Mathematics Exam > Mathematics Notes > Topic-wise Tests & Solved Examples for Mathematics > Analytic Function (Solved Questions)
Analytic Function (Solved Questions) | Topic-wise Tests & Solved Examples for Mathematics PDF Download
| Download, print and study this document offline |
Please wait while the PDF view is loading
Page 1
Free coaching of B.Sc (h) maths & JAM
For more 8130648819
Exercise
Q pg 3 / S how that f
lim
, f ( z h ) f ( z )
h
, f
lim
, f ( z i h ) f ( z )
h
Provided the limit exist.
Sol
n
:- Let us suppose that limit exist.
Let z=x+iy, also h be real
? z h ( x h,y) , or ( x h) iy
Z+ih =(x, y+h) or x+i (y+h)
? lim
, f ( x h , y ) f ( x , y )
h
lim
, f ( z h ) f ( z )
h
… … … . . ( i )
lim
, f ( x , y h ) f ( x , y )
h
lim
, f ( z ih ) f ( z )
h
… … … . . ( ii )
Hence f x & f y are given by eq
n
(i)& (ii) when the limit exist.
Que. 2 a. Show that f(z)=x
2
+iy
2
is diff at all points on the line y=x
b. Show that it is nowhere analytic.
Sol
n
:- given f(z)=x
2
+iy
2
? f(x, y ) x
2
+iy
2
? f x=2x & f y = i2y =2iy
f x & f y are cts everywhere
The fun
c
f is diff if it satisfy C-R eq
n
ie. f y= if x
2iy=(2x)i
Ie. y=x
Hence, f(z) is diff at all pts on the line y=x.
Qu e. S upp ose f is an a ly ti c in a reg ion f there. S how that f is const an t.
Sol
n
:- Let f=u+iv be analytic in a region
? Dif f w . r . t. ‘x ’ ‘y ’ p art ia lly .
? f x =u x+iv x
f y= u y+iv y
g iven f
{
u
iv
u
i v
u x=0, v x=0, u y=0, v y=0.
all the p artia l d eriv at ive are eq ual to z ero
Hence, f is constant.
Que. 6 Assume that fun
c
f is analytic in a region & that at the every point of the
r egion, ei ther f or f . S how that f is const an t. ,H in t consi d er f
2
.]
Sol
n
:- the fu n
n
f is analytic
? f
2
is also analytic
C onsider ( f
( z ) )
f ( z ) f
( z ) ( if f ( z ) or f ( z ) )
(f
2
) z
? y Prev ious que , f
2
is constant & hence f .
Que 8. Find all analytic function f = u + iv with u(x, y) = x
2
-y
2
.
So l
Giv en u ( x , y ) x
y
? u
x
u
y
we k now that C auch y Rie mann eq uati on is
u
v
? v
x
in te g r at e p artia lly w . r . t . y
Page 2
Free coaching of B.Sc (h) maths & JAM
For more 8130648819
Exercise
Q pg 3 / S how that f
lim
, f ( z h ) f ( z )
h
, f
lim
, f ( z i h ) f ( z )
h
Provided the limit exist.
Sol
n
:- Let us suppose that limit exist.
Let z=x+iy, also h be real
? z h ( x h,y) , or ( x h) iy
Z+ih =(x, y+h) or x+i (y+h)
? lim
, f ( x h , y ) f ( x , y )
h
lim
, f ( z h ) f ( z )
h
… … … . . ( i )
lim
, f ( x , y h ) f ( x , y )
h
lim
, f ( z ih ) f ( z )
h
… … … . . ( ii )
Hence f x & f y are given by eq
n
(i)& (ii) when the limit exist.
Que. 2 a. Show that f(z)=x
2
+iy
2
is diff at all points on the line y=x
b. Show that it is nowhere analytic.
Sol
n
:- given f(z)=x
2
+iy
2
? f(x, y ) x
2
+iy
2
? f x=2x & f y = i2y =2iy
f x & f y are cts everywhere
The fun
c
f is diff if it satisfy C-R eq
n
ie. f y= if x
2iy=(2x)i
Ie. y=x
Hence, f(z) is diff at all pts on the line y=x.
Qu e. S upp ose f is an a ly ti c in a reg ion f there. S how that f is const an t.
Sol
n
:- Let f=u+iv be analytic in a region
? Dif f w . r . t. ‘x ’ ‘y ’ p art ia lly .
? f x =u x+iv x
f y= u y+iv y
g iven f
{
u
iv
u
i v
u x=0, v x=0, u y=0, v y=0.
all the p artia l d eriv at ive are eq ual to z ero
Hence, f is constant.
