Find the derivatives F'(t) and F''(t) for each of the following complex-valued functions of the real variable t. (a) F(t)=e^{(1−i)t} (b) F(t)=e^{3 i t

remolatg 2021-06-23 Answered

Find the derivatives F'(t) and F''(t) for each of the following complex-valued functions of the real variable t.
(a)\(\displaystyle{F}{\left({t}\right)}={e}^{{{\left({1}−{i}\right)}{t}}}\)
(b)\(\displaystyle{F}{\left({t}\right)}={e}^{{{3}{i}{t}}}\)
(c) \(\displaystyle{F}{\left({t}\right)}={e}^{{{\left({2}+{3}{i}\right)}{t}}}\)

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Plainmath recommends

  • Ask your own question for free.
  • Get a detailed answer even on the hardest topics.
  • Ask an expert for a step-by-step guidance to learn to do it yourself.
Ask Question

Expert Answer

comentezq
Answered 2021-06-24 Author has 10515 answers

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

Have a similar question?
Ask An Expert
29
 

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Plainmath recommends

  • Ask your own question for free.
  • Get a detailed answer even on the hardest topics.
  • Ask an expert for a step-by-step guidance to learn to do it yourself.
Ask Question
...