find the constant c such that can be extended to be analytic at z =1 , when n∈ N is fixed.
Find the radius of convergence of the following series.
i) ii)
Find all the singularities of the function
Find the maximum modulus on the closed circular region defined by
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.
Find the zeros and singularities of the function Also find the residue at the poles.
IGNOU NEWS
Assignment Submission Last Date Extended Till 30 June 2026 Click Here★★★IGNOU June 2026 TEE Date Sheet Released Click Here★★★