Find L{F(t)} if F(t)={(sin t, text(between ) 0<t<pi),(0,text(between ) pi<t<2pi):}

Kamila Frye 2022-10-14 Answered
Find L { F ( t ) } if
F ( t ) = { sin t between  0 < t < π 0 between  π < t < 2 π
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Answers (1)

ehedem26
Answered 2022-10-15 Author has 13 answers
I presume the function is periodic. Have you tried considering
L [ f ( t ) ] = 0 e p t f ( t ) d t = n = 0 n π ( n + 1 ) π e p t f ( t ) = n = 0 2 n π ( 2 n + 1 ) π e p t sin ( t )
But this is just
n = 0 ( e p π + 1 ) ( e π ( ( 2 n + 1 ) ) p ) ( p sin ( 2 π n ) + cos ( 2 π n ) ) p 2 + 1
Since sin ( 2 π n ) = 0 and cos ( 2 π n ) = 1, this becomes
n = 0 ( e p π + 1 ) ( e π ( ( 2 n + 1 ) ) p ) p 2 + 1
And in fact we can pull out terms involving p to obtain
( e π + 1 ) e p π p 2 + 1 n = 0 e 2 n p π
And we evaluate the sum to obtain
( e π + 1 ) e p π p 2 + 1 ( e 2 π p e 2 π p 1 ) = ( e π p + 1 ) csch ( π p ) 2 ( p 2 + 1 )
If this function is not periodic, but 0 for t > 2 π, then
L [ f ( t ) ] = 0 e p t f ( t ) d t = 0 π e p t sin ( t ) d t = e p π + 1 p 2 + 1
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