1. **Stating the problem:** We want to express the complex function $$G^* = G_0 \left(1 + \left(i \frac{\omega}{\omega_n}\right)^{0.5}\right)$$ in terms of its real and imaginary parts. 2. **Recall the formula and rules:** The term $$\left(i \frac{\omega}{\omega_n}\right)^{0.5}$$ means the square root of a complex number. We use the fact that $$i = e^{i\pi/2}$$, so $$\left(i \frac{\omega}{\omega_n}\right)^{0.5} = \left(\frac{\omega}{\omega_n}\right)^{0.5} e^{i\pi/4} = \left(\frac{\omega}{\omega_n}\right)^{0.5} \left(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4}\right).$$ 3. **Calculate the trigonometric values:** $$\cos \frac{\pi}{4} = \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}.$$ 4. **Substitute back:** $$\left(i \frac{\omega}{\omega_n}\right)^{0.5} = \left(\frac{\omega}{\omega_n}\right)^{0.5} \left(\frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2}\right).$$ 5. **Rewrite $$G^*$$:** $$G^* = G_0 \left(1 + \left(\frac{\omega}{\omega_n}\right)^{0.5} \frac{\sqrt{2}}{2} + i G_0 \left(\frac{\omega}{\omega_n}\right)^{0.5} \frac{\sqrt{2}}{2}\right).$$ 6. **Separate real and imaginary parts:** - Real part: $$G_0 \left(1 + \left(\frac{\omega}{\omega_n}\right)^{0.5} \frac{\sqrt{2}}{2}\right)$$ - Imaginary part: $$G_0 \left(\frac{\omega}{\omega_n}\right)^{0.5} \frac{\sqrt{2}}{2}$$ **Final answer:** $$\boxed{G^* = G_0 \left[1 + \frac{\sqrt{2}}{2} \left(\frac{\omega}{\omega_n}\right)^{0.5}\right] + i G_0 \frac{\sqrt{2}}{2} \left(\frac{\omega}{\omega_n}\right)^{0.5}}.$$