Give definitions of \(e^z, \sin z, \cos z\) where \(z\) is a complex number and verify that \[ \sin(z_1+z_2) = \sin z_1 \cos z_2 + \cos z_1 \sin z_2. \] Prove that \[ x+\frac{x^4}{4!}+\frac{x^7}{7!}+\dots = \frac{1}{3}e^x+\frac{2}{3}e^{-\frac{x}{2}}\sin\left(\frac{x\sqrt{3}}{2}-\frac{\pi}{6}\right), \] where \(x\) is real.