# Laplace Sine Transform

$L[sin(at)]=\frac{a}{s^{2}+a^{2}}, Re>0$

$L[e^{kt}]=\frac{1}{s-k}, s>k$
$L[e^{-kt}]=\frac{1}{s+k}, s<-k$

$L[sin(at)]=\frac{1}{2i}L[e^{iat}-e^{-iat}]$
$=\frac{1}{2i}L[e^{iat}]-L[e^{-iat}]$
Using the above relations
$=\frac{1}{2i}[\frac{1}{s-ia}-\frac{1}{s+ia}], s>ia, s<-ia$

The problem is that I don't understand, how s>ia and s<-ia could imply that Real part of s>0?

## Answers and Replies

lurflurf
Homework Helper
Complex numbers are not ordered,
s>ia and s<-ia
does not make sense.
Real part of s>0
Is needed to assure the existence of the integral.