Page 1
COVERED TOPIC IN THIS FILE
Analysis V
Review of complex plane, sequences and series, connected sets and polygonally connected sets in the
complex plane, stereographic projection, analytic polynomials, power series, analytic functions, Cauchy-
Riemann equations, functions e
z
, sinz and cosz.
Reference:
[1]: Chapter 1, Chapter 2, Chapter 3.
Line integrals and their properties, closed curve theorem for entire functions, Cauchy integral formula and
T ayl or exp an sions for e nt ire f uncti ons, L iouv ill e’s t heor em an d the fu nd amen ta l theor em of al g eb r a.
Reference:
[1]: Chapter 4, Chapter 5.
Power series representation for functions analytic in a disc, analyticity in an arbitrary open set, uniqueness
theorem, definitions and examples of conformal mappings, bilinear transformations.
Reference:
[1]: Chapter 6 (Section 6.1-6.2, 6.3 (up to theorem 6.9), Chapter 9 (Section 9.2, 9.7-9.8, 9.9 (statement
only), 9.10, 9.11 (with examples), 9.13), Chapter 13 (Sections 13.1, 13.2 (up to theorem 13.11 including
examples)).
REFERENCES:
1. Joseph Bak and Donald J. Newman, Complex analysis (2
nd
Edition), Undergraduate Texts in
Mathematics, Springer-Verlag New York, Inc., New York, 1997.
Review
Sequence & Series of complex numbers
First consider the useful facts
if z x iy x , y
then Re ( z ) x , Im ( z ) y
z ¯ x iy , | z | v x
y
z z ¯ | z |
also , x z z ¯
y z z ¯
| Re ( z ) | , | Im ( z ) | | z | | Re ( z ) | | Im ( z ) |
T he ? in -equality : If z 1 & z 2 are arbitrary complex no., then
|z 1+z 2 | | z 1|+|z 2 | … …… . ( i)
Proof : | z
z
|
( z
z
) ( z
z
)
¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯
( z
z
) ( z
¯ z
¯ )
z
z
¯ z
z
¯ z
z
¯ z
z
¯
| z
|
z
z
¯ z
z
¯
¯ ¯ ¯ ¯ ¯ ¯
| z
|
| z
|
Re ( z
z
¯ ) | z
|
| z
|
| z
z
¯ | | z
|
}
| z
|
| z
| | z
¯ | | z
|
| z
|
| z
| | z
| | z
|
( | z
| | z
| )
Taking positive sq. root yields the desired inequality.
Note : |z 1| = |(z 1+z 2 ) ( z 2 ) | | z 1+z 2 | | z 2|
=|z 1+z 2|+|z 2|
? | z 1+z 2 | = | z 1 | | z 2|
?ly | z 1 z 2 | = | z 1 | | z 2|
Note equality occur in (i)
Re ( z
z
¯ ) | z
z
¯ | | z
| | z
|
z
z
¯ is a r ea l no . , or eq uiv ale nt ly
r g ( z
z
¯ ) i . e . r g ( z
) r g ( z
¯ )
r g ( z
) r g ( z
¯ ) g r ( z
¯ ) r g ( z
)
( z
z
¯ | z
|
)
r ea l r g ( z
z
¯ ) r g ( z
) r g ( z
¯ )
L . H . S v ( x
y
)
v ( x
y
)
x
y
??
| ?? |
Page 2
COVERED TOPIC IN THIS FILE
Analysis V
Review of complex plane, sequences and series, connected sets and polygonally connected sets in the
complex plane, stereographic projection, analytic polynomials, power series, analytic functions, Cauchy-
Riemann equations, functions e
z
, sinz and cosz.
Reference:
[1]: Chapter 1, Chapter 2, Chapter 3.
Line integrals and their properties, closed curve theorem for entire functions, Cauchy integral formula and
T ayl or exp an sions for e nt ire f uncti ons, L iouv ill e’s t heor em an d the fu nd amen ta l theor em of al g eb r a.
Reference:
[1]: Chapter 4, Chapter 5.
Power series representation for functions analytic in a disc, analyticity in an arbitrary open set, uniqueness
theorem, definitions and examples of conformal mappings, bilinear transformations.
Reference:
[1]: Chapter 6 (Section 6.1-6.2, 6.3 (up to theorem 6.9), Chapter 9 (Section 9.2, 9.7-9.8, 9.9 (statement
only), 9.10, 9.11 (with examples), 9.13), Chapter 13 (Sections 13.1, 13.2 (up to theorem 13.11 including
examples)).
REFERENCES:
1. Joseph Bak and Donald J. Newman, Complex analysis (2
nd
Edition), Undergraduate Texts in
Mathematics, Springer-Verlag New York, Inc., New York, 1997.
