How to find the real or imaginary part of an equation involving complex numbers? I am currently using the Debye model and need to find the real and imaginary parts of the equation. The Debye equation is $$ \epsilon_\text{r} = \epsilon_\infty + \frac{\epsilon_\text{s} - \epsilon_\infty}{1+i\frac{\omega\epsilon}{\sigma}} - i\frac{\sigma}{\omega\epsilon_0}, $$ where $i^2 = -1$. How would I find $\mathfrak{R}\\{\epsilon_\text{r}\\}$ and $\mathfrak{I}\\{\epsilon_\text{r}\\}$ from this equation?

Assuming all symbols are real except for the complex unit. Then the second term is fine; consider the first: \begin{eqnarray} \frac{\epsilon_{s}-\epsilon_{\infty}}{1+i\frac{\omega \epsilon}{\sigma}} &=& \frac{\epsilon_{s}-\epsilon_{\infty}}{1+i\frac{\omega \epsilon}{\sigma}}\frac{i}{i}\\\ &=& \frac{i(\epsilon_{s}-\epsilon_{\infty})}{i-\frac{\omega \epsilon}{\sigma}} \end{eqnarray} Now multiply both numerator and denominator by the conjugate of the denominator: \begin{eqnarray} \frac{(-i-\frac{\omega \epsilon}{\sigma})}{(-i-\frac{\omega \epsilon}{\sigma})}\frac{i(\epsilon_{s}-\epsilon_{\infty})}{i-\frac{\omega \epsilon}{\sigma}} &=& \frac{i(\epsilon_{s}-\epsilon_{\infty})(-i-\frac{\omega \epsilon}{\sigma})}{1+\frac{\omega^{2}}{\epsilon^{2}\sigma^{2}}}\\\ \end{eqnarray} Now multiply out the brackets on the numerator and include the second term I omitted above and then collect real and imaginary parts.

Good luck!