# Laplace transform of complex exponential

EvLer
I just want to be sure I understand this correctly, usually L[f(t)] = 1/(s-a), where $$f(t) = e^{at}$$, but if it is a complex number would still be 1/(s - complex_number)?
techinically, i think it should be, since every number can be reprsented as complex number. Just want to be sure about this with Laplace transform.
thanks much.

Homework Helper
Definitely! You can demonstrate it by direct integration.

Homework Helper
EvLer said:
I just want to be sure I understand this correctly, usually L[f(t)] = 1/(s-a), where $$f(t) = e^{at}$$, but if it is a complex number would still be 1/(s - complex_number)?
techinically, i think it should be, since every number can be reprsented as complex number. Just want to be sure about this with Laplace transform.
thanks much.
Yes and it is a nice way to find the laplace transforms of sin and cos.
L[cos(a x)+i sin(a x)]=L[exp(i x)]=1/(s-a i)=(s+a i)/(s^2+a^2)
hence (equating real and imaginary parts)
L[cos(a x)]=s/(s^2+a^2)
L[sin(a x)]=a/(s^2+a^2)