Review
Sequence & Series of complex numbers
First consider the useful facts
if z x iy x , y
then Re ( z ) x , Im ( z ) y
z ¯ x iy , | z | v x
y
z z ¯ | z |
also , x z z ¯
y z z ¯
| Re ( z ) | , | Im ( z ) | | z | | Re ( z ) | | Im ( z ) |
T he ? in -equality : If z 1 & z 2 are arbitrary complex no., then
|z 1+z 2 | | z 1|+|z 2 | … …… . ( i)
Proof : | z
z
|
( z
z
) ( z
z
)
¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯
( z
z
) ( z
¯ z
¯ )
z
z
¯ z
z
¯ z
z
¯ z
z
¯
| z
|
z
z
¯ z
z
¯
¯ ¯ ¯ ¯ ¯ ¯
| z
|
| z
|
Re ( z
z
¯ ) | z
|
| z
|
| z
z
¯ | | z
|
}
| z
|
| z
| | z
¯ | | z
|
| z
|
| z
| | z
| | z
|
( | z
| | z
| )
Taking positive sq. root yields the desired inequality.
Note : |z 1| = |(z 1+z 2 ) ( z 2 ) | | z 1+z 2 | | z 2|
=|z 1+z 2|+|z 2|
? | z 1+z 2 | = | z 1 | | z 2|
?ly | z 1 z 2 | = | z 1 | | z 2|
Note equality occur in (i)
Re ( z
z
¯ ) | z
z
¯ | | z
| | z
|
z
z
¯ is a r ea l no . , or eq uiv ale nt ly
r g ( z
z
¯ ) i . e . r g ( z
) r g ( z
¯ )
r g ( z
) r g ( z
¯ ) g r ( z
¯ ) r g ( z
)
( z
z
¯ | z
|
)
r ea l r g ( z
z
¯ ) r g ( z
) r g ( z
¯ )
L . H . S v ( x
y
)
v ( x
y
)
x
y
??
| ?? |
Note | z
| | z |
, | x
y
ixy | | x iy |
Def
n
A seq. {z n} of complex no. is said to converge to a complex no. z if the seq. {|z n z|}
of real no. converge to 0
i.e. z
z if | z
z | as n
z
z iff Re ( z
) Re ( z ) Im ( z
) Im ( z )
Proof forward Part : Assume that z n z then | z n z |
But |Re(z n ) Re( z ) | | Re ( z
z ) | | z
z |
and | Im ( z
) Im ( z ) | | Im ( z
z ) | | z
z |
Implies * Re ( z
) + * Im ( z
) + cg s to Re ( z ) , Im ( z ) r esp ect ively .
Converse Part : Assume that Re(z n ) Re ( z ) I m( z n ) Im ( z )
Consider,
| z
z | | ( x
i y
) ( x iy ) | w her e , z
x
i y
z x iy
= | ( x
x ) i ( y
y ) |
| x
x | | i ( y
y ) |
= | x
x | | y
y | ( | i ( y
y ) | | i | | y
y | )
= | Re ( z
) Re ( z ) | | Im ( z
) Im ( z ) |
as n
|z n z | as n imp lies z n z
Note: {z n + can ’t conv erg e t o m or e t han one limit .
If exists is unique.
( if possible z n z an d z n z
*
[TS : z = z
*
]
| z z
| | ( z
z
) ( z
z ) |
| z
z
| | z
z | by ? in eq ualit y
as n
? | z z
| z z
e . g . ( i ) |
n
n i
| | i
n i
| v n
2
n
n i
3
( ii ) |
n
( n ) i
n
i | n
| i | v n
8
( n ) i
n
9 i
( iii ) < z
> if | z |
| z
| | z
| | z |
as n ( | z | )
Def
n
A seq. {z n} of complex no. is called a Cauchy seq. if for each ?? > 0, there exists (N=N( ?? )) an integer N s.t.
| z
z
n , m
| z
z
| as n , m
Cauchy Criteria for convergence in complex plane
{z n} converges iff {z n} is a Cauchy seq.
Forward part : Assume that z n z then
Re(z n ) Re ( z ) Im ( z n ) Im ( z )
Proved before
Here {Re(z n)} & {Im(z n)} both are Cauchy seq. being convergent seq. of real no.
? | Re ( z
) Re ( z
) | as n , m
i . e . | Re ( z
z
) | as n , m
ly, | Im ( z
z
) | as n , m
But | z
z
| | Re ( z
z
) | | Im ( z
z
) |
as n , m
? * z n} is a cauchy seq.
Converse part: suppose {z m} is a Cauchy seq. , then using
| Re ( z
z
) | , | Im ( z
z
) | | z
z
| m , n
imp lies b oth * Re ( z
) + * Im ( z
) + a r e cauch y seq . of r ea l no . hence b oth conv erg es ,
Page 3
COVERED TOPIC IN THIS FILE
Analysis V
Review of complex plane, sequences and series, connected sets and polygonally connected sets in the
complex plane, stereographic projection, analytic polynomials, power series, analytic functions, Cauchy-
Riemann equations, functions e
z
, sinz and cosz.
Reference:
[1]: Chapter 1, Chapter 2, Chapter 3.
Line integrals and their properties, closed curve theorem for entire functions, Cauchy integral formula and
T ayl or exp an sions for e nt ire f uncti ons, L iouv ill e’s t heor em an d the fu nd amen ta l theor em of al g eb r a.
Reference:
[1]: Chapter 4, Chapter 5.
Power series representation for functions analytic in a disc, analyticity in an arbitrary open set, uniqueness
theorem, definitions and examples of conformal mappings, bilinear transformations.
