4.49 Complex solutions

Note that the roots in case (iii) are $\alpha\pm i\beta$, where by Euler’s formula

$$e^{(\alpha+i\beta)x}=e^{\alpha x}e^{i\beta x}=e^{\alpha x}(\cos\beta x+i\sin\beta x),$$

$$e^{(\alpha-i\beta)x}=e^{\alpha x}e^{i\beta x}=e^{\alpha x}(\cos\beta x-i\sin\beta x);$$

hence we can write the real solutions from case (iii) as

$$e^{\alpha x}\cos\beta x=2^{-1}\bigl(e^{(\alpha+i\beta)x}+e^{(\alpha-i\beta)x}\bigr),$$

$$e^{\alpha x}\sin\beta x=(2i)^{-1}\bigl(e^{(\alpha+i\beta)x}-e^{(\alpha-i\beta)x}\bigr).$$

Note that $e^{i\beta x}$ goes round the unit circle as $x$ increases, while $e^{\alpha x}$ is a real exponential functions such that $e^{\alpha x}\rightarrow\infty$ as $x\rightarrow\infty$ for $\alpha>0$, $e^{\alpha x}\rightarrow 0$ as $x\rightarrow\infty$ for $\alpha<0$.