What is Laplace of cos t log t delta(t−pi)?

Nyasia Flowers

Nyasia Flowers

Answered question

2022-09-11

What is Laplace of cos t log t   δ ( t π )?

Answer & Explanation

Azul Lang

Azul Lang

Beginner2022-09-12Added 20 answers

In general
δ ( t c ) f ( t ) d t = f ( c )
the trick is to notice that
δ ( t c ) f ( t ) d t = c ϵ c + ϵ δ ( t c ) f ( t ) d t
for ϵ > 0. If we can assume that f(t) is continuous in the interval ( c ϵ , c + ϵ ), that is, for every η > 0 there exists an ϵ such that | f ( t ) f ( c ) | < η whenever | t c | < ϵ, then
c ϵ c + ϵ δ ( t c ) f ( t ) d t = f ( c ) c ϵ c + ϵ δ ( t c ) d t = f ( c )
This tells you that
L { δ ( t c ) f ( t ) } = f ( c ) e c s
In your case f ( t ) = cos ( t ) log ( t ) δ ( t π ), with c = π. Thus
L { cos ( t ) log ( t ) δ ( t π ) } = log ( π ) e π s

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