Reference:
[1]: Chapter 6 (Section 6.1-6.2, 6.3 (up to theorem 6.9), Chapter 9 (Section 9.2, 9.7-9.8, 9.9 (statement
only), 9.10, 9.11 (with examples), 9.13), Chapter 13 (Sections 13.1, 13.2 (up to theorem 13.11 including
examples)).
REFERENCES:
1. Joseph Bak and Donald J. Newman, Complex analysis (2
nd
Edition), Undergraduate Texts in
Mathematics, Springer-Verlag New York, Inc., New York, 1997.
Review
Sequence & Series of complex numbers
First consider the useful facts
if z x iy x , y
then Re ( z ) x , Im ( z ) y
z ¯ x iy , | z | v x
y
z z ¯ | z |
also , x z z ¯
y z z ¯
| Re ( z ) | , | Im ( z ) | | z | | Re ( z ) | | Im ( z ) |
T he ? in -equality : If z 1 & z 2 are arbitrary complex no., then
|z 1+z 2 | | z 1|+|z 2 | … …… . ( i)
Proof : | z
z
|
( z
z
) ( z
z
)
¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯
( z
z
) ( z
¯ z
¯ )
z
z
¯ z
z
¯ z
z
¯ z
z
¯
| z
|
z
z
¯ z
z
¯
¯ ¯ ¯ ¯ ¯ ¯
| z
|
| z
|
Re ( z
z
¯ ) | z
|
| z
|
| z
z
¯ | | z
|
}
| z
|
| z
| | z
¯ | | z
|
| z
|
| z
| | z
| | z
|
( | z
| | z
| )
Taking positive sq. root yields the desired inequality.
Note : |z 1| = |(z 1+z 2 ) ( z 2 ) | | z 1+z 2 | | z 2|
=|z 1+z 2|+|z 2|
? | z 1+z 2 | = | z 1 | | z 2|
?ly | z 1 z 2 | = | z 1 | | z 2|
Note equality occur in (i)
Re ( z
z
¯ ) | z
z
¯ | | z
| | z
|
z
z
¯ is a r ea l no . , or eq uiv ale nt ly
r g ( z
z
¯ ) i . e . r g ( z
) r g ( z
¯ )
r g ( z
) r g ( z
¯ ) g r ( z
¯ ) r g ( z
)
( z
z
¯ | z
|
)
r ea l r g ( z
z
¯ ) r g ( z
) r g ( z
¯ )
L . H . S v ( x
y
)
v ( x
y
)
x
y
??
| ?? |
Note | z
| | z |
, | x
y
ixy | | x iy |
Def
n
A seq. {z n} of complex no. is said to converge to a complex no. z if the seq. {|z n z|}
of real no. converge to 0
i.e. z
z if | z
z | as n
z
z iff Re ( z
) Re ( z ) Im ( z
) Im ( z )
Proof forward Part : Assume that z n z then | z n z |
But |Re(z n ) Re( z ) | | Re ( z
z ) | | z
z |
and | Im ( z
) Im ( z ) | | Im ( z
z ) | | z
z |
Implies * Re ( z
) + * Im ( z
) + cg s to Re ( z ) , Im ( z ) r esp ect ively .
Converse Part : Assume that Re(z n ) Re ( z ) I m( z n ) Im ( z )
Consider,
| z
z | | ( x
i y
) ( x iy ) | w her e , z
x
i y
z x iy
= | ( x
x ) i ( y
y ) |
| x
x | | i ( y
y ) |
= | x
x | | y
y | ( | i ( y
y ) | | i | | y
y | )
= | Re ( z
) Re ( z ) | | Im ( z
) Im ( z ) |
as n
|z n z | as n imp lies z n z
Note: {z n + can ’t conv erg e t o m or e t han one limit .
If exists is unique.
( if possible z n z an d z n z
*
[TS : z = z
*
]
| z z
| | ( z
z
) ( z
z ) |
| z
z
| | z
z | by ? in eq ualit y
as n
? | z z
| z z
e . g . ( i ) |
n
n i
| | i
n i
| v n
2
n
n i
3
( ii ) |
n
( n ) i
n
i | n
| i | v n
8
( n ) i
n
9 i
( iii ) < z
> if | z |
| z
| | z
| | z |
as n ( | z | )
Def
n
A seq. {z n} of complex no. is called a Cauchy seq. if for each ?? > 0, there exists (N=N( ?? )) an integer N s.t.
| z
z
n , m
| z
z
| as n , m
Cauchy Criteria for convergence in complex plane
{z n} converges iff {z n} is a Cauchy seq.
Forward part : Assume that z n z then
Re(z n ) Re ( z ) Im ( z n ) Im ( z )
Proved before
Here {Re(z n)} & {Im(z n)} both are Cauchy seq. being convergent seq. of real no.
? | Re ( z
) Re ( z
) | as n , m
i . e . | Re ( z
z
) | as n , m
ly, | Im ( z
z
) | as n , m
But | z
z
| | Re ( z
z
) | | Im ( z
z
) |
as n , m
? * z n} is a cauchy seq.
