LaTeX Notation for Equations

Eq. 1
$$ P \left(\lambda,\gamma,\psi_0,\Delta\lambda,I_{\rm B},\Delta\psi,\Delta\theta \right)
= \int_{-\psi_0-\Delta\psi}^{+\psi_0+\Delta\psi} \frac{2}{3} \frac{e_0
\Delta\lambda \Delta\theta I_{\rm B} \rho^2}{\varepsilon_0 \beta \lambda^4 \gamma^4}
\left[1 + \left( \gamma \psi \right)^2 \right]^2 \left[
K_{2/3}\left[\xi\left(\lambda,\psi \right)  \right]^2 + \frac{\left(\gamma
\psi \right)^2}{1+\left(\gamma \psi \right)^2}
K_{1/3}\left[\xi(\lambda,\psi)  \right]^2 \right] $$




Eq. 2
$$ P_\sigma \left(\lambda,\gamma,\psi,\rho,\Delta\lambda,I_{\rm B},
\Delta\theta\right) = \frac{2}{3} ~ \frac{e_0 \Delta\lambda \Delta\theta I_{\rm B}
\rho^2}{\varepsilon_0 \beta \lambda^4 \gamma^4} \left[1 + \left( \gamma \psi
\right)^2 \right]^2 K_{2/3}\left[\xi\left(\lambda,\psi \right) \right]^2 $$




Eq. 3
$$ P_\pi \left(\lambda,\gamma,\psi,\rho,\Delta\lambda,I_{\rm B},\Delta\theta
\right) = \frac{2}{3} ~ \frac{e_0 \Delta\lambda \Delta\theta I_{\rm B}
\rho^2}{\varepsilon_0 \beta \lambda^4 \gamma^4} \left[1 + \left( \gamma \psi
\right)^2 \right] (\gamma\psi)^2 \, 
K_{1/3}\left[\xi(\lambda,\gamma)\right]^2 $$

Orbital Frequency
$$\nu_0 = \frac{\nu_{\rm RF}}{h} $$

Gamma factor
$$\gamma =\frac{E_0}{m_{\rm e} \,c_I^2} $$

Beta factor
$$\beta \sqrt{1-\gamma^2}$$

Orbital radius
$$\rho_0 = \frac{\beta \, c_I \, h}{2\pi \,\nu_{\rm RF}} $$

Characteristic Wavelength
$$\lambda_c = \frac{4\pi \, \rho_0}{3 \lambda^3} $$

Number of Electrons
$$ N_e = \frac{I_{\rm B} \, 2\pi\, \rho_0}{e_0 \, \beta \, c}$$

Energy loss per turn
$$ U_0 = \frac{e_0^2 \, \gamma^4}{3\varepsilon_0 \, \rho_0}$$

Radiated power per electron
$$P_\gamma = \frac{c_I \, e_0^2 \, \gamma^4}{6\pi \,\varepsilon_0 \, \rho_0}$$

Total radiated power
$$P_{\rm tot} - N_e \, P_\gamma$$

Synchrotron Oscillation frequency
$$u_s = \nu_0 
\sqrt{\frac{h\,e_0 V_{\rm RF} \cos(\psi_s)}{2\pi \, \beta \, E_0(n-1)}}$$

Natural energy spread
$$\frac{\sigma_E}{E_0} = \sqrt{\frac{C_q\, \gamma^2}{\rho_0} ~ 
\frac{1-n}{3-4n}}$$