For the function $$G(w) = \frac{\sqrt2}{2}-\frac{\sqrt2}{2}e^{iw},$$ show that $$G(w) = -\sqrt2ie^{iw/2} \sin(w/2).$$
Hey everyone, I'm very new to this kind of maths and would really appreciate any help. Hopefully I can get an idea from this and apply it to other similar questions. Thank you.
Use the definition for the complex sine: $$ \sin(z)=\frac{ e^{iz}-e^{-iz} } {2i} $$ Thus, $$-\sqrt{2}ie^{i\frac{w}{2}}\sin\frac{w}{2} =-\sqrt{2}ie^{i\frac{w}{2}}(\frac{1}{2i}(e^{i\frac{w}{2}} - e^{-i\frac{w}{2}})) $$
Now simplify to get your result.