Converse part: suppose {z m} is a Cauchy seq. , then using
| Re ( z
z
) | , | Im ( z
z
) | | z
z
| m , n
imp lies b oth * Re ( z
) + * Im ( z
) + a r e cauch y seq . of r ea l no . hence b oth conv erg es ,
say Re ( z
) x Im ( z
) y then
| z
( x iy ) | | ( x
x ) i ( y
y ) | , z
x
i y
| x
x | | y
y |
= | Re ( ( z
) x ) | | Im ( z
) y |
as n
implies {z n} converges
Def
n
The infinite series ? z
of com p lex nos . is sai d to be conv erg e if the seq . * S
+ ,
S
z
z
of p ariti al sum S is conv erg en t .
f r om the Cauc hy c r it erion, i. e. “ a se q . is co nv. i f f it is a C auch y seq ” , w e se e t hat
? z
conv erg es iff * S
+ is a cauch y seq .
For each ?? , ? s. t .
| S
S
| | ? z
| or
| S
S
| | ? z
| f or all m , n
f r om this .
using z
S
S
f or n / it f oll ow s that conv erg en ce of ? z
imp lies lim
z
( i . e . a ne ccessary con d
f or the series ? z
to conv erg e is that z
as n
Remark s in the case of seq uence we hav e , ? z
conv erg es to z iff ? Re ( z
)
conv erg es to Re ( z ) ? Im ( z
)
conv erg e to Im ( z ) .
suf f ici en t con d
f or cg ces of ? z
is that ? | z
|
cg s .
i.e. absolutely convergent series is cgt.
also , as w it h r ea l series , we say a series ? z
is sai d to be ab solu te ly conv erg en t if the
series ? | z
|
, of ve r ea l no . is conv erg en t . F uther using the f act that
| S
S
| | ? z
| ? | z
|
, n m
I mplie s that “ ever y ab solu te ly conv er g en t se r ie s is conver g en t. ” C onver se i s not tru e e . g .
?
i
n
i
( )
conv erg es ,
sin ce , |
i
n
i
| v n
and ?
v n
cg s
( ?
n
*
? ?
i
n
i
cg s ab solu te ly and hence cg t .
Gene r ali z ed C auch y ’s n
th
root test : let ? z
be a series of com p lex te r m suc h that lim
S up | z
|
Page 4
COVERED TOPIC IN THIS FILE
Analysis V
Review of complex plane, sequences and series, connected sets and polygonally connected sets in the
complex plane, stereographic projection, analytic polynomials, power series, analytic functions, Cauchy-
Riemann equations, functions e
z
, sinz and cosz.
Reference:
[1]: Chapter 1, Chapter 2, Chapter 3.
Line integrals and their properties, closed curve theorem for entire functions, Cauchy integral formula and
T ayl or exp an sions for e nt ire f uncti ons, L iouv ill e’s t heor em an d the fu nd amen ta l theor em of al g eb r a.
Reference:
[1]: Chapter 4, Chapter 5.
Power series representation for functions analytic in a disc, analyticity in an arbitrary open set, uniqueness
theorem, definitions and examples of conformal mappings, bilinear transformations.
Reference:
[1]: Chapter 6 (Section 6.1-6.2, 6.3 (up to theorem 6.9), Chapter 9 (Section 9.2, 9.7-9.8, 9.9 (statement
only), 9.10, 9.11 (with examples), 9.13), Chapter 13 (Sections 13.1, 13.2 (up to theorem 13.11 including
examples)).
REFERENCES:
1. Joseph Bak and Donald J. Newman, Complex analysis (2
nd
Edition), Undergraduate Texts in
Mathematics, Springer-Verlag New York, Inc., New York, 1997.
Review
Sequence & Series of complex numbers
First consider the useful facts
if z x iy x , y
then Re ( z ) x , Im ( z ) y
z ¯ x iy , | z | v x
y
z z ¯ | z |
also , x z z ¯
y z z ¯
| Re ( z ) | , | Im ( z ) | | z | | Re ( z ) | | Im ( z ) |
T he ? in -equality : If z 1 & z 2 are arbitrary complex no., then
|z 1+z 2 | | z 1|+|z 2 | … …… . ( i)
Proof : | z
z
|
( z
z
) ( z
z
)
¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯
( z
z
) ( z
¯ z
¯ )
z
z
¯ z
z
¯ z
z
¯ z
z
¯
| z
|
z
z
¯ z
z
¯
¯ ¯ ¯ ¯ ¯ ¯
| z
|
| z
|
Re ( z
z
¯ ) | z
|
| z
|
| z
z
¯ | | z
|
}
| z
|
| z
| | z
¯ | | z
|
| z
|
| z
| | z
| | z
|
( | z
| | z
| )
Taking positive sq. root yields the desired inequality.
Note : |z 1| = |(z 1+z 2 ) ( z 2 ) | | z 1+z 2 | | z 2|
=|z 1+z 2|+|z 2|
? | z 1+z 2 | = | z 1 | | z 2|
?ly | z 1 z 2 | = | z 1 | | z 2|
Note equality occur in (i)
Re ( z
z
¯ ) | z
z
¯ | | z
| | z
|
z
z
¯ is a r ea l no . , or eq uiv ale nt ly
r g ( z
z
¯ ) i . e . r g ( z
) r g ( z
¯ )
r g ( z
) r g ( z
¯ ) g r ( z
¯ ) r g ( z
)
( z
z
¯ | z
|
)
r ea l r g ( z
z
¯ ) r g ( z
) r g ( z
¯ )
L . H . S v ( x
y
)
v ( x
y
)
x
y
??
| ?? |
Note | z
| | z |
, | x
y
ixy | | x iy |
Def
n
A seq. {z n} of complex no. is said to converge to a complex no. z if the seq. {|z n z|}
of real no. converge to 0
i.e. z
z if | z
z | as n
z
z iff Re ( z
) Re ( z ) Im ( z
) Im ( z )
Proof forward Part : Assume that z n z then | z n z |
But |Re(z n ) Re( z ) | | Re ( z
z ) | | z
z |
and | Im ( z
) Im ( z ) | | Im ( z
z ) | | z
z |
Implies * Re ( z
) + * Im ( z
) + cg s to Re ( z ) , Im ( z ) r esp ect ively .
