(a)Let \(\displaystyle{F}{\left({t}\right)}={e}^{{{\left({1}-{i}\right)}{t}}}\). Then \(F'(t)=(1-i)e^{(1-i)t}\) and \(\displaystyle{F}'{\left({t}\right)}={\left({1}-{i}\right)}^{{{2}}}{e}^{{{\left({1}-{i}\right)}{t}}}=-{2}{i}{e}^{{{\left({1}-{i}\right)}{t}}}\).

(b) Let \(\displaystyle{F}{\left({t}\right)}={e}^{{{3}{i}{t}}}\). Then \(\displaystyle{F}'{\left({t}\right)}={3}{i}{e}^{{{3}{i}{t}}}\) and \(\displaystyle{F}{''}{\left({t}\right)}={\left({3}{i}\right)}^{{{2}}}{e}^{{{3}{i}{t}}}=-{9}{e}^{{{3}{i}{t}}}\).

(c) Let \(\displaystyle{F}{\left({t}\right)}={e}^{{{\left({2}+{3}{I}\right)}{t}}}\). Then \(\displaystyle{F}'{\left({t}\right)}={\left({2}+{3}{i}\right)}{e}^{{{\left({2}+{3}{i}\right)}{t}}}\) and \(\displaystyle{F}{''}{\left({t}\right)}={\left({2}+{3}{i}\right)}^{{{2}}}{e}^{{{\left({2}+{3}{i}\right)}{t}}}={\left(-{5}+{12}{i}\right)}{e}^{{{\left({2}+{3}{i}\right)}{t}}}\).