Question

1. If f(z) = x+ay+i(bx+cy) is analytic then a,b,c equals to

A. c=1and a=b B. a=1 and c=-b C. b=1 and a=-c D.a=b=c=1

2. A point at which a function ceases to be analytic is called a

A.Singular point B. Non-Singular point C.Regular point D.

Non-Regular point ​

Answer 1

Given:

1. Given: f(z) = x+ay+i(bx+cy) is analytic

2. A point at a function ceases to be analytic

To find:

1. If f(z) = x+ay+i(bx+cy) is analytic then a,b,c equals to

2. A point at which a function ceases to be analytic is called a

Solution:

1.

From given, we have a function,

f(z) = x + ay + i(bx + cy)

Let, w = f(z) = x + ay + i(bx + cy),

so we get,

w = u + iv

⇒ u + iv = x + ay + i(bx + cy)

⇒ u = x + ay

differentiating w.r.t x ⇒ ux = 1

differentiating w.r.t y ⇒ uy = a

⇒ v = bx + cy

differentiating w.r.t x ⇒ vx = b

differentiating w.r.t y ⇒ vy = c

Since the given function is analytic, we have,

ux = vy ⇒ 1 = c

uu = vx ⇒ a = b

Option A. c=1and a=b is correct.

2.

From given, we have a statement,

A point at which a function ceases to be analytic is called a singular point.

Option A. Singular point is correct.