Question

prove that real and imaginary parts of an analytic function justify lapalce equation.

Answer 1

Answer:

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Explanation:

The function Φ(x,y)Φ(x,y) which has continuous partial derivatives of first and second order and which satisfied Laplace's equation in two dimensions is called a harmonic function.

i.e. ∂2ϕ∂x2+∂2ϕ∂y2=0∂2ϕ∂x2+∂2ϕ∂y2=0 or V¯¯¯¯2ϕ=0V¯2ϕ=0

Given f(z) = u + iv is an analytic function

By Cauchy Riemann equations.

∂u∂x=∂v∂y∂u∂x=∂v∂y and ∂u∂y=−∂v∂x∂u∂y=−∂v∂x

Now

∂2u∂x2+∂2u∂y2=∂∂x(∂u∂x)+∂∂y(∂u∂y)∂2u∂x2+∂2u∂y2=∂∂x(∂u∂x)+∂∂y(∂u∂y)

=∂∂x(∂v∂y

Answer 2

Answer:

Prove that real and imaginary parts of an analytic function f(z) = u + iv are harmonic function. Harmonic Function: The function Φ(x,y) which has continuous partial derivatives of first and second order and which satisfied Laplace's equation in two dimensions is called a harmonic function.

Explanation:

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