I would like to calculate the real part of the following expression:
$$-\lambda_1 \psi_1 \mathrm{e}^{-\lambda_1 t}\left[\frac{\mathrm{e}^{\left(\lambda_1+i\omega\right)t}-1}{\lambda_1+i\omega}\right]$$
It is some time since I have had to perform such calculations, and would appreciate a worked solution with some explanation if possible!
(n.b. This is not for any form of assessed work)
If we denote $z$ the complexe number, and we suppose that all coefficients $\lambda_1,\psi_1,\omega,t$ are real numbers, we have:$$z=\frac{\lambda_1\psi_1}{\lambda_1^2+\omega^2}(e^{-\lambda_1 t}-e^{i\omega t})(\lambda_1-i\omega)=\frac{\lambda_1\psi_1}{\lambda_1^2+\omega^2}(e^{-\lambda_1 t}-\cos(\omega t)- i\sin (\omega t))(\lambda_1-i\omega)$$ Then : $$\mathcal Re(z)=\frac{\lambda_1\psi_1}{\lambda_1^2+\omega^2}(\lambda_1(e^{-\lambda_1 t}-\cos(\omega t))-\omega \sin(\omega t))$$