A test of Equation References

Here is a labeled equation: $$x+1\over\sqrt{1-x^2}\label{ref1}$$ with a reference to ref1: \ref{ref1}, and another numbered one with no label: $$x+1\over\sqrt{1-x^2}$$ This one uses \nonumber: $$x+1\over\sqrt{1-x^2}\nonumber$$

Here's one using the equation environment: \begin{equation} x+1\over\sqrt{1-x^2} \end{equation} and one with equation* environment: \begin{equation*} x+1\over\sqrt{1-x^2} \end{equation*}

This is a forward reference [\ref{ref2}] and another \eqref{ref2} for the following equation: $$x+1\over\sqrt{1-x^2}\label{ref2}$$ More math: $$x+1\over\sqrt{1-x^2}$$ Here is a ref inside math: $$\ref{ref2}+1$$ and text after it. \begin{align} x& = y_1-y_2+y_3-y_5+y_8-\dots && \text{by \eqref{ref1}}\\ & = y'\circ y^* && \text{(by \eqref{ref3})}\\ & = y(0) y' && \text {by Axiom 1.} \end{align} Here's a bad ref [\ref{ref4}] to a nonexistent label.

An alignment: \begin{align} a&=b\label{ref3}\cr &=c+d \end{align} and a starred one: \begin{align*} a&=b\cr &=c+d \end{align*}