Author Topic: Cheat-sheet  (Read 615 times)

Victor Ivrii

  • Administrator
  • Elder Member
  • *****
  • Posts: 1332
  • Karma: 0
    • View Profile
    • Personal website of Victor Ivrii
Cheat-sheet
« on: December 09, 2015, 05:18:59 AM »
In the final exam the following "cheat sheet" will be attached as the last page. You will be able to detach it. No other aids are allowed

Appendix: Some useful formulas
.


  • The two dimensional  Laplacian in polar coordinates:
    \begin{equation*}
    \Delta  f =\frac {\partial^2 f}  {\partial r^2}  +\frac{1}{r}  \frac{\partial f}   {\partial r}  + \frac{1}{r^2} \frac {\partial^2 f}  {\partial \theta^2}.
    \end{equation*}
  • The Stokes theorem
    \begin{equation*}
    \int _D  \frac {\partial f } {\partial x_i}  \, dx = \int_{\partial D}f  n_i \, d \sigma
    \end{equation*}
    where $ n $ (with components $n_i$) is the unit normal vector pointing outside.
  • The complex Fourier series of a periodic function $f(x)$  of period $2l$, defined on the interval $(-l,l)$ is
    $$f(x)= \sum_{n=-\infty}^{+\infty} c_n e^{\pi i nx/l}$$
    with the coefficients $c_n$ given by the formula
    \begin{equation*}
    c_n=\frac{1}{2l} \int_{-l}^{l} f(x) e^{-\pi i nx/l} dx
    \end{equation*}
  • The Fourier transform of a function  $f(x)$ is defined by
    \begin{equation*}
    \hat{f} (k) = \frac{1}{2\pi}\int_{-\infty}^\infty e^{-ikx} f(x) dx.
    \end{equation*}
    The inverse Fourier transform is
    \begin{equation*}
     f(x) =  \int_{-\infty}^\infty e^{ikx} \hat{f} (k) dk.
    \end{equation*}
  • Here some of its properties: 
    • if  $g(x) =f(ax)$ , then  $\displaystyle{ \hat{g} (k) =\frac{1}{|a|} \hat f( \frac{k}{a})}$;
    • $\widehat{f'}(k)=ik\hat{f}(k)$;
    • if $g(x)=xf(x)$ then $\hat{g}(k)= -i\hat{f}\,'(k)$;
    • if $g(x) =f(x-a)$, then $\hat g(k) = e^{-iak} \hat f(k)$;
    • if $h= f*g$, then $\hat h(k) =2\pi  \hat f(k) \hat g(k)$;
    • if $f(x)=e^{-x^2/2}$, then $\hat f(k) =\frac{1}{\sqrt{2\pi} }e^{-k^2/2}$.
« Last Edit: December 09, 2015, 05:20:35 AM by Victor Ivrii »