Unit-4

Complex Variable – Differentiation

Q1: Find-

A1. Here we have-

Divide numerator and denominator by , we get-

Q2: If w = log z, then find . Also determine where w is non-analytic.

A2. Here we have

Therefore-

and

Again-

Hence the C-R conditions are satisfied also the partial derivatives are continuous except at (0 , 0).

So that w is analytic everywhere but not at z = 0

Q3: Prove that the function is an analytical function.

A3. Let =u+iv

Let =u and =v

Hence C-R-Equation satisfied.

Q4: Prove that

A4. Given that

Since

V=2xy

Now

But

Hence

Q5: Prove that and are harmonic functions of (x, y).

A5.

We have

Now

Here it satisfies Laplace equation so that u (x, y) is harmonic.

Now-

On adding the above results-

We get-

So that v(x, y) is also a harmonic function.

Q6: Find the harmonic conjugate function of the function U (x, y) = 2x (1 – y).

A6.

We have,

U(x, y) = 2x (1 – y)

Let V is the harmonic conjugate of U.

So that by total differentiation,

Hence the harmonic conjugate of U is

Q7: If

Then find f(z)

A7.

Here-

Which is the required answer.

Q8: Show that the mapping is conformal in the whole of the z plane.

A8.

Let z=x+iy

Then

Consider the mapping of the straight line x=a in z plane the w plane which gives which is a circle in the w plane in the anticlockwise direction similarly the straight line y=b is mapped into which is a radius vector in the w plane.

The angle between the line x=a and y=b in the z plane is a right angle. The corresponding angle in the w plane between the circle e = constant and the radius vector is also a right angle which establishes that the mapping is conformal.

Q9: Show that the curve u = constant and v = constant cut orthogonally at all intersections but the transformation w = u + iv is not conformal. Where-

A9.

Let …………. (1)

Differentiate (1), we get-

…………… (2)

Now-

…………….. (3)

Differentiate (3), we get-

………. (4)

As we know that for the condition for orthogonallity, from (2) and (4)

So that these two curves cut orthogonally.

Here,

And

Here the C-R equation is not satisfied so that the function u + iv is not analytic.

Hence the transformation is not conformal.

Q10: How that the bilinear transformation w= transforms in the z-plane to 4u+3=0 in w-plane.

A10.

Consider the circle in z-plane

= 0

Thus, centre of the circle is (h,k)c(2,0) and radius r=2.

Thus in z-plane it is given as =2....(1)

Consider w=

W(z-4) = 2z+3

Wz-4w=2z+3

Wz-2z=4w+3

Z(w-2) = (4w+3)

z =

z-2 = - 2