Math examples

$$\begin{eqnarray} \phi(\lambda) & = & \frac{1} {2 \pi i}\int^{c+i\infty}_{c-i\in... ...\epsilon , \bar{\epsilon} & = & \mbox{actual/average energy loss} \end{eqnarray}$$

Since (6) or (7d) should 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).

$$\begin{eqnarray}V \mathbf{\pi}^{sr} & = & \left< \sum_i M_i \mathbf{V}_i \mathbf... ...m_i \sum_\alpha \mathbf{p}_{i\alpha} \mathbf{f}_{i\alpha} \right> \end{eqnarray}$$

$$\begin{align}B_{ij}^\alpha & = \left(B_{ij}^\alpha\right)_0 + \le... ...} + \frac{\d N_k^\beta}{\d X_i} \frac{\d N_k^\alpha}{\d X_j} \right)\end{align}$$