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<table style="max-width: 60vw; margin: auto;"><tr><th colspan="2">Differentials</th></tr><tr><td>\bigg(\frac{\partial^2} {\partial x^2} + \frac {\partial^2}{\partial y^2} \bigg){\big\lvert\varphi (x+iy)\big\rvert}^2</td><td style="position: relative"><math display="block"><mo symmetric="true" stretchy="true"minsize="2.4em" maxsize="2.4em">(</mo><mfrac><mrow><msup><mi>∂</mi><mn>2</mn></msup></mrow><mrow><mi>∂</mi><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mi>∂</mi><mn>2</mn></msup></mrow><mrow><mi>∂</mi><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac><mo symmetric="true" stretchy="true"minsize="2.4em" maxsize="2.4em">)</mo><msup><mrow><mo symmetric="true" stretchy="true"minsize="1.2em" maxsize="1.2em">|</mo><mi>φ</mi><mo symmetric="false" stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>i</mi><mi>y</mi><mo symmetric="false" stretchy="false">)</mo><mo symmetric="true" stretchy="true"minsize="1.2em" maxsize="1.2em">|</mo></mrow><mn>2</mn></msup></math></td></tr><tr><th colspan="2">Environments</th></tr><tr><td>f(x) = \begin{cases}1/3 & \text{if }0 \le x \le 1; \\ 2/3 & \text{if }3\le x \le 4;\\ 0 &\text{elsewhere.} \end{cases}</td><td style="position: relative"><math display="block"><mi>f</mi><mo symmetric="false" stretchy="false">(</mo><mi>x</mi><mo symmetric="false" stretchy="false">)</mo><mo>=</mo><mrow><mo stretchy="true">{</mo><mtable class="menv-cells-left menv-cases"><mtr><mtd><mn>1</mn><mo>/</mo><mn>3</mn></mtd><mtd><mtext>if </mtext><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>1</mn><mo>;</mo></mtd></mtr><mtr><mtd><mn>2</mn><mo>/</mo><mn>3</mn></mtd><mtd><mtext>if </mtext><mn>3</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>4</mn><mo>;</mo></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mtext>elsewhere.</mtext></mtd></mtr></mtable></mrow></math></td></tr><tr><td>\begin{pmatrix}
\begin{pmatrix}a&b\\c&d
\end{pmatrix} &
\begin{pmatrix}e&f\\g&h
\end{pmatrix} \\ 0 &
\begin{pmatrix}i&j\\k&l
\end{pmatrix}
\end{pmatrix}</td><td style="position: relative"><math display="block"><mrow><mo stretchy="true">(</mo><mtable class="menv-arraylike"><mtr><mtd><mrow><mo stretchy="true">(</mo><mtable class="menv-arraylike"><mtr><mtd><mi>a</mi></mtd><mtd><mi>b</mi></mtd></mtr><mtr><mtd><mi>c</mi></mtd><mtd><mi>d</mi></mtd></mtr></mtable><mo stretchy="true">)</mo></mrow></mtd><mtd><mrow><mo stretchy="true">(</mo><mtable class="menv-arraylike"><mtr><mtd><mi>e</mi></mtd><mtd><mi>f</mi></mtd></mtr><mtr><mtd><mi>g</mi></mtd><mtd><mi>h</mi></mtd></mtr></mtable><mo stretchy="true">)</mo></mrow></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mrow><mo stretchy="true">(</mo><mtable class="menv-arraylike"><mtr><mtd><mi>i</mi></mtd><mtd><mi>j</mi></mtd></mtr><mtr><mtd><mi>k</mi></mtd><mtd><mi>l</mi></mtd></mtr></mtable><mo stretchy="true">)</mo></mrow></mtd></mtr></mtable><mo stretchy="true">)</mo></mrow></math></td></tr><tr><td>\det\begin{vmatrix}
c_0&c_1&c_2&\dots& c_n\\
c_1 & c_2 & c_3 & \dots &
c_{n+1}\\ c_2 & c_3 & c_4
&\dots & c_{n+2}\\ \vdots
&\vdots&\vdots & &\vdots
\\c_n & c_{n+1} & c_{n+2}
&\dots&c_{2n}
