Fall Sem. 2022-23
Tutorial– 1
Programme Name & Branch: B. Tech.
Course Name & Code: Complex Variables and Linear Algebra (BMAT201L)
Slot: C1+TC1+TCC1
Practice all the questions
 x3 (1  i )  y 3 (1  i )
, if z  0

1. Show that the function f ( z )  
x2  y 2
0,
if z  0

is not analytic at origin although C-R equations are satisfied at the origin.
2.
If
w  f ( z )  x 2  ay 2  2 xy  i (bx 2  y 2  2 xy ) is analytic, then find the values of a and b.
Answer: a=-1 and b=1
3.
w  f ( z )    i represents the complex potential function for an electric field and
If
 ( x, y )  e x
Answer:
4.
If
2
 y2
cos 2 xy , then find  ( x, y).
 ( x, y)  e x
2
 y2
sin 2 xy  k
f ( z )  u  iv is an analytic function of z and 2u  v  e2 x ((2 x  y ) cos 2 y  ( x  2 y)sin 2 y),
then find f(z) in terms of z.
Answer:
f ( z )  ze2 z  c .