This laplace Transform are true L(e^(it))=[(e^((i−s)t))/(i−s)]_0^(oo)=(1)/(s−i)?

cimithe4c 2022-10-15 Answered
Laplace Transform Example: L ( e i t ) = [ e ( i s ) t i s ] 0 = 1 s i ?
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Answers (2)

RamPatWeese2w
Answered 2022-10-16 Author has 15 answers
Note that De Moivre's formula tells you that e i t for any t R is bounded because sin t and cos t are bounded.
L ( e i t ) = 0 e ( i s ) t d t = [ e ( i s ) t i s ] 0 = lim t e i t e s t ( i s ) 0 1 i s 1 s i
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duandaTed05
Answered 2022-10-17 Author has 6 answers
Note that for any real t and complex s, we have
| e ( i s ) t | = e Re ( ( i s ) t ) = e a t ,
where a = Re ( s ). (Recall in general that | e z | = e Re ( z ) for any complex z.) So you can see that if a = Re ( s ) > 0, we have lim t | e ( i s ) t | = 0, and so
lim t e ( i s ) t i s = 0.
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