\end{vmatrix} > 0</td><td style="position: relative"><math display="block"><mi>det</mi><mo>â¡</mo><mspace width="0.1667em" /><mrow><mo stretchy="true">|</mo><mtable class="menv-arraylike"><mtr><mtd><msub><mi>c</mi><mn>0</mn></msub></mtd><mtd><msub><mi>c</mi><mn>1</mn></msub></mtd><mtd><msub><mi>c</mi><mn>2</mn></msub></mtd><mtd><mi>⋯</mi></mtd><mtd><msub><mi>c</mi><mi>n</mi></msub></mtd></mtr><mtr><mtd><msub><mi>c</mi><mn>1</mn></msub></mtd><mtd><msub><mi>c</mi><mn>2</mn></msub></mtd><mtd><msub><mi>c</mi><mn>3</mn></msub></mtd><mtd><mi>⋯</mi></mtd><mtd><msub><mi>c</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mi>c</mi><mn>2</mn></msub></mtd><mtd><msub><mi>c</mi><mn>3</mn></msub></mtd><mtd><msub><mi>c</mi><mn>4</mn></msub></mtd><mtd><mi>⋯</mi></mtd><mtd><msub><mi>c</mi><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></msub></mtd></mtr><mtr><mtd><mi>â‹®</mi></mtd><mtd><mi>â‹®</mi></mtd><mtd><mi>â‹®</mi></mtd><mtd></mtd><mtd><mi>â‹®</mi></mtd></mtr><mtr><mtd><msub><mi>c</mi><mi>n</mi></msub></mtd><mtd><msub><mi>c</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd><mtd><msub><mi>c</mi><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></msub></mtd><mtd><mi>⋯</mi></mtd><mtd><msub><mi>c</mi><mrow><mn>2</mn><mi>n</mi></mrow></msub></mtd></mtr></mtable><mo stretchy="true">|</mo></mrow><mo>></mo><mn>0</mn></math></td></tr><tr><th colspan="2">Everything</th></tr><tr><td>\sum_{p\text{ prime}} f(p)=\int_{t>1} f(t)d\pi(t)</td><td style="position: relative"><math display="block"><munder><mo movablelimits="false">∑</mo><mrow><mi>p</mi><mtext> prime</mtext></mrow></munder><mi>f</mi><mo symmetric="false" stretchy="false">(</mo><mi>p</mi><mo symmetric="false" stretchy="false">)</mo><mo>=</mo><msub><mo movablelimits="false">∫</mo><mrow><mi>t</mi><mo>></mo><mn>1</mn></mrow></msub><mi>f</mi><mo symmetric="false" stretchy="false">(</mo><mi>t</mi><mo symmetric="false" stretchy="false">)</mo><mi>d</mi><mi>Ï€</mi><mo symmetric="false" stretchy="false">(</mo><mi>t</mi><mo symmetric="false" stretchy="false">)</mo></math></td></tr><tr><td>\lim_{n \to +\infty} \frac{\sqrt{2\pi n}}{n!} \genfrac (){}{}n{e}^n = 1</td><td style="position: relative"><math display="block"><munder><mi>lim</mi><mrow><mi>n</mi><mo>→</mo><mi>+</mi><mi>∞</mi></mrow></munder><mo>â¡</mo><mspace width="0.1667em" /><mfrac><mrow><msqrt><mrow><mn>2</mn><mi>Ï€</mi><mi>n</mi></mrow></msqrt></mrow><mrow><mi>n</mi><mi>!</mi></mrow></mfrac><msup><mrow><mo stretchy="true">(</mo><mfrac><mi>n</mi><mrow><mi>e</mi></mrow></mfrac><mo stretchy="true">)</mo></mrow><mi>n</mi></msup><mo>=</mo><mn>1</mn></math></td></tr><tr><td>\det(A) = \sum_{\sigma \in S_n} \epsilon(\sigma) \prod_{i=1}^n a_{i, \sigma_i}</td><td style="position: relative"><math display="block"><mi>det</mi><mo>â¡</mo><mo symmetric="false" stretchy="false">(</mo><mi>A</mi><mo symmetric="false" stretchy="false">)</mo><mo>=</mo><munder><mo movablelimits="false">∑</mo><mrow><mi>σ</mi><mo>∈</mo><msub><mi>S</mi><mi>n</mi></msub></mrow></munder><mi>ϵ</mi><mo symmetric="false" stretchy="false">(</mo><mi>σ</mi><mo symmetric="false" stretchy="false">)</mo><munderover><mo movablelimits="false">âˆ</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>a</mi><mrow><mi>i</mi><mo>,</mo><msub><mi>σ</mi><mi>i</mi></msub></mrow></msub></math></td></tr><tr><th colspan="2">Fractions</th></tr><tr><td>\frac{x+y^2}{k+1}</td><td style="position: relative"><math display="block"><mfrac><mrow><mi>x</mi><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfrac></math></td></tr><tr><td>x+y^\frac{2}{k + 1}</td><td style="position: relative"><math display="block"><mi>x</mi><mo>+</mo><msup><mi>y</mi><mfrac><mrow><mn>2</mn></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfrac></msup></math></td></tr><tr><td>\frac{a}{b/2}</td><td style="position: relative"><math