Hello:
I have this function:
$\displaystyle f(z)=\frac{cos(t)+isin(t)}{e^{t}}$
whose domain is the line:
$\displaystyle z=(1+i)t$
I need to find the analytic continuation for the whole complex plane. I don't know how to start it.
Thank you!
Hello:
I have this function:
$\displaystyle f(z)=\frac{cos(t)+isin(t)}{e^{t}}$
whose domain is the line:
$\displaystyle z=(1+i)t$
I need to find the analytic continuation for the whole complex plane. I don't know how to start it.
Thank you!
You need to express $\displaystyle f(z)=\frac{\cos t+i\sin t}{e^{t}}$ as a function of $\displaystyle z = \frac t{1+i}$. Start by using that fact that $\displaystyle \cos t+i\sin t = e^{it}$ and then use the index laws. Finally, substitute $\displaystyle z = \frac t{1+i}$ (and simplify the result).
Hello, thanks for answering, here what i've done:
$\displaystyle f(z)=\frac{cos(t)+isin(t)}{e^{t}}$
as
$\displaystyle e^{it}=cost+isint$
so
$\displaystyle f(z)=\frac{cos(t)+isin(t)}{e^{t}}=\frac{e^{it}}{e^ t}=e^{it-t}=e^{t(i-1)}$
we know
$\displaystyle t=(1+i)z$
so
$\displaystyle f(z)=e^{t(i-1)}=e^{-2z}$
This is what you meant? If it is, I don't understand why we have to do this to obtain the analytic continuation for the whole complex plane.
Thank you!
Yes, that is what I meant (apart from the mistake at the end). The reason for doing it that way is that if you want f to be an analytic function of z then you have to express it in terms of z. Fortunately, you end up with the expression $\displaystyle f(z) = e^{iz}$, which certainly is analytic on the whole complex plane, and therefore gives the required analytic continuation for f(z).