Looking for a more appropriate version of \cases

\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage{mathtools}
\def\FD{\text{FD}}
\def\for{\text{for }}
\begin{document}

\[
\begin{cases}
\phantom{2}\dfrac{n(x)}{1+j^2F^2}                           & \for j=0\\[15pt]
         2 \dfrac{\cos(jk)}{1+j^2F^2}f_j^{\FD}(n(x))        & \for j>0\\[15pt]
         2 \dfrac{\sin(jk)jF/\tau}{1+j^2F^2}f_j^{\FD}(n(x)) & \for j<0 
\end{cases}
\]

\[\left\{
\begin{matrix*}[l]
   & \dfrac{n(x)}{1+j^2F^2}                           & \for j=0\\[15pt]
 2 & \dfrac{\cos(jk)}{1+j^2F^2}f_j^{\FD}(n(x))        & \for j>0\\[15pt]
 2 & \dfrac{\sin(jk)jF/\tau}{1+j^2F^2}f_j^{\FD}(n(x)) & \for j<0 
\end{matrix*}
\right.\]

\end{document}

No magic here:

\documentclass{article}
\usepackage{amsmath}
\usepackage{mathtools}
\newcommand\FD{\mathrm{FD}}
\begin{document}
\[
\begin{dcases}
\phantom{2}\frac{n(x)}{1+j^2F^2} & \text{for } j=0\\
2\frac{\cos(jk)}{1+j^2F^2}f_j^{\FD}(n(x))& \text{for } j>0\\
2\frac{\sin(jk)jF/\tau}{1+j^2F^2}f_j^{\FD}(n(x)) & \text{for } j<0 
\end{dcases}
\]
\end{document}

Here is a "solution" with the package IEEEtranstool :

\documentclass{minimal}
\usepackage{mathtools}
\usepackage[retainorgcmds]{IEEEtrantools}
\def\FD{\text{FD}}
\begin{document}
\begin{IEEEeqnarray*}{rl"t"rC}
 &\frac{n(x)}{1+j^2F^2} & for &  j&=0\\
 2&\frac{\cos(jk)}{1+j^2F^2}f_j^{\FD}(n(x))& for & j&>0\\
 2&\frac{\sin(jk)jF/\tau}{1+j^2F^2}f_j^{\FD}(n(x)) & for& j&<0
\end{IEEEeqnarray*}
\end{document}

The syntax is yours, but these solution lacks the vertical brace. Maybe a TeX wizard may helps.

For the explanation : rcl for right center left column in math mode, RCL is for math operator (extra-space), stu for text column and " for extra-space

The `retainorgcmds option prevents IEEEtrantools from overriding existing LaTeX commands (itemize, enumerate and description)

See also the documentation : p.17 and more