Math examples

$$\phi(\lambda) = \frac{1} {2 \pi i}\int^{c+i\infty}_{c-i\infty} \exp \left( u \ln u + \lambda u \right ) du \hspace{1cm}\mbox{for } c \geq 0$$ (1)

$$\lambda = \frac{\epsilon -\bar{\epsilon} }{\xi} - \gamma' - \beta^2 - \ln \frac{\xi} {E_{\rm max}}$$ (2)

$$\gamma = 0.577215\dots \mathrm{\hspace{5mm}(Euler's\ constant)}$$ (3)

$$\gamma' = 0.422784\dots = 1 - \gamma$$ (4)

$$\epsilon , \bar{\epsilon} = \mbox{actual/average energy loss}$$ (5)

Since 2 and 6 hold for arbitrary $\delta\mathbf{c}$-vectors, it is clear that $\mathcal{N}(A) = \mathcal{R}(B)$ and that when $y=B(x)$ one has...
...the Pythagorians knew infinitely many solutions in integers to $a^2+b^2=c^2$. That no non-trivial integer solutions exist for $a^n+b^n=c^n$ with integers $n>2$ has long been suspected (Fermat, c.1637). Only during the current decade has this been proved (Wiles, 1995).

$$V \mathbf{\pi}^{sr} = \left< \sum_i M_i \mathbf{V}_i \mathbf{V}_i + \sum_i \sum_{j>i} \mathbf{R}_{ij} \mathbf{F}_{ij}\right>$$ (6)

$$= \left< \sum_i M_i \mathbf{V}_i \mathbf{V}_i + \sum_{i}\sum_{j>i}\... ... j\beta} - \sum_i \sum_\alpha \mathbf{p}_{i\alpha} \mathbf{f}_{i\alpha} \right>$$