Fourier Transforms of Harmonic Functions - Physics Stack Exchange most recent 30 from physics.stackexchange.com 2022-01-25T22:24:34Z https://physics.stackexchange.com/feeds/question/411474 https://creativecommons.org/licenses/by-sa/4.0/rdf https://physics.stackexchange.com/q/411474 0 Fourier Transforms of Harmonic Functions Lone Wolf https://physics.stackexchange.com/users/69164 2018-06-12T19:35:35Z 2018-06-12T19:35:35Z <p>Suppose you are presented with the equation ($D=3$) \begin{equation} \nabla^2 A(x) = \nabla^2 B(x). \end{equation} Decompose $A$ and $B$ into their Fourier components, \begin{equation} A(x) = \int d^3k \ e^{ikx} \tilde A(k),\qquad B(x) = \int d^3k\ e^{ikx} \tilde B(k), \end{equation} to yield \begin{equation} \int d^3k \ e^{ikx}\left[-k^2 (\tilde A - \tilde B)\right]=0. \end{equation} Assuming the inverse Fourier transform of zero is uniquely zero (or based on the answers <a href="https://physics.stackexchange.com/questions/100794/fouriertransforming-the-klein-gordon-equation">Fourier Transforming the Klein Gordon Equation</a>), we conclude \begin{equation} \tilde A = \tilde B \quad\Rightarrow\quad A(x) = B(x). \end{equation} Yet we know $A(x)$ may vary up to arbitrary harmonic functions.</p> <p>What gives? </p> <p>Also consider this:<br> \begin{equation} 0 = \int d^3k\ e^{ikx} (k^2)^n \delta (k),\qquad\text{for}\qquad n&gt;0. \end{equation}</p> <p>A closely related problem may be posed simply as \begin{equation} \nabla^2 A(x) = 0 \quad\Rightarrow\quad -k^2 \tilde A(k) = 0\quad\Rightarrow\quad \tilde A(k) = 0\quad\Rightarrow\quad A(x) = 0. \end{equation}</p>