# Math Help - CAUCHY - RIEMANN Help please

1. ## CAUCHY - RIEMANN Help please

Why did Cauchy - RIemann invented this.. arrghh... help

Show that sin(z) satisfies the CAUCHY - RIEMANN conditions for analyticity for all values of z. Does [1/sin(z)] satisfy similar conditions? Calculate the derivative of [1/sin(z)] at z=0, $\pm \Pi /2,\pm \Pi,\pm 3\Pi/2$. Comment on each answer.

2. ## Re: CAUCHY - RIEMANN Help please

Have you tried going through the Cauchy-Riemann conditions with this function? If not, you should give it a go!

3. ## Re: CAUCHY - RIEMANN Help please

Originally Posted by timeforchg
Why did Cauchy - RIemann invented this.. arrghh... help
So that you know when a complex function is differentiable...

Have you started by writing \displaystyle \begin{align*} \sin{(z)} = \sin{(x + i\,y)} = \sin{x}\cosh{y} + i \cos{x}\sinh{y} = u(x, y) + i\,v(x, y) \end{align*} and evaluating their partial derivatives?

4. ## Re: CAUCHY - RIEMANN Help please

f(z) = sin (z)
= sin (x + iy)
= sin x cosh y + i cos x sinh y

thus,

u(x,y)=sin x cosh y ....... v(x,y)= cos x sinh y

du/dx = cos x ............ dv/dx = -sin x
du/dy = -sinh y ........... dv/dy = cosh y

therefore I will compare it with the Cauchy Riemann formula.

Am I right to say it doesn't satisfies the condition??

5. ## Re: CAUCHY - RIEMANN Help please

Originally Posted by timeforchg
f(z) = sin (z)
= sin (x + iy)
= sin x cosh y + i cos x sinh y

thus,

u(x,y)=sin x cosh y ....... v(x,y)= cos x sinh y

du/dx = cos x ............ dv/dx = -sin x
du/dy = -sinh y ........... dv/dy = cosh y

therefore I will compare it with the Cauchy Riemann formula.

Am I right to say it doesn't satisfies the condition??
Your calculation of the partial derivatives is incorrect. When calculating a partial derivative, all other variables are held constant, so are treated as constants. So with \displaystyle \begin{align*} u = \sin{x}\cosh{y} \end{align*}, when evaluating \displaystyle \begin{align*} \frac{\partial u}{\partial x} \end{align*}, the \displaystyle \begin{align*} \cosh{y} \end{align*} is treated as constant, and constant multiples in derivatives stay where they are... So it should be \displaystyle \begin{align*} \frac{\partial u}{\partial x} = \cos{x}\cosh{y} \end{align*}. Fix the rest.

6. ## Re: CAUCHY - RIEMANN Help please

thanks for notifying me the error and yes they satisfy the condition.

7. ## Re: CAUCHY - RIEMANN Help please

If I were to find whether 1/sin(z) satisfy the C.R condition,
Do I use 1/sin(x) cosh(y) + j cos(x) sinh(y) or the one with inverse [sin(x) cosh(y) + j cos(x) sinh(y)]^-1.
How do I change it to (x+iy)?

8. ## Re: CAUCHY - RIEMANN Help please

Originally Posted by timeforchg
If I were to find whether 1/sin(z) satisfy the C.R condition,
Do I use 1/sin(x) cosh(y) + j cos(x) sinh(y) or the one with inverse [sin(x) cosh(y) + j cos(x) sinh(y)]^-1.
How do I change it to (x+iy)?
Multiply top and bottom by the bottom's conjugate.

9. ## Re: CAUCHY - RIEMANN Help please

Originally Posted by Prove It
Multiply top and bottom by the bottom's conjugate.
when i try to conjugate. I get 0. so does it meant it does not exist?

10. ## Re: CAUCHY - RIEMANN Help please

If the denominator is \displaystyle \begin{align*} \sin{x}\cosh{y} + j\cos{x}\sinh{y} \end{align*}, the conjugate is \displaystyle \begin{align*} \sin{x}\cosh{y} - j\cos{x}\sinh{y} \end{align*}. Multiplying them together will give \displaystyle \begin{align*} \sin^2{x}\cosh^2{y} + \cos^2{x}\sinh^2{y} \end{align*}. This is not 0.

11. ## Re: CAUCHY - RIEMANN Help please

Originally Posted by Prove It
If the denominator is \displaystyle \begin{align*} \sin{x}\cosh{y} + j\cos{x}\sinh{y} \end{align*}, the conjugate is \displaystyle \begin{align*} \sin{x}\cosh{y} - j\cos{x}\sinh{y} \end{align*}. Multiplying them together will give \displaystyle \begin{align*} \sin^2{x}\cosh^2{y} + \cos^2{x}\sinh^2{y} \end{align*}. This is not 0.
I use the wrong equation that is 1\sin(x+iy) thus I conjugate it. Oops. Once again thanks. will try it now.

12. ## Re: CAUCHY - RIEMANN Help please

Yes your right. I did not get 0 when I do it manually. By using my calculator my answer will get 0.

btw am i right to say that 1/sin (z) in (a+jb) format is

sin(x) cosh(y)/ sin^2 (x) cosh^2 (y) + cos^2 (x) sinh^2 (y) - j cos(x) sinh(y)/ sin^2 (x) cosh^2 (y) + cos^2 (x) sinh^2 (y) ??

Correct me if I'm wrong

13. ## Re: CAUCHY - RIEMANN Help please

Originally Posted by timeforchg
Yes your right. I did not get 0 when I do it manually. By using my calculator my answer will get 0.

btw am i right to say that 1/sin (z) in (a+jb) format is

sin(x) cosh(y)/ sin^2 (x) cosh^2 (y) + cos^2 (x) sinh^2 (y) - j cos(x) sinh(y)/ sin^2 (x) cosh^2 (y) + cos^2 (x) sinh^2 (y) ??

Correct me if I'm wrong
It's nasty to do it manually..

If by theorem, 1/sin(z) will satisfy Cauchy - Riemann conditions for all values of z except z = k pi + pi/2, ( k=0, +- 1, +- 2 ...), where the denominator of the function equals to zero. ??

but I try it out, it seems it doesn't satisfy the equation. hmm...

14. ## Re: CAUCHY - RIEMANN Help please

SOLVED!! Thanks a lot guys!! Appreciate it.