Que. 6 Assume that fun
c
f is analytic in a region & that at the every point of the
r egion, ei ther f or f . S how that f is const an t. ,H in t consi d er f
2
.]
Sol
n
:- the fu n
n
f is analytic
? f
2
is also analytic
C onsider ( f
( z ) )
f ( z ) f
( z ) ( if f ( z ) or f ( z ) )
(f
2
) z
? y Prev ious que , f
2
is constant & hence f .
Que 8. Find all analytic function f = u + iv with u(x, y) = x
2
-y
2
.
So l
Giv en u ( x , y ) x
y
? u
x
u
y
we k now that C auch y Rie mann eq uati on is
u
v
? v
x
in te g r at e p artia lly w . r . t . y
Free coaching of B.Sc (h) maths & JAM
For more 8130648819
? v xy c
? f ( x
y
) i ( xy c )
f ( x iy )
ic
f ( z ) z
c
, c
.
Qu e . S how that there is no an aly ti c f uncti on f u iv w it h u ( x , y ) x
y
.
So l
L et if p ossi b le there is an an aly ti c f uncti on
f u iv w it h u ( x , y ) x
y
if the f uncti on f is an aly ti c then it sat i sf ie s C R eq uati on .
?
u
x
x v
y
v xy c
… … . . ( )
u
y
y v
x
v xy c
… … . . ( )
Which is a contradiction.
? we can not f in d an aly ti c f uncti on f u iv w it h u ( x , y ) x
y
.
lte r na te
also , ( v
)
( v
)
usin g ( i ) ( ii )
b ut v
v
sin ce f is an aly ti c
? a contrad ict ion .
en ti r e an aly C R e q
Qu e . S upp ose f is an en t ire f uncti on of the f or m
f ( x , y ) u ( x ) iv ( y )
S how that f is a line ar p oly nomia l .
So l
L et f ( x , y ) u ( x ) iv ( y ) is an en t ire f uncti on . … ( i )
? it is an aly ti c ov er w hol e d isc .
? it sat isfy C auch y Rie mann eq uati on
ote en ti r e an aly ti c C R eq uati on sa ti sf ie d
So , u
v
i . e . u
( x ) v
( y ) , u
( x ) is only f uncti on of x v
( y ) is the only f uncti on of y -
u
( x ) and v
( y ) b oth are consta nt .
L et u
( x ) v
( y ) a
now , {
( ) ( )
in te g r at e
u ( x ) ax c
v ( y ) ay c
Put the value of u & v in equation (i)
? f ( x , y ) ( ax c
) i ( ay c
)
a ( x iy ) ( c
i c
)
a ( x iy ) ( c
i c
)
az b w her e b c
i c
Hence, f is a linear polynomial.
Qu e . . a . S how that e
is en ti r e by v erif y in g the C auch y Rie mann eq uati on f or its r ea l and imagina r y p arts .
or Pro v e that e
is an a ly ti c in the case of r ea l and imagina r y p art .
b . Pro v e
e
e
e
So l
L et f ( z ) e
e
e
. e
u iv e
( cos y isi n y )
{
cos v e
sin y
? 8
u
e
cos y , u
e
sin y
v
e
sin y , v
e
cos y
Hence, C-R equation satisfied.
Qu e . . a . S how | e
| e
.
Page 3
Free coaching of B.Sc (h) maths & JAM
For more 8130648819
Exercise
Q pg 3 / S how that f
lim
, f ( z h ) f ( z )
h
, f
lim
, f ( z i h ) f ( z )
h
Provided the limit exist.
Sol
n
:- Let us suppose that limit exist.
Let z=x+iy, also h be real
? z h ( x h,y) , or ( x h) iy
Z+ih =(x, y+h) or x+i (y+h)
? lim
, f ( x h , y ) f ( x , y )
h
lim
, f ( z h ) f ( z )
h
… … … . . ( i )
lim
, f ( x , y h ) f ( x , y )
h
lim
, f ( z ih ) f ( z )
h
… … … . . ( ii )
Hence f x & f y are given by eq
n
(i)& (ii) when the limit exist.
Que. 2 a. Show that f(z)=x
2
+iy
2
is diff at all points on the line y=x
b. Show that it is nowhere analytic.
Sol
n
:- given f(z)=x
2
+iy
2
? f(x, y ) x
2
+iy
2
? f x=2x & f y = i2y =2iy
f x & f y are cts everywhere
The fun
c
f is diff if it satisfy C-R eq
n
ie. f y= if x
2iy=(2x)i
Ie. y=x
Hence, f(z) is diff at all pts on the line y=x.
