In a complex function
f(x, y) = u(x, y) + i v(x, y)
where i is the imaginary unit, and x, y, u(x, y), and v(x, y) are real.
If f(x, y) is analytic, then which of the following equations is/are TRUE?
(a) ∂²u/∂x² + ∂²u/∂y² = 0
(b) ∂²v/∂x² + ∂²v/∂y² = 0
(c) ∂²u/∂x² + ∂²v/∂y² = 0
(d) (∂u/∂x)(∂v/∂x) + (∂u/∂y)(∂v/∂y) = 0
Answer (1)
To determine which of the expressions are true when the complex function f ( x , y ) = u ( x , y ) + i v ( x , y ) is analytic, we can use the properties of analytic functions and Cauchy-Riemann equations.
A complex function f ( x , y ) is said to be analytic if it is differentiable at every point in its domain. This is equivalent to the function satisfying the Cauchy-Riemann equations:
∂ x ∂ u = ∂ y ∂ v
∂ y ∂ u = − ∂ x ∂ v
Additionally, if a function is analytic, its real and imaginary components u ( x , y ) and v ( x , y ) are harmonic functions. A function g ( x , y ) is harmonic if it satisfies Laplace's equation:
∂ x 2 ∂ 2 g + ∂ y 2 ∂ 2 g = 0
Therefore, for f ( x , y ) being analytic:
u ( x , y ) is harmonic: ∂ x 2 ∂ 2 u + ∂ y 2 ∂ 2 u = 0
v ( x , y ) is harmonic: ∂ x 2 ∂ 2 v + ∂ y 2 ∂ 2 v = 0
Now, let's match these with the given options:
(a) ∂ x 2 ∂ 2 u + ∂ y 2 ∂ 2 u = 0 - This is TRUE since u ( x , y ) is harmonic.
(b) ∂ x 2 ∂ 2 v + ∂ y 2 ∂ 2 v = 0 - This is TRUE since v ( x , y ) is harmonic.
(c) ∂ x 2 ∂ 2 u + ∂ y 2 ∂ 2 v = 0 - This is NOT necessarily true as it doesn't follow from the properties of harmonic functions.
(d) ( ∂ x ∂ u ) ( ∂ x ∂ v ) + ( ∂ y ∂ u ) ( ∂ y ∂ v ) = 0 - This is TRUE. It's derived from the Cauchy-Riemann equations.
So, the true statements are (a), (b), and (d).
