Course Code: EMA51002
Course Name: Analytical Mathematics
Module 2: COMPLEX VARIABLES
PART-A
1 Define an Entire (or) an Integral function BTL-1
2 BTL-2
Determine whether the function z is analytic or not
3 State the necessary conditions for f(z) to be analytic BTL-2
4 State the Sufficient conditions for f(z) to be analytic BTL-2
5 Prove that the following functions are analytic f(z) =z2 BTL-2
6 z
2
BTL-2
Show that f (z) = is differentiable at z = 0 but not analytic at z = 0
7 Define Harmonic function BTL-1
8 Show that the function u = 2x – x3 + 3xy2 is harmonic BTL-2
9 Define an Analytic function (or) Holomorphic function (or) Regular BTL-1
function
10 Find a function w such that w = u + iv is analytic, if u = ex siny BTL-2
11 Define Conformal transformation BTL-1
12 Define Cross Ratio of four points BTL-1
13 Define Bilinear transformation BTL-1
14 Prove that f(z) = sin z is analytic BTL-2
15 2 BTL-2
Find the critical points of the transformation w (z -α)(z -β) .
16 1 BTL-2
w=
Obtain the invariant points of the transformation z
17 z 3 BTL-2
Find the map of the circle under the transformation w = 2z .
18 Find the image of the hyperbola x2 – y2 = 10 under the transformation w = BTL-2
z2
19 z 1 BTL-2
Find the image of the circle by the transformation w = z + 2 + 4i
Part B
1 Show that the function 𝑓(𝑧) = |𝑧|2 is differentiable only at the orgin BTL-3
2 Find the image of the circle z 1 1 under the transformation w z 2 . BTL-3
3 sin 2 x BTL-3
Determine the analytic function f(z) = u+iv, if u
cosh 2 y cos 2 x .
𝑦
4 Construct the analytic function whose real part is 𝑢 = 𝑥 2 +𝑦2 BTL-3
5 1 BTL-3
Prove that 𝑢 = log(𝑥 2 + 𝑦 2 ) is harmonic
2
6 Find the region of the w-plane into the rectangular region in z plane BTL-3
bounded by the lines x=0, x=2 and y=0, y=1 in the region mapped under
the transformation w=z+(1+2i).
7 Determine the region of the w-plane into which the region in the z-plane BTL-3
bounded by the lines x=0, y=0, x=1, y=1 is mapped under the
transformation w = (1+i)z.
Course Name: Analytical Mathematics
Module 2: COMPLEX VARIABLES
PART-A
1 Define an Entire (or) an Integral function BTL-1
2 BTL-2
Determine whether the function z is analytic or not
3 State the necessary conditions for f(z) to be analytic BTL-2
4 State the Sufficient conditions for f(z) to be analytic BTL-2
5 Prove that the following functions are analytic f(z) =z2 BTL-2
6 z
2
BTL-2
Show that f (z) = is differentiable at z = 0 but not analytic at z = 0
7 Define Harmonic function BTL-1
8 Show that the function u = 2x – x3 + 3xy2 is harmonic BTL-2
9 Define an Analytic function (or) Holomorphic function (or) Regular BTL-1
function
10 Find a function w such that w = u + iv is analytic, if u = ex siny BTL-2
11 Define Conformal transformation BTL-1
12 Define Cross Ratio of four points BTL-1
13 Define Bilinear transformation BTL-1
14 Prove that f(z) = sin z is analytic BTL-2
15 2 BTL-2
Find the critical points of the transformation w (z -α)(z -β) .
16 1 BTL-2
w=
Obtain the invariant points of the transformation z
17 z 3 BTL-2
Find the map of the circle under the transformation w = 2z .
18 Find the image of the hyperbola x2 – y2 = 10 under the transformation w = BTL-2
z2
19 z 1 BTL-2
Find the image of the circle by the transformation w = z + 2 + 4i
Part B
1 Show that the function 𝑓(𝑧) = |𝑧|2 is differentiable only at the orgin BTL-3
2 Find the image of the circle z 1 1 under the transformation w z 2 . BTL-3
3 sin 2 x BTL-3
Determine the analytic function f(z) = u+iv, if u
cosh 2 y cos 2 x .
𝑦
4 Construct the analytic function whose real part is 𝑢 = 𝑥 2 +𝑦2 BTL-3
5 1 BTL-3
Prove that 𝑢 = log(𝑥 2 + 𝑦 2 ) is harmonic
2
6 Find the region of the w-plane into the rectangular region in z plane BTL-3
bounded by the lines x=0, x=2 and y=0, y=1 in the region mapped under
the transformation w=z+(1+2i).
7 Determine the region of the w-plane into which the region in the z-plane BTL-3
bounded by the lines x=0, y=0, x=1, y=1 is mapped under the
transformation w = (1+i)z.