Converse Part : Assume that Re(z n ) Re ( z ) I m( z n ) Im ( z )
Consider,
| z
z | | ( x
i y
) ( x iy ) | w her e , z
x
i y
z x iy
= | ( x
x ) i ( y
y ) |
| x
x | | i ( y
y ) |
= | x
x | | y
y | ( | i ( y
y ) | | i | | y
y | )
= | Re ( z
) Re ( z ) | | Im ( z
) Im ( z ) |
as n
|z n z | as n imp lies z n z
Note: {z n + can ’t conv erg e t o m or e t han one limit .
If exists is unique.
( if possible z n z an d z n z
*
[TS : z = z
*
]
| z z
| | ( z
z
) ( z
z ) |
| z
z
| | z
z | by ? in eq ualit y
as n
? | z z
| z z
e . g . ( i ) |
n
n i
| | i
n i
| v n
2
n
n i
3
( ii ) |
n
( n ) i
n
i | n
| i | v n
8
( n ) i
n
9 i
( iii ) < z
> if | z |
| z
| | z
| | z |
as n ( | z | )
Def
n
A seq. {z n} of complex no. is called a Cauchy seq. if for each ?? > 0, there exists (N=N( ?? )) an integer N s.t.
| z
z
n , m
| z
z
| as n , m
Cauchy Criteria for convergence in complex plane
{z n} converges iff {z n} is a Cauchy seq.
Forward part : Assume that z n z then
Re(z n ) Re ( z ) Im ( z n ) Im ( z )
Proved before
Here {Re(z n)} & {Im(z n)} both are Cauchy seq. being convergent seq. of real no.
? | Re ( z
) Re ( z
) | as n , m
i . e . | Re ( z
z
) | as n , m
ly, | Im ( z
z
) | as n , m
But | z
z
| | Re ( z
z
) | | Im ( z
z
) |
as n , m
? * z n} is a cauchy seq.
Converse part: suppose {z m} is a Cauchy seq. , then using
| Re ( z
z
) | , | Im ( z
z
) | | z
z
| m , n
imp lies b oth * Re ( z
) + * Im ( z
) + a r e cauch y seq . of r ea l no . hence b oth conv erg es ,
say Re ( z
) x Im ( z
) y then
| z
( x iy ) | | ( x
x ) i ( y
y ) | , z
x
i y
| x
x | | y
y |
= | Re ( ( z
) x ) | | Im ( z
) y |
as n
implies {z n} converges
Def
n
The infinite series ? z
of com p lex nos . is sai d to be conv erg e if the seq . * S
+ ,
S
z
z
of p ariti al sum S is conv erg en t .
f r om the Cauc hy c r it erion, i. e. “ a se q . is co nv. i f f it is a C auch y seq ” , w e se e t hat
? z
conv erg es iff * S
+ is a cauch y seq .
For each ?? , ? s. t .
| S
S
| | ? z
| or
| S
S
| | ? z
| f or all m , n
f r om this .
using z
S
S
f or n / it f oll ow s that conv erg en ce of ? z
imp lies lim
z
( i . e . a ne ccessary con d
f or the series ? z
to conv erg e is that z
as n
Remark s in the case of seq uence we hav e , ? z
conv erg es to z iff ? Re ( z
)
conv erg es to Re ( z ) ? Im ( z
)
conv erg e to Im ( z ) .
suf f ici en t con d
f or cg ces of ? z
is that ? | z
|
cg s .
i.e. absolutely convergent series is cgt.
also , as w it h r ea l series , we say a series ? z
is sai d to be ab solu te ly conv erg en t if the
series ? | z
|
, of ve r ea l no . is conv erg en t . F uther using the f act that
| S
S
| | ? z
| ? | z
|
, n m
I mplie s that “ ever y ab solu te ly conv er g en t se r ie s is conver g en t. ” C onver se i s not tru e e . g .
?
i
n
i
( )
conv erg es ,
sin ce , |
i
n
i
| v n
and ?
v n
cg s
( ?
n
*
? ?
i
n
i
cg s ab solu te ly and hence cg t .
Gene r ali z ed C auch y ’s n
th
root test : let ? z
be a series of com p lex te r m suc h that lim
S up | z
|
( a ) if , then series ? z
cg s ab s .
( b ) if , then series ? z
d iv .
( c ) if , the series may or may not conv .
Gene r ali sed D’ lembe r t Ra ti o te st
if lim
S up |
z
z
| L lim
|
z
z
| , then
( a ) if L , then the series cg s ab solu te ly
( b ) if l , the series d iv .