Qu e. S upp ose f is an a ly ti c in a reg ion f there. S how that f is const an t.
Sol
n
:- Let f=u+iv be analytic in a region
? Dif f w . r . t. ‘x ’ ‘y ’ p art ia lly .
? f x =u x+iv x
f y= u y+iv y
g iven f
{
u
iv
u
i v
u x=0, v x=0, u y=0, v y=0.
all the p artia l d eriv at ive are eq ual to z ero
Hence, f is constant.
Que. 6 Assume that fun
c
f is analytic in a region & that at the every point of the
r egion, ei ther f or f . S how that f is const an t. ,H in t consi d er f
2
.]
Sol
n
:- the fu n
n
f is analytic
? f
2
is also analytic
C onsider ( f
( z ) )
f ( z ) f
( z ) ( if f ( z ) or f ( z ) )
(f
2
) z
? y Prev ious que , f
2
is constant & hence f .
Que 8. Find all analytic function f = u + iv with u(x, y) = x
2
-y
2
.
So l
Giv en u ( x , y ) x
y
? u
x
u
y
we k now that C auch y Rie mann eq uati on is
u
v
? v
x
in te g r at e p artia lly w . r . t . y
Free coaching of B.Sc (h) maths & JAM
For more 8130648819
? v xy c
? f ( x
y
) i ( xy c )
f ( x iy )
ic
f ( z ) z
c
, c
.
Qu e . S how that there is no an aly ti c f uncti on f u iv w it h u ( x , y ) x
y
.
So l
L et if p ossi b le there is an an aly ti c f uncti on
f u iv w it h u ( x , y ) x
y
if the f uncti on f is an aly ti c then it sat i sf ie s C R eq uati on .
?
u
x
x v
y
v xy c
… … . . ( )
u
y
y v
x
v xy c
… … . . ( )
Which is a contradiction.
? we can not f in d an aly ti c f uncti on f u iv w it h u ( x , y ) x
y
.
lte r na te
also , ( v
)
( v
)
usin g ( i ) ( ii )
b ut v
v
sin ce f is an aly ti c
? a contrad ict ion .
en ti r e an aly C R e q
Qu e . S upp ose f is an en t ire f uncti on of the f or m
f ( x , y ) u ( x ) iv ( y )
S how that f is a line ar p oly nomia l .
So l
L et f ( x , y ) u ( x ) iv ( y ) is an en t ire f uncti on . … ( i )
? it is an aly ti c ov er w hol e d isc .
? it sat isfy C auch y Rie mann eq uati on
ote en ti r e an aly ti c C R eq uati on sa ti sf ie d
So , u
v
i . e . u
( x ) v
( y ) , u
( x ) is only f uncti on of x v
( y ) is the only f uncti on of y -
u
( x ) and v
( y ) b oth are consta nt .
L et u
( x ) v
( y ) a
now , {
( ) ( )
in te g r at e
u ( x ) ax c
v ( y ) ay c
Put the value of u & v in equation (i)
? f ( x , y ) ( ax c
) i ( ay c
)
a ( x iy ) ( c
i c
)
a ( x iy ) ( c
i c
)
az b w her e b c
i c
Hence, f is a linear polynomial.
Qu e . . a . S how that e
is en ti r e by v erif y in g the C auch y Rie mann eq uati on f or its r ea l and imagina r y p arts .
or Pro v e that e
is an a ly ti c in the case of r ea l and imagina r y p art .
b . Pro v e
e
e
e
So l
L et f ( z ) e
e
e
. e
u iv e
( cos y isi n y )
{
cos v e
sin y
? 8
u
e
cos y , u
e
sin y
v
e
sin y , v
e
cos y
Hence, C-R equation satisfied.
Qu e . . a . S how | e
| e
.
Free coaching of B.Sc (h) maths & JAM
For more 8130648819
b . e
f or any z .
c . e
cis y
d . e
has in f in it ely many solu ti on f or any .
So l
e
is w ell d efie d f uncti on .
e
e
e
. e
e
( cos i sin )
? | e
| v ( e
)
( cos
y sin
y ) e
( b ) e
f or any z
as | e
| e
| e
|
e
.
( c ) let x
? e
e
cos y i sin cisy
( d ) L et r ( cos isi n )
? e
r e
e
r e
e
e
r e
{
r e
x log r and
e
e
y k w her e k , , , …
Since, k has infinitely many values.
has in f in it ely many v alues .
e
has in f in it e ly many v alues p r ov id ed .