display="block"><mfrac><mrow><mi>a</mi></mrow><mrow><mi>b</mi><mo>/</mo><mn>2</mn></mrow></mfrac></math></td></tr><tr><td>a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{a_4}}}}</td><td style="position: relative"><math display="block"><msub><mi>a</mi><mn>0</mn></msub><mo>+</mo><mrow displaystyle="true" scriptlevel="0"><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mrow displaystyle="true" scriptlevel="0"><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mi>a</mi><mn>2</mn></msub><mo>+</mo><mrow displaystyle="true" scriptlevel="0"><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mi>a</mi><mn>3</mn></msub><mo>+</mo><mrow displaystyle="true" scriptlevel="0"><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mi>a</mi><mn>4</mn></msub></mrow></mfrac></mrow></mrow></mfrac></mrow></mrow></mfrac></mrow></mrow></mfrac></mrow></math></td></tr><tr><td>a_0+\frac{1}{a_1+\frac{1}{a_2+\frac{1}{a_3+ \frac{1}{a_4}}}}</td><td style="position: relative"><math display="block"><msub><mi>a</mi><mn>0</mn></msub><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mi>a</mi><mn>2</mn></msub><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mi>a</mi><mn>3</mn></msub><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mi>a</mi><mn>4</mn></msub></mrow></mfrac></mrow></mfrac></mrow></mfrac></mrow></mfrac></math></td></tr><tr><td>\binom{p}{2} x^2 y^{p-2} - \frac{1}{1-x} \frac{1}{1-x^2}</td><td style="position: relative"><math display="block"><mrow><mo stretchy="true">(</mo><mfrac linethickness="0em"><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo stretchy="true">)</mo></mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mo>−</mo><mi>x</mi></mrow></mfrac><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></math></td></tr><tr><th colspan="2">Integrals</th></tr><tr><td>\int_1^x \frac{dt}{t}</td><td style="position: relative"><math display="block"><msubsup><mo movablelimits="false">∫</mo><mn>1</mn><mi>x</mi></msubsup><mfrac><mrow><mi>d</mi><mi>t</mi></mrow><mrow><mi>t</mi></mrow></mfrac></math></td></tr><tr><td>\int\!\!\!\int_D dx,dy</td><td style="position: relative"><math display="block"><mo movablelimits="false">∫</mo><mspace width="-0.16666667em" style="margin-left: -0.16666667em" /><mspace width="-0.16666667em" style="margin-left: -0.16666667em" /><mspace width="-0.16666667em" style="margin-left: -0.16666667em" /><msub><mo movablelimits="false">∫</mo><mi>D</mi></msub><mi>d</mi><mi>x</mi><mo>,</mo><mi>d</mi><mi>y</mi></math></td></tr><tr><th colspan="2">Over Under Braces</th></tr><tr><td>\overbrace{x +\cdots + x} ^{k \text{ times}}</td><td style="position: relative"><math display="block"><mover><mover><mrow><mi>x</mi><mo>+</mo><mi>⋯</mi><mo>+</mo><mi>x</mi></mrow><mo stretchy="true">âž</mo></mover><mrow><mi>k</mi><mtext> times</mtext></mrow></mover></math></td></tr><tr><td>{\underbrace{\overbrace{ \mathstrut a,\dots,a}^{k ,a\rq\text{s}}, \overbrace{ \mathstrut b,\dots,b}^{l, b\rq\text{s}}}_{k+l \text{ elements}}}</td><td style="position: relative"><math display="block"><mrow><munder><munder><mrow><mover><mover><mrow> height="0.7em" /><mi>a</mi><mo>,</mo><mi>…</mi><mo>,</mo><mi>a</mi></mrow><mo stretchy="true">âž</mo></mover><mrow><mi>k</mi><mo>,</mo><mi>a</mi><mi>′</mi><mtext>s</mtext></mrow></mover><mo>,</mo><mover><mover><mrow> height="0.7em" /><mi>b</mi><mo>,</mo><mi>…</mi><mo>,</mo><mi>b</mi></mrow><mo stretchy="true">âž</mo></mover><mrow><mi>l</mi><mo>,</mo><mi>b</mi><mi>′</mi><mtext>s</mtext></mrow></mover></mrow><mo stretchy="true">âŸ</mo></munder><mrow><mi>k</mi><mo>+</mo><mi>l</mi><mtext> elements</mtext></mrow></munder></mrow></math></td></tr><tr><th