( c ) if l L , no conclu sion .
com p lex an aly sis ( ak ew mann )
T op olo g y of the com p lex p lane
f or
z x iy x , y z
x
i y
x
, y
} z , z
, the com p lex p lane .
| z z
| v ( x x
)
( y y
)
1.5 Definition
( i ) D ( z
r ) * z | z z
| r + is called an op en d isc of r ad ius
r cen te r ed at z
, also called nb hd of z
.
( ii ) C irc le c ( z
, r ) * z | z z
| +
= circle with center z 0 , radius r.
( iii ) subset S is called op en ( in ) . If f or ever y z
S , there exists
r . It mean s that som e d isc arr ound z
lies en ti r ely in S . f or in sta nce , the
in te r ior of a cir cle ( * z | z z
| r + ) , the en ti r e com p lex p lane , half p lane ( Re ( z ) , Im ( z )
, Re ( z ) ) ect . are op en set s . n op en d isc is an op en set .
( iv ) set S is s . t . b op en iff f or ea ch z S , ? s . that D ( z , ) S .
ote S * z | z | + is not op en .
ote S * x x ( , ) + is op en in , b ut not in .
( v ) set S is called clo sed if its com p lement
S * z z S + is op e n in
e . g . * z | z z
| r + is clo sed .
( vi ) S
S s * z z S +
( v ii ) set is clo sed iff < z
> and < z
> z
z
( v ii i ) S * z z is a b d r y pt of S +
C ol l
of p oint s w hos e ne igh b our hoo d hav e a non emp ty in te r sect ion
w it h b oth S and S
( n b hd )
( ix ) S
¯
S S cl osu r e of S
( x ) S is b d r y iff S D ( , r ) f or som e r .
( xi ) S is com p a ct iff S is clo sed b d d ( in )
( xii ) set S is sai d to be d isconn ece d if there exists two d isj oint op en se t s . t .
S , , S , S
( xiii ) S is s. t . b . d isconn ect ed iff S is a union of two non emp ty d isj oit op en subse ts .
( xiv ) S is called conne cted if it is not d isconn ect ed .
in other w or d s , S is conne cted if and only if , ea ch p ai r of p oint s z
, z
of S can be
conne cted by an arc ly in g in S , as sho w n in the f oll
f ig .
the fu n
, , - d efine d by
Page 5
COVERED TOPIC IN THIS FILE
Analysis V
Review of complex plane, sequences and series, connected sets and polygonally connected sets in the
complex plane, stereographic projection, analytic polynomials, power series, analytic functions, Cauchy-
Riemann equations, functions e
z
, sinz and cosz.
Reference:
[1]: Chapter 1, Chapter 2, Chapter 3.
Line integrals and their properties, closed curve theorem for entire functions, Cauchy integral formula and
T ayl or exp an sions for e nt ire f uncti ons, L iouv ill e’s t heor em an d the fu nd amen ta l theor em of al g eb r a.
Reference:
[1]: Chapter 4, Chapter 5.
Power series representation for functions analytic in a disc, analyticity in an arbitrary open set, uniqueness
theorem, definitions and examples of conformal mappings, bilinear transformations.
Reference:
[1]: Chapter 6 (Section 6.1-6.2, 6.3 (up to theorem 6.9), Chapter 9 (Section 9.2, 9.7-9.8, 9.9 (statement
only), 9.10, 9.11 (with examples), 9.13), Chapter 13 (Sections 13.1, 13.2 (up to theorem 13.11 including
examples)).
REFERENCES:
1. Joseph Bak and Donald J. Newman, Complex analysis (2
nd
Edition), Undergraduate Texts in
Mathematics, Springer-Verlag New York, Inc., New York, 1997.
Review
Sequence & Series of complex numbers
First consider the useful facts
if z x iy x , y
then Re ( z ) x , Im ( z ) y
z ¯ x iy , | z | v x
y
z z ¯ | z |
also , x z z ¯
y z z ¯
| Re ( z ) | , | Im ( z ) | | z | | Re ( z ) | | Im ( z ) |
T he ? in -equality : If z 1 & z 2 are arbitrary complex no., then
|z 1+z 2 | | z 1|+|z 2 | … …… . ( i)
Proof : | z
z
|
( z
z
) ( z
z
)
¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯
( z
z
) ( z
¯ z
¯ )
z
z
¯ z
z
¯ z
z
¯ z
z
¯
| z
|
z
z
¯ z
z
¯
¯ ¯ ¯ ¯ ¯ ¯
| z
|
| z
|
Re ( z
z
¯ ) | z
|
| z
|
| z
z
¯ | | z
|
}
| z
|
| z
| | z
¯ | | z
|
| z
|
| z
| | z
| | z
|
( | z
| | z
| )
Taking positive sq. root yields the desired inequality.
