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

real part u is provided.
4) discuss the Milne-Thomson’s method to construct an analytic function F(z) = u + iv when the
imaginary part v is provided.
5) obtain an analytic function F(z) = u + iv and express it in terms of z if
u = x3 - 3xy2 + 3x2 - 3y2 + 1.
6) obtain an analytic function F(z) = u + iv if, v = e-ysinx and F(0) = 1.
7) obtain an analytic function F(z) = u + iv where the real part is e-2xsin (x2 – y2).
8) If F(z) = u + iv is analytic function of z = x + iy and u - v = y y
y
x e e
x x e
-
-
- -
+ -
2cos
cos sin , obtain F(z) if
f(p / 2) = 0.
9) Show that the function F(z) = e-ysinx is harmonic and obtain its harmonic conjugate.
10) Use Milne-Thomson’s method to construct an analytic function F(z) = u + iv where
u = ex ( xcosy - ysiny).
11) Use Milne-Thomson’s method to construct an analytic function F(z) = u + iv where
v = tan-1 (y / x).
12) Determine the analytic function F(z) = u + iv if u = x2 - y2 and F(0) = 1.
13) obtain by Milne-Thomson’s method the an analytic function F(z) = u + iv where
v = ex ( xsiny + ycosy).
14) If u = x2 - y2 and v = x2 y2
y
+
-
, then show that u and v satisfy Laplace formula but u + iv is
not an analytic function of z.
15) Show that if the harmonic functions u and v satisfy C.R. equations, then u + iv is an analytic
function.
16) If F(z) is analytic in a simply connected region R then ?
b
a
F(z) dz is independent of the path
of the integration in R joing the points a and b.
17) Evaluate ?
C
z dz where C is the arc of the parabola y2 = 4ax (a >0) in the 1st quadrant from
the vertex to the end point of its latus rectum.
18) Evaluate ?
C
dz
z - a
1 where C is circle z - a = 2.
19) Evaluate ?
C
( y – x – 3x2 i) dz where C is the straight line joining 0 to one + i.
20) Evaluate ?
C
( y – x – 3x2 i ) dz where C is the straight line joining 0 to i 1st and then i to one + i.
21) Show that the integral of one / z along a semicircular are from -1 to one has the value pi or –pi
according as the arc lies beneath or above the real axis.
6
22)
Show that if F(z) is an analytic function in a region bounded by 2 simple closed curves C1
and C2 and also on C1 and C2, then ?
C1
F(z) dz = ?
C2
F(z) dz .
23) State Cauchy’s theorem for integrals and verify it for F(z) = z + one rounder the contour z = 1.
24) If C is a circle z - a = r, prove that ?
C
( z – a )n dz = 0; n being an integer other than -1.
25)
Evaluate ?
C z
dz where C is the circle with centre at origin and radius a.
26) Verify Cauchy-Goursat Theorem for F(z) =z + two taken round the unit circle z = 1.
27) Verify Cauchy’s integral Theorem for F(z) = z2 round the circle z = 1.
28) Verify Cauchy’s Theorem for F(z) = z around a closed curve C. where c is the rectangle
bounded by the lines : x = 0, x = 1, y = 0, y= 1,
29)
Use Cauchy-Goursat Theorem to find the value of ?
C
e2 dz, where C is the circle z = one and
deduce that (i) sin( sin ) 0
2
0
? cos ? + ? ? =
p
e ? d (ii) cos( sin ) 0
2
0
? cos ? + ? ? =
p
e ? d .

Earning:   Approval pending.