Qu e . . F in d all the solu ti on of
( a ) e
(b) e
i (c) e
3 (d) e
i
So l
e
e
e
e
e
( cosy isi ny )
e
cosy , e
sin y …… . ( i) on comparing real and imaginary part of both side
squa r in g and ad d in g
( e
)
e
x log
? f r om e q
( i )
cos sin
y k k be any in te g er i . e . k
? z x iy ikz w her e k
i . e . z * ik k be any in te g er +
( b ) e
i
e
i
e
. e
i
e
( cosy isi ny ) i
e
cosy , e
sin y …. ( i)
squa r in g and ad d in g
Page 4
Free coaching of B.Sc (h) maths & JAM
For more 8130648819
Exercise
Q pg 3 / S how that f
lim
, f ( z h ) f ( z )
h
, f
lim
, f ( z i h ) f ( z )
h
Provided the limit exist.
Sol
n
:- Let us suppose that limit exist.
Let z=x+iy, also h be real
? z h ( x h,y) , or ( x h) iy
Z+ih =(x, y+h) or x+i (y+h)
? lim
, f ( x h , y ) f ( x , y )
h
lim
, f ( z h ) f ( z )
h
… … … . . ( i )
lim
, f ( x , y h ) f ( x , y )
h
lim
, f ( z ih ) f ( z )
h
… … … . . ( ii )
Hence f x & f y are given by eq
n
(i)& (ii) when the limit exist.
Que. 2 a. Show that f(z)=x
2
+iy
2
is diff at all points on the line y=x
b. Show that it is nowhere analytic.
Sol
n
:- given f(z)=x
2
+iy
2
? f(x, y ) x
2
+iy
2
? f x=2x & f y = i2y =2iy
f x & f y are cts everywhere
The fun
c
f is diff if it satisfy C-R eq
n
ie. f y= if x
2iy=(2x)i
Ie. y=x
Hence, f(z) is diff at all pts on the line y=x.
Qu e. S upp ose f is an a ly ti c in a reg ion f there. S how that f is const an t.
Sol
n
:- Let f=u+iv be analytic in a region
? Dif f w . r . t. ‘x ’ ‘y ’ p art ia lly .
? f x =u x+iv x
f y= u y+iv y
g iven f
{
u
iv
u
i v
u x=0, v x=0, u y=0, v y=0.
all the p artia l d eriv at ive are eq ual to z ero
Hence, f is constant.
Que. 6 Assume that fun
c
f is analytic in a region & that at the every point of the
r egion, ei ther f or f . S how that f is const an t. ,H in t consi d er f
2
.]
Sol
n
:- the fu n
n
f is analytic
? f
2
is also analytic
C onsider ( f
( z ) )
f ( z ) f
( z ) ( if f ( z ) or f ( z ) )
(f
2
) z
? y Prev ious que , f
2
is constant & hence f .
Que 8. Find all analytic function f = u + iv with u(x, y) = x
2
-y
2
.
So l
Giv en u ( x , y ) x
y
? u
x
u
y
we k now that C auch y Rie mann eq uati on is
u
v
? v
x
in te g r at e p artia lly w . r . t . y
Free coaching of B.Sc (h) maths & JAM
For more 8130648819
? v xy c
? f ( x
y
) i ( xy c )
f ( x iy )
ic
f ( z ) z
c
, c
.
Qu e . S how that there is no an aly ti c f uncti on f u iv w it h u ( x , y ) x
y
.
So l
L et if p ossi b le there is an an aly ti c f uncti on
f u iv w it h u ( x , y ) x
y
if the f uncti on f is an aly ti c then it sat i sf ie s C R eq uati on .
?
u
x
x v
y
v xy c
… … . . ( )
u
y
y v
x
v xy c
… … . . ( )
Which is a contradiction.
? we can not f in d an aly ti c f uncti on f u iv w it h u ( x , y ) x
y
.
lte r na te
also , ( v
)
( v
)
usin g ( i ) ( ii )
b ut v
v
sin ce f is an aly ti c
? a contrad ict ion .
en ti r e an aly C R e q
Qu e . S upp ose f is an en t ire f uncti on of the f or m
f ( x , y ) u ( x ) iv ( y )
S how that f is a line ar p oly nomia l .
So l
L et f ( x , y ) u ( x ) iv ( y ) is an en t ire f uncti on . … ( i )
? it is an aly ti c ov er w hol e d isc .
? it sat isfy C auch y Rie mann eq uati on
ote en ti r e an aly ti c C R eq uati on sa ti sf ie d
So , u
v
i . e . u
( x ) v
( y ) , u
( x ) is only f uncti on of x v
( y ) is the only f uncti on of y -
u
( x ) and v
( y ) b oth are consta nt .