Note : |z 1| = |(z 1+z 2 ) ( z 2 ) | | z 1+z 2 | | z 2|
=|z 1+z 2|+|z 2|
? | z 1+z 2 | = | z 1 | | z 2|
?ly | z 1 z 2 | = | z 1 | | z 2|
Note equality occur in (i)
Re ( z
z
¯ ) | z
z
¯ | | z
| | z
|
z
z
¯ is a r ea l no . , or eq uiv ale nt ly
r g ( z
z
¯ ) i . e . r g ( z
) r g ( z
¯ )
r g ( z
) r g ( z
¯ ) g r ( z
¯ ) r g ( z
)
( z
z
¯ | z
|
)
r ea l r g ( z
z
¯ ) r g ( z
) r g ( z
¯ )
L . H . S v ( x
y
)
v ( x
y
)
x
y
??
| ?? |
Note | z
| | z |
, | x
y
ixy | | x iy |
Def
n
A seq. {z n} of complex no. is said to converge to a complex no. z if the seq. {|z n z|}
of real no. converge to 0
i.e. z
z if | z
z | as n
z
z iff Re ( z
) Re ( z ) Im ( z
) Im ( z )
Proof forward Part : Assume that z n z then | z n z |
But |Re(z n ) Re( z ) | | Re ( z
z ) | | z
z |
and | Im ( z
) Im ( z ) | | Im ( z
z ) | | z
z |
Implies * Re ( z
) + * Im ( z
) + cg s to Re ( z ) , Im ( z ) r esp ect ively .
Converse Part : Assume that Re(z n ) Re ( z ) I m( z n ) Im ( z )
Consider,
| z
z | | ( x
i y
) ( x iy ) | w her e , z
x
i y
z x iy
= | ( x
x ) i ( y
y ) |
| x
x | | i ( y
y ) |
= | x
x | | y
y | ( | i ( y
y ) | | i | | y
y | )
= | Re ( z
) Re ( z ) | | Im ( z
) Im ( z ) |
as n
|z n z | as n imp lies z n z
Note: {z n + can ’t conv erg e t o m or e t han one limit .
If exists is unique.
( if possible z n z an d z n z
*
[TS : z = z
*
]
| z z
| | ( z
z
) ( z
z ) |
| z
z
| | z
z | by ? in eq ualit y
as n
? | z z
| z z
e . g . ( i ) |
n
n i
| | i
n i
| v n
2
n
n i
3
( ii ) |
n
( n ) i
n
i | n
| i | v n
8
( n ) i
n
9 i
( iii ) < z
> if | z |
| z
| | z
| | z |
as n ( | z | )
Def
n
A seq. {z n} of complex no. is called a Cauchy seq. if for each ?? > 0, there exists (N=N( ?? )) an integer N s.t.
| z
z
n , m
| z
z
| as n , m
Cauchy Criteria for convergence in complex plane
{z n} converges iff {z n} is a Cauchy seq.
Forward part : Assume that z n z then
Re(z n ) Re ( z ) Im ( z n ) Im ( z )
Proved before
Here {Re(z n)} & {Im(z n)} both are Cauchy seq. being convergent seq. of real no.
? | Re ( z
) Re ( z
) | as n , m
i . e . | Re ( z
z
) | as n , m
ly, | Im ( z
z
) | as n , m
But | z
z
| | Re ( z
z
) | | Im ( z
z
) |
as n , m
? * z n} is a cauchy seq.
Converse part: suppose {z m} is a Cauchy seq. , then using
| Re ( z
z
) | , | Im ( z
z
) | | z
z
| m , n
imp lies b oth * Re ( z
) + * Im ( z
) + a r e cauch y seq . of r ea l no . hence b oth conv erg es ,
say Re ( z
) x Im ( z
) y then
| z
( x iy ) | | ( x
x ) i ( y
y ) | , z
x
i y
| x
x | | y
y |
= | Re ( ( z
) x ) | | Im ( z
) y |
as n
implies {z n} converges
Def
n
The infinite series ? z
of com p lex nos . is sai d to be conv erg e if the seq . * S
+ ,
S
z
z
of p ariti al sum S is conv erg en t .
f r om the Cauc hy c r it erion, i. e. “ a se q . is co nv. i f f it is a C auch y seq ” , w e se e t hat
? z
conv erg es iff * S
+ is a cauch y seq .
For each ?? , ? s. t .
| S
S
| | ? z
| or
| S
S
| | ? z
| f or all m , n
f r om this .
using z
S
S
f or n / it f oll ow s that conv erg en ce of ? z
imp lies lim
z
( i . e . a ne ccessary con d
f or the series ? z
to conv erg e is that z
as n
Remark s in the case of seq uence we hav e , ? z
conv erg es to z iff ? Re ( z
)
conv erg es to Re ( z ) ? Im ( z
)
conv erg e to Im ( z ) .
suf f ici en t con d
f or cg ces of ? z
is that ? | z
|
cg s .
i.e. absolutely convergent series is cgt.
also , as w it h r ea l series , we say a series ? z
is sai d to be ab solu te ly conv erg en t if the
series ? | z
|
, of ve r ea l no . is conv erg en t . F uther using the f act that
| S
S
| | ? z
| ? | z
|
, n m
I mplie s that “ ever y ab solu te ly conv er g en t se r ie s is conver g en t. ” C onver se i s not tru e e . g .
?
i
n
i
( )
conv erg es ,
sin ce , |
i
n
i
| v n
and ?
v n
cg s
( ?
n
*
? ?
i
n
i
cg s ab solu te ly and hence cg t .
Gene r ali z ed C auch y ’s n
th
root test : let ? z
be a series of com p lex te r m suc h that lim
S up | z
|
( a ) if , then series ? z
cg s ab s .
