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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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FAQs on Complex Analysis (Part - 1) Solved Example - Topic-wise Tests & Solved Examples for Mathematics
| 1. What is complex analysis and why is it important in mathematics? |  |
Ans. Complex analysis is a branch of mathematics that deals with the study of functions of complex numbers. It is important because it provides a powerful tool for solving problems in various fields such as physics, engineering, and computer science. Complex analysis helps in understanding the behavior of functions and their properties in the complex plane.
| 2. What are the basic concepts in complex analysis? | |
Ans. The basic concepts in complex analysis include complex numbers, complex functions, analyticity, contour integration, and the Cauchy-Riemann equations. Complex numbers are numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit. Complex functions are functions that operate on complex numbers, and analyticity refers to the property of a function being differentiable in a region. Contour integration involves integrating complex functions along a closed curve, and the Cauchy-Riemann equations relate the real and imaginary parts of an analytic function.
| 3. How is complex analysis used in solving problems in physics? | |
Ans. Complex analysis is extensively used in physics for solving problems related to wave propagation, electrical circuits, quantum mechanics, and fluid dynamics, among others. The use of complex analysis allows physicists to analyze and understand the behavior of physical systems by representing them in the complex plane. Complex analysis techniques, such as contour integration, help in calculating complex integrals and solving differential equations that arise in physics.
| 4. What is the significance of the Cauchy-Riemann equations in complex analysis? | |
Ans. The Cauchy-Riemann equations are fundamental equations in complex analysis that relate the partial derivatives of a complex function to its analyticity. These equations provide necessary and sufficient conditions for a function to be analytic in a region. They form the foundation for many results and techniques in complex analysis, including the computation of complex integrals using contour integration and the development of power series representations for analytic functions.
| 5. Can complex analysis be applied to real-valued functions as well? | |
Ans. Yes, complex analysis can also be applied to real-valued functions. Many real-valued functions can be extended to complex functions by considering them as functions of a complex variable with an imaginary part of zero. By doing so, complex analysis techniques can be used to study the properties and behavior of these real functions in a more general and powerful way. Complex analysis provides a unified framework that encompasses both real and complex analysis.
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Additional Information about Complex Analysis (Part - 1) Solved Example for Mathematics Preparation
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