## Section6Some Advanced Ideas

The multi-row displayed mathematics in the proof of the Fundamental Theorem had equations aligned on the equals signs via the & character. Sometimes you don't want that. Here is an example with some differential equations, with each equation centered and unnumbered,

\begin{gather*} {\mathcal L}(y')(s) = s {\mathcal L}(y)(s) - y(0) = s Y(s) - y(0)\\ {\mathcal L}(y'')(s) = s^2 {\mathcal L}(y)(s) - sy(0) - y'(0)= s^2 Y(s) - sy(0) - y'(0)\text{.} \end{gather*}LaTeX has a device where you can interrupt a sequence of equations with a small amout of text and preserve the equation alignment on either side. Here are two tests of that device, with aligned equations and non-aligned equations. Study the source to see use and differences. (The math does not make sense.)

Aligned and numbered first.

\begin{align} {\mathcal L}(y')(s) &= s {\mathcal L}(y)(s) - y(0) = s Y(s) - y(0)\tag{6.1}\\ {\mathcal L}(y'')(s) &= s^2 {\mathcal L}(y)(s) - sy(0) - y'(0)= s^2 Y(s) - sy(0) - y'(0).\tag{6.2}\\ \end{align}And so it follows that,

\begin{align} {\mathcal L}(y')(s)^{++} &= s {\mathcal L}(y)(s) - y(0) = s Y(s) - y(0)\tag{6.3}\\ {\mathcal L}(y'')(s)^{5} &= s^2 {\mathcal L}(y)(s) - sy(0) - y'(0)= s^2 Y(s) - sy(0) - y'(0).\tag{6.4} \end{align}Now with no numbers and no alignment. We include two cross-references in the `intertext`

portion for testing.

First an external reference to `http://example.com`

and internal cross-reference to Corollary 4.1. And so it follows that,

Tables can get quite complex. Simple ones are simpler, such as this example of numerical computations for Euler's method.

\(i\) | \(t_i\) | \(x_i\) | \(y_i\) |

0 | 0.00 | 0.0000 | 0.5000 |

1 | 0.20 | 0.1000 | 0.4800 |

2 | 0.40 | 0.1960 | 0.4560 |

3 | 0.60 | 0.2872 | 0.4295 |

4 | 0.80 | 0.3731 | 0.4027 |

5 | 1.00 | 0.4536 | 0.3783 |

6 | 1.20 | 0.5293 | 0.3591 |

7 | 1.40 | 0.6011 | 0.3480 |

8 | 1.60 | 0.6707 | 0.3474 |

9 | 1.80 | 0.7402 | 0.3603 |

10 | 2.00 | 0.8123 | 0.3900 |