( b ) if , then series ? z
d iv .
( c ) if , the series may or may not conv .
Gene r ali sed D’ lembe r t Ra ti o te st
if lim
S up |
z
z
| L lim
|
z
z
| , then
( a ) if L , then the series cg s ab solu te ly
( b ) if l , the series d iv .
( c ) if l L , no conclu sion .
com p lex an aly sis ( ak ew mann )
T op olo g y of the com p lex p lane
f or
z x iy x , y z
x
i y
x
, y
} z , z
, the com p lex p lane .
| z z
| v ( x x
)
( y y
)
1.5 Definition
( i ) D ( z
r ) * z | z z
| r + is called an op en d isc of r ad ius
r cen te r ed at z
, also called nb hd of z
.
( ii ) C irc le c ( z
, r ) * z | z z
| +
= circle with center z 0 , radius r.
( iii ) subset S is called op en ( in ) . If f or ever y z
S , there exists
r . It mean s that som e d isc arr ound z
lies en ti r ely in S . f or in sta nce , the
in te r ior of a cir cle ( * z | z z
| r + ) , the en ti r e com p lex p lane , half p lane ( Re ( z ) , Im ( z )
, Re ( z ) ) ect . are op en set s . n op en d isc is an op en set .
( iv ) set S is s . t . b op en iff f or ea ch z S , ? s . that D ( z , ) S .
ote S * z | z | + is not op en .
ote S * x x ( , ) + is op en in , b ut not in .
( v ) set S is called clo sed if its com p lement
S * z z S + is op e n in
e . g . * z | z z
| r + is clo sed .
( vi ) S
S s * z z S +
( v ii ) set is clo sed iff < z
> and < z
> z
z
( v ii i ) S * z z is a b d r y pt of S +
C ol l
of p oint s w hos e ne igh b our hoo d hav e a non emp ty in te r sect ion
w it h b oth S and S
( n b hd )
( ix ) S
¯
S S cl osu r e of S
( x ) S is b d r y iff S D ( , r ) f or som e r .
( xi ) S is com p a ct iff S is clo sed b d d ( in )
( xii ) set S is sai d to be d isconn ece d if there exists two d isj oint op en se t s . t .
S , , S , S
( xiii ) S is s. t . b . d isconn ect ed iff S is a union of two non emp ty d isj oit op en subse ts .
( xiv ) S is called conne cted if it is not d isconn ect ed .
in other w or d s , S is conne cted if and only if , ea ch p ai r of p oint s z
, z
of S can be
conne cted by an arc ly in g in S , as sho w n in the f oll
f ig .
the fu n
, , - d efine d by
( t ) ( t ) z
z
is calle the line segm en t w it h and p oint s z
z
and d en ote d by
, z
, z
-
? if ( t ) S f or ea ch t , , - , then the line se g ment , z
, z
- , ( w her e z
, z
S )
is sai d to be conta in e in S .
by a p ol y g ona l line f r om z
to z
… . a f in it e union of line segm en ts of the f or m
, z
, z
- , z
, z
- … , z
z
-
.z
and z
are sai d to be p oly g ona lly conne cted /
set S is sai d to be p oly g ona lly conne cted if any two p oint s of S can be conne cte d by
a p oly g ona l line ( b asi cal ly hor iz ona l or v ertica ly ) conta in ed in S .
Fig
For example, any open disc D(z 0; r) is polygonally connected; for if z 1, z 2 D( z 0; r) and
( t) ( t ) z 1+tz 2 , the n for t
| ( t ) z
| | ( t ) z
t z
z
|
| ( t ) z
t z
t z
t z
z
|
= | ( t ) z
z
t z
t z
t z
|
= | ( t ) z
z
( t ) t ( z
z
) |
= | ( t ) ( z
z
) t ( z
z
) |
( t ) | z
z
| t | z
z
|
< ( t ) r tr r
nd so f or ea ch t , , - , r ( t) D( z 0; r)
cir cle is conn ect ed b ut not polygonally connected.
e. g . * z Re( z ) a , a + is a r egion.
set S is ca lled b d d if there is a n r s. t. S * z | z | r +
S et s w hich are closed as well as bdd are called compact.
1.6 Definition
A domain is an open connected set, and is also called a region.
Or An open connected set will be called a region.
Imp. 1.7 Proposition :
A region S is polygonally connected.
i.e. Que 16. any open conneted set is polygonally connected.
Proof Suppose z 0 S ( f i xed ) , w her e S is giv en ope n conn . S et
L et b e t he set of p oi nt s z S w hich can b e p oly g ona lly conne cted to z 0 in S.
i . e . * z S z can be p oly g ona lly conne ct ed to z
+
* z S z ca n
t be p oly g ona lly conne cted to z
+
i.e. B = S\A
? S , wh ere , Since z 0
Claim: B=
First, we show that A is open set.
so let z ( ? z can be p oly . conne cted to z
) … . . ( )
ote z S an d S b ei ng o p en set .
? D( z r ) S for some r>0. Drow figures
We show D ( z r )
So let t D ( z r )
? t is conne cted to z by mean s of a sin g le line segm en t … … . ( )
? t is conne cted to z
by mean s of a p oly g ona l line . using and
i. e. t
i. e. D ( z r )
is op en ( z w as arbi trar y )
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