Question

The only function among the following that is analytic, is
1. f(z) = Re(z)
2. \(f\left( z \right) = \bar z\)
3. f(z) = Im (z)
4. f(z) = sin(z)

Answer 1

Correct Answer - Option 4 : f(z) = sin(z)

i) f(z) = Re(z) = Re(x + iy) = x

here u = x, v = 0

u_x = 1, v_y = 0

f(z) is not satisfying CR equations Hence f(z) is not analytic.

ii) f(z) = z̅ = x - fy

u = x, v = -y

u_x = 1, v_y = -1

u_x ≠ v_y

f(x) is not satisfying CR equations.

Hence f(z) is not analytic

iii) f(z) = Im (z) = Im (x + iy) = y

u = 0, v = y

u_x = 0, v_y = 1

f(x) is not satisfying CR equations.

Hence f(z) is not analytic

iv) f(z) = sin z = (x + iy)

= sin x cos iy + cos x sin iy

= sin x cos hy + i cos x sin hy

u = sin x cos hy, v = cos x sin hy

u_x = cos x cos hy

v_y = cos x cos hy

u_y = sin x sin hy

v_x = - sin x sin hy

u_x = v_y and u_y = - v_x

f(z) is satisfying CR equation.

Hence f(z) is analytic.