Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Laplace's Eqn and Cauchy's Integral Formula

  1. Oct 21, 2005 #1
    Is there a connection between Laplace's Equation and Cauchy's integral formula? There seems to be quite a similarity, eg, solutions of Laplaces Eqn are determined by their values at the boundary.
  2. jcsd
  3. Oct 21, 2005 #2


    User Avatar
    Gold Member

    Yes, there is a connection. Cauchy's integral formula assumes that the function in question is analytic. A function is analytic if and only if it satisfies the Cauchy-Riemann equations:
    If f(z)=u(x,y)+iv(x,y), then
    [tex]\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}[/tex]
    [tex]\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}[/tex]
    Since the function is analytic, then u and y have continous partial derivatives of all orders, so we may differentiate the above expressions to obtain:
    [tex]\frac{\partial^2 u}{\partial x^2}=\frac{\partial^2 v}{\partial x \partial y}[/tex]
    [tex]\frac{\partial^2 u}{\partial y^2}=-\frac{\partial^2 v}{\partial y \partial x}[/tex]
    Since these derivatives are continuos, then:
    [tex]\frac{\partial^2 v}{\partial y \partial x}=\frac{\partial^2 v}{\partial x \partial y}[/tex]
    [tex]\frac{\partial^2 u}{\partial x^2}=-\frac{\partial^2 u}{\partial y^2}[/tex]
    [tex]\rightarrow \frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0[/tex]
    Which is Laplace's equation. It can be proven similarly that the imaginary part of f also satisfies Laplace's equation.
    Last edited: Oct 21, 2005
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - Laplace's Cauchy's Integral Date
I Laplace's equation Sunday at 1:04 PM
I Laplace's equation in 3 D Mar 13, 2018
A Applying boundary conditions on an almost spherical body Feb 15, 2018
A Causality in differential equations Feb 10, 2018
I Intuition behind Cauchy–Riemann equations Dec 12, 2017