North Maharashtra University 2008 B.Sc Mathematics S.Y. - MTH – 221 Functions of a Complex Variable. - Question Paper

8) A Milne – Thomson method is used to construct
a) analytic function , b) Continuous function
c) differentiable function, d) None of these.
III) ques. for 4 marks ;
1) describe he continuity of f(z) at z = 0 z and examine for continuity at z=0 the function
f(z) = 0 eight 2
4
?
+
- ; z
( x y )z
x y( iy x )
1) describe limit of a function f(Z).
Evaluate ; lim
4 16
8
4 2
3
+ +
+
z z
z
z
3
2?i
?
2) Prove that a differentiable function is always continuous . Is the converse actual ? Justify by an
example.
3) Use the definition of limit to prove that, lim [x + i( 2x + y )] = 1+ i
z ? 1- i
5) Show that if lim f(z) exists, it is unique
z 0 ? z
6) Prove that lim
z
z does not exist
z ? 0
4) Prove that lim
( x y ).z
x y( y ix )
6 2
3
+
-
does not exist , where z ? 0
z? 0
5) Evaluate : lim
z i
z i
+
5 -
z ? i
6) 9) Evaluate : lim
z i
z
- -
+
1
4 4
z ? 1+ i
10) Examine for continuity the function , f(z) =
z i
z
2
4 4
-
+
at z ? 2i
= 3+4i ; z=2i, at z = 2i
11)If f(z) =
z i
z z z z
-
3 four - two three + eight two - two + 5
when z ? i and f(i) = 2+3i , examine f(z) for continuity
At z =i..
12)Show that the function f(z) = z is continuous everywhere but not differentiable .
13) describe an analytic function . provide 2 examples of an analytic function.
14) Show that f(z) = two z is not analytic at any point in the z-plane .
3
15) State and prove the necessary condition for the f(z) to be analytic . Are these conditions
sufficient ?
16) State and prove the sufficient conditions for the function f(z) to be analytic.
17) Prove that a necessary condition for a complex function w = f(z) = u(x,y)+iv(x,y) to be analytic
at a point z =x+iy of its domain D is that at (x,y) the 1st order partial derivatives of u and v with
respect to x and y exist and satisfy the Cauchy – Riemann equations : u x y = v and u v . y x = -
18)
Prove that for the function F(z) = U(x,y) + V(x, y), if the 4 partial derivatives Ux, Uy, Vxand
Vy exist and are continuous at a point z = x + iy in the domain D and that they satisfy Cauchy-
Riemann equartions: Ux = Vy; Uy = -Vx at (x, y), then F(z) is analytic at the point z = x + iy.
19) Show that the function described by F(z) = ?xy? , when z ? 0 and F(0) = 0, is not analytic at
z = 0 even though the C-R equations are satisfied at z = 0.
20)
describe F(z) = z5 I z I-4 ; if z ? 0
= 0 ; if z = 0.
Show that F(z) is not analytic at the origin even though it satisfies C-R equations at the origin.
21) Show that the function F(z) = ( )
2 2
3 one ) three (1 )
x y
x i y i
+
+ - -
when z ? 0 and F(0) = 0 is continuous at
z = 0 and C-R equations are satisfied at the origin.
22) If F(z) and
_____
F(z) are analytic functions of z, show that F(z) is a constant function.
23) If F(z) is an analytic function with constant modulus, then prove that F(z) is a constant
function.
24) Show that -
? ?
?
=
?
?
+
?
?
x y z z
2
2
2
2
2
4 .
25) Show that F(z) = e z is not analytic for any z.
26) Show that four 2
2
0
lim
x y
x y
z ? +
does not exist.
27) Show that if W = F(z) = 3x - 2iy, then
dz
dW does not exist.
28) Show that the function F(z) = ( )
2 2
3 one ) three (1 )
x y
x i y i
+
+ - -
when z ? 0 and F(0) = 0 is continuous and
that C-R equations are satisfied at the origin but F1(0) does not exist .
29)
Show that the function described by F(z) = two 4
2 ( )
x y
xy x iy
+
+
; z ? 0
= 0 ; z = 0.
satisfies C.R. equations at z = 0 but not analytic there at.
30)
If F(z) is an analytic function of z then show that
(i) two 2
2
2
2
2
F ( z ) four F ( z )
x y
= '
?
?
+
?
?
(ii) [ ]2 2
2
2
2
2
RF ( z ) two F ( z )
x y
= '
?
?
+
?
?

Earning:   Approval pending.