Complex exponentials - homework

[Poetria]

Could you give me a hint how to attack this problem?

Find a complex number z = a+i*b such that f(t)=Re e^(z*t) where f(t)=cos(2*pi*t)

I have begun as follows:

e^((a+i*b)*t)=e^(a*t)*(cos(b)+i*sin(b))

Re e^(z*t)= e^(a*t)*cos(b)

What to do now?

 

[Ssnow]

Hi, $t$ is real? If is yes so you must find $a,b$ from $e^{at}\cos{b}=cos(2\pi t)$ ...

 

[Poetria]

Hi, $t$ is real? If is yes so you must find $a,b$ from $e^{at}\cos{b}=cos(2\pi t)$ ...

Yes, t is a real variable.I know I must but how? Thank you very much. :)

 

[Ssnow]

another hint: $e^{at}cos{(b)}=e^{0t}cos{(2\pi t)}$, you can read now $a=...$ and $b=...$

 

[Poetria]

another hint: $e^{at}cos{(b)}=e^{0t}cos{(2\pi t)}$, you can read now $a=...$ and $b=...$

Ok, I got it. :) Great. Thank you very much. :)