L et u
( x ) v
( y ) a
now , {
( ) ( )
in te g r at e
u ( x ) ax c
v ( y ) ay c
Put the value of u & v in equation (i)
? f ( x , y ) ( ax c
) i ( ay c
)
a ( x iy ) ( c
i c
)
a ( x iy ) ( c
i c
)
az b w her e b c
i c
Hence, f is a linear polynomial.
Qu e . . a . S how that e
is en ti r e by v erif y in g the C auch y Rie mann eq uati on f or its r ea l and imagina r y p arts .
or Pro v e that e
is an a ly ti c in the case of r ea l and imagina r y p art .
b . Pro v e
e
e
e
So l
L et f ( z ) e
e
e
. e
u iv e
( cos y isi n y )
{
cos v e
sin y
? 8
u
e
cos y , u
e
sin y
v
e
sin y , v
e
cos y
Hence, C-R equation satisfied.
Qu e . . a . S how | e
| e
.
Free coaching of B.Sc (h) maths & JAM
For more 8130648819
b . e
f or any z .
c . e
cis y
d . e
has in f in it ely many solu ti on f or any .
So l
e
is w ell d efie d f uncti on .
e
e
e
. e
e
( cos i sin )
? | e
| v ( e
)
( cos
y sin
y ) e
( b ) e
f or any z
as | e
| e
| e
|
e
.
( c ) let x
? e
e
cos y i sin cisy
( d ) L et r ( cos isi n )
? e
r e
e
r e
e
e
r e
{
r e
x log r and
e
e
y k w her e k , , , …
Since, k has infinitely many values.
has in f in it ely many v alues .
e
has in f in it e ly many v alues p r ov id ed .
Qu e . . F in d all the solu ti on of
( a ) e
(b) e
i (c) e
3 (d) e
i
So l
e
e
e
e
e
( cosy isi ny )
e
cosy , e
sin y …… . ( i) on comparing real and imaginary part of both side
squa r in g and ad d in g
( e
)
e
x log
? f r om e q
( i )
cos sin
y k k be any in te g er i . e . k
? z x iy ikz w her e k
i . e . z * ik k be any in te g er +
( b ) e
i
e
i
e
. e
i
e
( cosy isi ny ) i
e
cosy , e
sin y …. ( i)
squa r in g and ad d in g
Free coaching of B.Sc (h) maths & JAM
For more 8130648819
( e
)
e
x log
? f r om eq uati on ( )
cos , sin
? y ( k )
w her e k be any in te g er
? y 2 i ( k )
k be any in te g er 3
( d ) e
i
e
i
e
e
i
e
( cos i sin ) i
e
cos , e
sin …… ( i)
squa r in g and ad d in g
( e
)
e
( )
x log
f r om eq uati on ( )
v cos , v sin
cosy v , sin y v
? y k k be any in te g er
? z {
log i . k / k be any in te g er }
( e ) e
3
e
3
e
. e
3
e
( cos i sin ) 3
e
cos 3 , e
sin … . . ( i )
squa r in g and ad d in g .
( e
)
e
3 d oubt w hy d id we ta k e ve 3 , not 3 ?
x log 3
From equation (1)
3 cos 3 , 3 sin
cosy , sin y
? y ( k ) k be any in te g er
Page 5
Free coaching of B.Sc (h) maths & JAM
For more 8130648819
Exercise
Q pg 3 / S how that f
lim
, f ( z h ) f ( z )
h
, f
lim
, f ( z i h ) f ( z )
h
Provided the limit exist.
Sol
n
:- Let us suppose that limit exist.
Let z=x+iy, also h be real
? z h ( x h,y) , or ( x h) iy
Z+ih =(x, y+h) or x+i (y+h)
? lim
, f ( x h , y ) f ( x , y )
h
lim
, f ( z h ) f ( z )
h
… … … . . ( i )
lim
, f ( x , y h ) f ( x , y )
h
lim
, f ( z ih ) f ( z )
h
… … … . . ( ii )
Hence f x & f y are given by eq
n
(i)& (ii) when the limit exist.
Que. 2 a. Show that f(z)=x
2
+iy
2
is diff at all points on the line y=x
b. Show that it is nowhere analytic.
Sol
n
:- given f(z)=x
2
+iy
2
? f(x, y ) x
2
+iy
2
? f x=2x & f y = i2y =2iy
f x & f y are cts everywhere
The fun
c
f is diff if it satisfy C-R eq
n
ie. f y= if x
2iy=(2x)i
Ie. y=x
Hence, f(z) is diff at all pts on the line y=x.
