Question

Easy Question :)

Derive Laplace Equation for complex analytical function

Hint :- Use Cauchy Riemann Equation , Differentiate partially of Cauchy Riemann Equations in order to get same RHS with different signs , add them to get the required result :) ​

Answer 1

Answer:

In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as



Pierre-Simon Laplace

{\displaystyle \nabla ^{2}\!f=0\qquad {\mbox{or}}\qquad \Delta f=0,}

where {\displaystyle \Delta =\nabla \cdot \nabla =\nabla ^{2}}is the Laplace operator,[note 1] {\displaystyle \nabla \cdot } is the divergence operator (also symbolized "div"), {\displaystyle \nabla } is the gradient operator (also symbolized "grad"), and {\displaystyle f(x,y,z)} is a twice-differentiable real-valued function. The Laplace operator therefore maps a scalar function to another scalar function.

If the right-hand side is specified as a given function, {\displaystyle h(x,y,z)}, we have

{\displaystyle \Delta f=h.}

This is called Poisson's equation, a generalization of Laplace's equation. Laplace's equation and Poisson's equation are the simplest examples of elliptic partial differential equations. Laplace's equation is also a special case of the Helmholtz equation.

Answer 2

Step-by-step explanation:

${\displaystyle \nabla ^{2}\!f=0\qquad {\fbox{or}}\qquad \Delta f=0,}</p><p>where {\displaystyle \Delta =\nabla \cdot \nabla =\nabla ^{2}}{\displaystyle \Delta =\nabla \cdot \nabla =\nabla ^{2}}$is the Laplace operator,[note 1] ${\displaystyle \nabla \cdot }\nabla \cdot$ is the divergence operator (also symbolized "div"),${\displaystyle \nabla }\nabla$ is the gradient operator (also symbolized "grad"), and${\displaystyle f(x,y,z)}{\displaystyle f(x,y,z)}$is a twice-differentiable real-valued function. The Laplace operator therefore maps a scalar function to another scalar function.

If the right-hand side is specified as a given function,${\displaystyle h(x,y,z)}{\displaystyle h(x,y,z)}$, we have

${\displaystyle \Delta f=h.}{\displaystyle \Delta f=h.}$

This is called Poisson's equation, a generalization of Laplace's equation. Laplace's equation and Poisson's equation are the simplest examples of elliptic partial differential equations. Laplace's equation is also a special case of the Helmholtz equation