colspan="2">Roots</th></tr><tr><td>\sqrt{1+\sqrt{1+\sqrt{1+ \sqrt{1+\sqrt{1+\sqrt{1+ \sqrt{1+x}}}}}}}</td><td style="position: relative"><math display="block"><msqrt><mrow><mn>1</mn><mo>+</mo><msqrt><mrow><mn>1</mn><mo>+</mo><msqrt><mrow><mn>1</mn><mo>+</mo><msqrt><mrow><mn>1</mn><mo>+</mo><msqrt><mrow><mn>1</mn><mo>+</mo><msqrt><mrow><mn>1</mn><mo>+</mo><msqrt><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></msqrt></mrow></msqrt></mrow></msqrt></mrow></msqrt></mrow></msqrt></mrow></msqrt></mrow></msqrt></math></td></tr><tr><th colspan="2">Scripts</th></tr><tr><td>x^2y^2</td><td style="position: relative"><math display="block"><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>2</mn></msup></math></td></tr><tr><td>_2F_3</td><td style="position: relative"><math display="block"><msub><mrow></mrow><mn>2</mn></msub><msub><mi>F</mi><mn>3</mn></msub></math></td></tr><tr><td>x^{2y}</td><td style="position: relative"><math display="block"><msup><mi>x</mi><mrow><mn>2</mn><mi>y</mi></mrow></msup></math></td></tr><tr><td>2^{2^{2^x}}</td><td style="position: relative"><math display="block"><msup><mn>2</mn><mrow><msup><mn>2</mn><mrow><msup><mn>2</mn><mi>x</mi></msup></mrow></msup></mrow></msup></math></td></tr><tr><td>y_{x^2}</td><td style="position: relative"><math display="block"><msub><mi>y</mi><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></msub></math></td></tr><tr><td>y_{x_2}</td><td style="position: relative"><math display="block"><msub><mi>y</mi><mrow><msub><mi>x</mi><mn>2</mn></msub></mrow></msub></math></td></tr><tr><td>x_{92}^{31415} + \pi</td><td style="position: relative"><math display="block"><msubsup><mi>x</mi><mrow><mn>92</mn></mrow><mrow><mn>31415</mn></mrow></msubsup><mo>+</mo><mi>Ï€</mi></math></td></tr><tr><td>x_{y^a_b}^{z^c_d}</td><td style="position: relative"><math display="block"><msubsup><mi>x</mi><mrow><msubsup><mi>y</mi><mi>b</mi><mi>a</mi></msubsup></mrow><mrow><msubsup><mi>z</mi><mi>d</mi><mi>c</mi></msubsup></mrow></msubsup></math></td></tr><tr><td>y_3'''</td><td style="position: relative"><math display="block"><msub><mi>y</mi><mn>3</mn></msub><mi>′</mi><mi>′</mi><mi>′</mi></math></td></tr><tr><th colspan="2">Summations</th></tr><tr><td>\sum_{\genfrac{}{}{0mu}{2}{0 \le i \le m}{0 < j < n}} P(i, j)</td><td style="position: relative"><math display="block"><munder><mo movablelimits="false">∑</mo><mrow><mrow displaystyle="false" scriptlevel="1"><mfrac linethickness="0mu"><mrow><mn>0</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>m</mi></mrow><mrow><mn>0</mn><mo><</mo><mi>j</mi><mo><</mo><mi>n</mi></mrow></mfrac></mrow></mrow></munder><mi>P</mi><mo symmetric="false" stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo symmetric="false" stretchy="false">)</mo></math></td></tr><tr><td>\sum_{i=1}^p \sum_{j=1}^q \sum_{k=1}^r a_{ij}b_{jk}c_{ki}</td><td style="position: relative"><math display="block"><munderover><mo movablelimits="false">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>p</mi></munderover><munderover><mo movablelimits="false">∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>q</mi></munderover><munderover><mo movablelimits="false">∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>r</mi></munderover><msub><mi>a</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mi>b</mi><mrow><mi>j</mi><mi>k</mi></mrow></msub><msub><mi>c</mi><mrow><mi>k</mi><mi>i</mi></mrow></msub></math></td></tr></table></body>
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