# Thread: Complex analysis function.

1. ## Complex analysis function.

What i have to do in these case?!

Show that the function f(z)=lnz is differentiable(express z in polar form)

and another question is

Try to differentiate the trigonometric functon
f(z)=cosz and f(z)=sinz using the exponential definition.

Thanks,
NaNa

2. Originally Posted by NaNa
What i have to do in these case?!

Show that the function f(z)=lnz is differentiable(express z in polar form)
In polar form, $z= r e^{i\theta}$ so that $f(z)= ln(r e^{i\theta})$. The fact that any integer multiple of $2\pi$ can be added to $\theta$ giving the same z makes this a multivalued function.

and another question is

Try to differentiate the trigonometric functon
f(z)=cosz and f(z)=sinz using the exponential definition.

Thanks,
NaNa
Well, the "exponential definitions" are $cos(z)= \frac{e^z+ e^{-z}}{2}$ and $sin(z)= \frac{e^z- e^{-z}}{2i}$.

3. Originally Posted by NaNa
What i have to do in these case?!

Show that the function f(z)=lnz is differentiable(express z in polar form)

and another question is

Try to differentiate the trigonometric functon
f(z)=cosz and f(z)=sinz using the exponential definition.

Thanks,
NaNa
The first, if it's differentiable, it's analytic and if it's analytic it satisfies the Cauchy-Riemann equations. So, using:

$Log(z)=\ln(r)+i\Theta$

use the Cauchy-Riemann equations so show it's analytic. For the second, just do what it said: express:

$\cos(z)=\frac{e^{iz}+e^{-iz}}{2}$ and just differentiate it with respect to z. Same dif for $\sin(z)$

4. thanks a lot!!!

5. How I use the Cauchy-Riemann here?!?!

6. is ok I found it