Qu e. S upp ose f is an a ly ti c in a reg ion f there. S how that f is const an t.
Sol
n
:- Let f=u+iv be analytic in a region
? Dif f w . r . t. ‘x ’ ‘y ’ p art ia lly .
? f x =u x+iv x
f y= u y+iv y
g iven f
{
u
iv
u
i v
u x=0, v x=0, u y=0, v y=0.
all the p artia l d eriv at ive are eq ual to z ero
Hence, f is constant.
Que. 6 Assume that fun
c
f is analytic in a region & that at the every point of the
r egion, ei ther f or f . S how that f is const an t. ,H in t consi d er f
2
.]
Sol
n
:- the fu n
n
f is analytic
? f
2
is also analytic
C onsider ( f
( z ) )
f ( z ) f
( z ) ( if f ( z ) or f ( z ) )
(f
2
) z
? y Prev ious que , f
2
is constant & hence f .
Que 8. Find all analytic function f = u + iv with u(x, y) = x
2
-y
2
.
So l
Giv en u ( x , y ) x
y
? u
x
u
y
we k now that C auch y Rie mann eq uati on is
u
v
? v
x
in te g r at e p artia lly w . r . t . y
Free coaching of B.Sc (h) maths & JAM
For more 8130648819
? v xy c
? f ( x
y
) i ( xy c )
f ( x iy )
ic
f ( z ) z
c
, c
.
Qu e . S how that there is no an aly ti c f uncti on f u iv w it h u ( x , y ) x
y
.
So l
L et if p ossi b le there is an an aly ti c f uncti on
f u iv w it h u ( x , y ) x
y
if the f uncti on f is an aly ti c then it sat i sf ie s C R eq uati on .
?
u
x
x v
y
v xy c
… … . . ( )
u
y
y v
x
v xy c
… … . . ( )
Which is a contradiction.
? we can not f in d an aly ti c f uncti on f u iv w it h u ( x , y ) x
y
.
lte r na te
also , ( v
)
( v
)
usin g ( i ) ( ii )
b ut v
v
sin ce f is an aly ti c
? a contrad ict ion .
en ti r e an aly C R e q
Qu e . S upp ose f is an en t ire f uncti on of the f or m
f ( x , y ) u ( x ) iv ( y )
S how that f is a line ar p oly nomia l .
So l
L et f ( x , y ) u ( x ) iv ( y ) is an en t ire f uncti on . … ( i )
? it is an aly ti c ov er w hol e d isc .
? it sat isfy C auch y Rie mann eq uati on
ote en ti r e an aly ti c C R eq uati on sa ti sf ie d
So , u
v
i . e . u
( x ) v
( y ) , u
( x ) is only f uncti on of x v
( y ) is the only f uncti on of y -
u
( x ) and v
( y ) b oth are consta nt .
L et u
( x ) v
( y ) a
now , {
( ) ( )
in te g r at e
u ( x ) ax c
v ( y ) ay c
Put the value of u & v in equation (i)
? f ( x , y ) ( ax c
) i ( ay c
)
a ( x iy ) ( c
i c
)
a ( x iy ) ( c
i c
)
az b w her e b c
i c
Hence, f is a linear polynomial.
Qu e . . a . S how that e
is en ti r e by v erif y in g the C auch y Rie mann eq uati on f or its r ea l and imagina r y p arts .
or Pro v e that e
is an a ly ti c in the case of r ea l and imagina r y p art .
b . Pro v e
e
e
e
So l
L et f ( z ) e
e
e
. e
u iv e
( cos y isi n y )
{
cos v e
sin y
? 8
u
e
cos y , u
e
sin y
v
e
sin y , v
e
cos y
Hence, C-R equation satisfied.
Qu e . . a . S how | e
| e
.
Free coaching of B.Sc (h) maths & JAM
For more 8130648819
b . e
f or any z .
c . e
cis y
d . e
has in f in it ely many solu ti on f or any .
So l
e
is w ell d efie d f uncti on .
e
e
e
. e
e
( cos i sin )
? | e
| v ( e
)
( cos
y sin
y ) e
( b ) e
f or any z
as | e
| e
| e
|
e
.
( c ) let x
? e
e
cos y i sin cisy
( d ) L et r ( cos isi n )
? e
r e
e
r e
e
e
r e
{
r e
x log r and
e
e
y k w her e k , , , …
Since, k has infinitely many values.
has in f in it ely many v alues .
e
has in f in it e ly many v alues p r ov id ed .
