\documentclass[a4paper]{jarticle}
\usepackage{amsmath,amssymb}
\begin{document}
\textbf{AmS-LaTeX 変換例}
\newline
\textbf{英文変換例}
\begin{equation}
f(x,y)=\dfrac{1}{4\pi ^{2}}\displaystyle \int_{-\infty }^{\infty } \displaystyle \int_{-\infty }^{\infty } F(u,\upsilon )e^{j(ux+\upsilon y)}\,du \,d\upsilon \tag{$4.2$}
\end{equation}
$P(\phi ,\omega )$ is actually $F(u,\upsilon )$ in polar coordinates with $u=\omega \sin \phi $ and $\upsilon =-\omega \cos \phi $. The transformation from cartesian into polar coordinates replaces the differential $dud\upsilon $ by $|\omega |d\omega d\phi $ where $|\omega |$ is the determinant of the Jacobian matrix of the trasformation. Thus we find,
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\hspace{2cm}
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\textbf{連立した式、フォントスタイル（ボールド体）の変換例}
\newline
ここで、格子点$(x,y)$における原子の横ずれ$\Delta _{x}(x,y),\Delta _{y}(x,y)$（以下$\Delta _{x},\Delta _{y}$とする）により歪んだ格子像は、
\begin{equation}
\begin{split}
g(x,y)&= g(\boldsymbol{x+\Delta _{x}\/},\boldsymbol{y+\Delta _{y}\/})\\
&=\displaystyle \sum_{n=-\infty }^{\infty } \displaystyle \sum_{m=-\infty }^{\infty } G_{nm}\exp \left[2\pi i\left\{\dfrac{n(\boldsymbol{x+\Delta _{x}\/})}{X}+\dfrac{m(\boldsymbol{y+\Delta _{y}\/})}{Y}\right\}\right]\\
&=\displaystyle \sum_{n=-\infty }^{\infty } \displaystyle \sum_{m=-\infty }^{\infty } G_{nm}\exp \left[2\pi i\left\{\dfrac{n\Delta _{x}}{X}+\dfrac{m\Delta _{y}}{Y}\right\}\right]\exp \left[2\pi i\left\{\dfrac{nx}{X}+\dfrac{my}{Y}\right\}\right]
\end{split} \tag{$1.2$}
\end{equation}
\textbf{シグマなどの上下限の複数行にわたる変換例}
\[ s=\displaystyle \sum_{k=1}^{n} \left(\displaystyle \sum_{\begin{subarray}{c}
j=1\\ j\neq k\end{subarray}
}^{n} \dfrac{xi-x_{j}}{x_{k}-x_{j}}\right)y_{k} \]
\textbf{文字修飾の変換例}
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一文字の文字修飾
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${}^{t}a$
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$\vec{a} $
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$\hat{\jmath } $
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$\tilde{\imath } $
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$\bar{x} $
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$\ddddot{x} $
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複数文字の文字修飾
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$\overrightarrow{ab} $
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$\overline{abc} $
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$\underline{abc} $
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\textbf{表の変換例（セル単位で右揃え、左揃え、センタリングの設定可）}
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\begin{table}[htbp]
\caption{東北}
\label{東北}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|} \hline
\multicolumn{1}{|c|}{県名} & \multicolumn{1}{|c|}{人口} & \multicolumn{1}{|c|}{世帯数} & \multicolumn{1}{|c|}{面積} & \multicolumn{1}{|c|}{人口密度} & \multicolumn{1}{|c|}{世帯当り人数} \\ \hline
\multicolumn{1}{|c|}{青森} & \multicolumn{1}{|r|}{$1508045$} & \multicolumn{1}{|r|}{$506198$} & \multicolumn{1}{|r|}{$9619$} & \multicolumn{1}{|r|}{$156.78$} & \multicolumn{1}{|r|}{$2.98$} \\ \hline
\multicolumn{1}{|c|}{岩手} & \multicolumn{1}{|r|}{$1430332$} & \multicolumn{1}{|r|}{$453404$} & \multicolumn{1}{|r|}{$15277$} & \multicolumn{1}{|r|}{$93.63$} & \multicolumn{1}{|r|}{$3.15$} \\ \hline
\multicolumn{1}{|c|}{宮城} & \multicolumn{1}{|r|}{$2299396$} & \multicolumn{1}{|r|}{$758450$} & \multicolumn{1}{|r|}{$7292$} & \multicolumn{1}{|r|}{$315.33$} & \multicolumn{1}{|r|}{$3.03$} \\ \hline
\multicolumn{1}{|c|}{秋田} & \multicolumn{1}{|r|}{$1225868$} & \multicolumn{1}{|r|}{$383269$} & \multicolumn{1}{|r|}{$11612$} & \multicolumn{1}{|r|}{$105.57$} & \multicolumn{1}{|r|}{$3.20$} \\ \hline
\multicolumn{1}{|c|}{山形} & \multicolumn{1}{|r|}{$1256481$} & \multicolumn{1}{|r|}{$358345$} & \multicolumn{1}{|r|}{$9327$} & \multicolumn{1}{|r|}{$134.71$} & \multicolumn{1}{|r|}{$3.51$} \\ \hline
\multicolumn{1}{|c|}{福島} & \multicolumn{1}{|r|}{$2135646$} & \multicolumn{1}{|r|}{$651402$} & \multicolumn{1}{|r|}{$13784$} & \multicolumn{1}{|r|}{$154.94$} & \multicolumn{1}{|r|}{$3.28$} \\ \hline
\end{tabular}
\end{center}
\end{table}
\end{document}
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