Weird error in typing latex eqn

Question

typing this:

\begin{equation} \label{u_cart}
 \begin{array}{ll}
  u(x,y)_{N=5}=
  \begin{cases}
    \begin{array}{ll}
 \frac{(x+i y)^5 \text{BesselJ}\left[\text{{``}1.5{''}} \sqrt{x^2+y^2},5\right]}{\left(x^2+y^2\right)^{5/2}} & 0<\sqrt{x^2+y^2}<\frac{1}{2} \\
 \frac{(x+i y)^5 \left(\text{BesselJ}\left[3 \sqrt{x^2+y^2},5\right]+\text{BesselY}\left[3 \sqrt{x^2+y^2},5\right]\right)}{\left(x^2+y^2\right)^{5/2}}
& \frac{1}{2}<\sqrt{x^2+y^2}<1 \\
 \frac{(x+i y)^5 \text{HankelH1}\left[\sqrt{x^2+y^2},5\right]}{\left(x^2+y^2\right)^{5/2}} & \sqrt{x^2+y^2}>1 \endgroup
 \end{cases}
\end{array}
\end{array}
\end{equation}

I get 37 weird errors, although it seems to show nicely. What is wrong?

Thanks!

Answer 1

You did not need the arrays, because the first one operated on a single column, and the 2nd one was redundant to what cases provides (further, one of the arrays was mismatched with cases). You also had a stray \endgroup thrown in for no reason, which was excised. I also added extra vertical space with \\[8pt].

\documentclass{article}
\usepackage{amsmath}
\usepackage[margin=1in]{geometry}
\begin{document}   
\begin{equation} \label{u_cart}
  u(x,y)_{N=5}=
  \begin{cases}
 \frac{(x+i y)^5 \text{BesselJ}\left[\text{{``}1.5{''}} 
 \sqrt{x^2+y^2},5\right]}{\left(x^2+y^2\right)^{5/2}} 
 & 0<\sqrt{x^2+y^2}<\frac{1}{2} \\[8pt]
%
 \frac{(x+i y)^5 \left(\text{BesselJ}\left[3 \sqrt{x^2+y^2},5\right]
 +\text{BesselY}\left[3 \sqrt{x^2+y^2},5\right]\right)}
 {\left(x^2+y^2\right)^{5/2}}
 & \frac{1}{2}<\sqrt{x^2+y^2}<1 \\[8pt]
%
 \frac{(x+i y)^5 \text{HankelH1}\left[\sqrt{x^2+y^2},5\right]}{\left(x^2+y^2\right)^{5/2}} & \sqrt{x^2+y^2}>1 
 \end{cases}
\end{equation}
\end{document}

Answer 2

Here's a variation on Steven Segletes's answer. First, use a dcases environment (provided by the mathtools package, which is a superset of the amsmath package) to show the fractions in display style. Second, get rid of the \left and \right directives, as their sizing operations are good for the material at hand. Third, define BesselJ, BesselY, and HankelH1 as math operators, to get the spacing right.

\documentclass{article}
\usepackage[margin=1in]{geometry}
\usepackage{mathtools} % for 'dcases' environment
\DeclareMathOperator{\BesselJ}{BesselJ}
\DeclareMathOperator{\BesselY}{BesselY}
\DeclareMathOperator{\HankelH}{HankelH1}
\begin{document}   

\begin{equation} \label{u_cart}
u(x,y)^{}_{N=5}=
\begin{dcases}
 \frac{(x+i y)^5 \BesselJ[\textnormal{``1.5''} 
   \sqrt{x^2+y^2},5]}{(x^2+y^2)^{5/2}} 
   & 0<\sqrt{x^2+y^2}<\frac{1}{2} \\[8pt]
%
 \frac{(x+i y)^5 \bigl(\BesselJ[3 \sqrt{x^2+y^2},5]
   +\BesselY[3 \sqrt{x^2+y^2},5]\bigr)}
   {(x^2+y^2)^{5/2}}
   & \frac{1}{2}<\sqrt{x^2+y^2}<1 \\[8pt]
%
 \frac{(x+i y)^5 \HankelH[\sqrt{x^2+y^2},5]}{(x^2+y^2)^{5/2}} 
   & \hphantom{\frac{1}{2}<{}}\sqrt{x^2+y^2}>1 
\end{dcases}
\end{equation}

\end{document}