Qu e . . F in d all the solu ti on of
( a ) e
(b) e
i (c) e
3 (d) e
i
So l
e
e
e
e
e
( cosy isi ny )
e
cosy , e
sin y …… . ( i) on comparing real and imaginary part of both side
squa r in g and ad d in g
( e
)
e
x log
? f r om e q
( i )
cos sin
y k k be any in te g er i . e . k
? z x iy ikz w her e k
i . e . z * ik k be any in te g er +
( b ) e
i
e
i
e
. e
i
e
( cosy isi ny ) i
e
cosy , e
sin y …. ( i)
squa r in g and ad d in g
Free coaching of B.Sc (h) maths & JAM
For more 8130648819
( e
)
e
x log
? f r om eq uati on ( )
cos , sin
? y ( k )
w her e k be any in te g er
? y 2 i ( k )
k be any in te g er 3
( d ) e
i
e
i
e
e
i
e
( cos i sin ) i
e
cos , e
sin …… ( i)
squa r in g and ad d in g
( e
)
e
( )
x log
f r om eq uati on ( )
v cos , v sin
cosy v , sin y v
? y k k be any in te g er
? z {
log i . k / k be any in te g er }
( e ) e
3
e
3
e
. e
3
e
( cos i sin ) 3
e
cos 3 , e
sin … . . ( i )
squa r in g and ad d in g .
( e
)
e
3 d oubt w hy d id we ta k e ve 3 , not 3 ?
x log 3
From equation (1)
3 cos 3 , 3 sin
cosy , sin y
? y ( k ) k be any in te g er
Free coaching of B.Sc (h) maths & JAM
For more 8130648819
? z * log 3 i ( k ) k be any in te g er +
…… …… …… … …… …… … … …
sin z e
e
i
and cos z e
e
cos ( ix ) e
( )
e
( )
e
e
not e | sin x | , | cosx | x
b ut sin ( z ) and cosz are not b ound ed
Complex sin & cos are not bdd
in sin x cos x
cos ( ix ) e
e
( as n )
| sin ( i ) | | e
e
|
not e ( sin z )
d
dz
[
i
( e
e
) ]
i
[ i e
( i ) e
]
[ e
e
]
cos z
Qu e . V erif y the id en ti ti es
( a ) sin z sin z cos z ( b ) sin
z cos
z ( c ) ( sin z )
cosz
( a ) we k now that sin z e
e
i
and cos z e
e
? {
LHS sin z i
( e
e
)
RHS sin cos
6 4
e
e
i
5 4
e
e
5 7 i
( e
e
)
? sin z sin z cosz
Qu e . F in d ( cos )
So l
d
dz
( cos ) d
dz
4
( e
e
) 5 ( i e
ie
) i
( e
e
)
Multiplying & dividing by i, we get
i
i
( e
e
) i
( e
e
) sin
Qu e . Pro v e that sin ( x iy ) sin cos hy i cos sin hy
Read More
|
27 docs|150 tests
|
FAQs on Analytic Function (Solved Questions) - Topic-wise Tests & Solved Examples for Mathematics
| 1. What is an analytic function? |  |
An analytic function is a complex-valued function that is locally given by a convergent power series. It is a function that can be expressed as a power series that converges in a neighborhood of each point in its domain.
| 2. What are some properties of analytic functions? | |
Some properties of analytic functions include the fact that they are infinitely differentiable, they preserve angles between curves, and their real and imaginary parts satisfy the Cauchy-Riemann equations. Additionally, the derivative of an analytic function is also analytic.
| 3. How can I determine if a function is analytic? | |
To determine if a function is analytic, one can check if it satisfies the Cauchy-Riemann equations. These equations relate the partial derivatives of the function's real and imaginary parts. If the partial derivatives satisfy these equations, then the function is analytic.
| 4. Can an analytic function have singularities? | |
Yes, an analytic function can have singularities. A singularity occurs when a function is not analytic at a certain point. There are different types of singularities, such as removable singularities, poles, and essential singularities. These occur when the function behaves in a certain way at the singularity point.
| 5. What are some applications of analytic functions? | |
Analytic functions have various applications in mathematics and physics. They are used in complex analysis to study properties of functions and their behavior. In physics, they are used to model and solve problems in fluid dynamics, electromagnetism, and quantum mechanics. Analytic functions also find applications in engineering, signal processing, and image processing.
Related Exams
About this Document
|
4.66/5 Rating |
|
Nov 08, 2024 Last updated |
Document Description: Analytic Function (Solved Questions) for Mathematics 2024 is part of Topic-wise Tests & Solved Examples for Mathematics preparation.
The notes and questions for Analytic Function (Solved Questions) have been prepared according to the Mathematics exam syllabus. Information about Analytic Function (Solved Questions) covers topics
like and Analytic Function (Solved Questions) Example, for Mathematics 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Analytic Function (Solved Questions).
Introduction of Analytic Function (Solved Questions) in English is available as part of our Topic-wise Tests & Solved Examples for Mathematics
for Mathematics & Analytic Function (Solved Questions) in Hindi for Topic-wise Tests & Solved Examples for Mathematics course.
Download more important topics related with notes, lectures and mock test series for Mathematics
Exam by signing up for free. Mathematics: Analytic Function (Solved Questions) | Topic-wise Tests & Solved Examples for Mathematics
Description
Full syllabus notes, lecture & questions for Analytic Function (Solved Questions) | Topic-wise Tests & Solved Examples for Mathematics - Mathematics | Plus excerises question with solution to help you revise complete syllabus for Topic-wise Tests & Solved Examples for Mathematics | Best notes, free PDF download
Information about Analytic Function (Solved Questions)
In this doc you can find the meaning of Analytic Function (Solved Questions) defined & explained in the simplest way possible. Besides explaining types of
Analytic Function (Solved Questions) theory, EduRev gives you an ample number of questions to practice Analytic Function (Solved Questions) tests, examples and also practice Mathematics
tests
|
27 docs|150 tests
|
Download as PDF
|
Explore Courses for Mathematics exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
Related Searches
Summary
,Objective type Questions
,Extra Questions
,Viva Questions
,ppt
,Sample Paper
,mock tests for examination
,practice quizzes
,study material
,MCQs
,video lectures
,Free
,Semester Notes
,Analytic Function (Solved Questions) | Topic-wise Tests & Solved Examples for Mathematics
,past year papers
,Important questions
,Analytic Function (Solved Questions) | Topic-wise Tests & Solved Examples for Mathematics
,shortcuts and tricks
,Previous Year Questions with Solutions
,Exam
,Analytic Function (Solved Questions) | Topic-wise Tests & Solved Examples for Mathematics
,
Additional Information about Analytic Function (Solved Questions) for Mathematics Preparation
Analytic Function (Solved Questions) Free PDF Download
The Analytic Function (Solved Questions) is an invaluable resource that delves deep into the core of the Mathematics exam.
These study notes are curated by experts and cover all the essential topics and concepts, making your preparation more efficient and effective.
With the help of these notes, you can grasp complex subjects quickly, revise important points easily,
and reinforce your understanding of key concepts. The study notes are presented in a concise and easy-to-understand manner,
allowing you to optimize your learning process. Whether you're looking for best-recommended books, sample papers, study material,
or toppers' notes, this PDF has got you covered. Download the Analytic Function (Solved Questions) now and kickstart your journey towards success in the Mathematics exam.
Importance of Analytic Function (Solved Questions)
The importance of Analytic Function (Solved Questions) cannot be overstated, especially for Mathematics aspirants.
This document holds the key to success in the Mathematics exam.
It offers a detailed understanding of the concept, providing invaluable insights into the topic.
By knowing the concepts well in advance, students can plan their preparation effectively.
Utilize this indispensable guide for a well-rounded preparation and achieve your desired results.
Analytic Function (Solved Questions) Notes
Analytic Function (Solved Questions) Notes offer in-depth insights into the specific topic to help you master it with ease.
This comprehensive document covers all aspects related to Analytic Function (Solved Questions).
It includes detailed information about the exam syllabus, recommended books, and study materials for a well-rounded preparation.
Practice papers and question papers enable you to assess your progress effectively.
Additionally, the paper analysis provides valuable tips for tackling the exam strategically.
Access to Toppers' notes gives you an edge in understanding complex concepts.
Whether you're a beginner or aiming for advanced proficiency, Analytic Function (Solved Questions) Notes on EduRev are your ultimate resource for success.
Analytic Function (Solved Questions) Mathematics
The "Analytic Function (Solved Questions) Mathematics Questions" guide is a valuable resource for all aspiring students preparing for the
Mathematics exam. It focuses on providing a wide range of practice questions to help students gauge
their understanding of the exam topics. These questions cover the entire syllabus, ensuring comprehensive preparation.
The guide includes previous years' question papers for students to familiarize themselves with the exam's format and difficulty level.
Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.
Study Analytic Function (Solved Questions) on the App
Students of Mathematics can study Analytic Function (Solved Questions) alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the Analytic Function (Solved Questions),
students can also utilize the EduRev App for other study materials such as previous year question papers, syllabus, important questions, etc.
The EduRev App will make your learning easier as you can access it from anywhere you want.
The content of Analytic Function (Solved Questions) is prepared as per the latest Mathematics syllabus.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup