- #1
ognik
- 643
- 2
Looking for someone to check my working & answers please. Problem is 'find all the zeros of sin z'
A) sin z = sin(x+iy) = sin(x)cosh(y) + i cos(x)sinh(y)
Roots are when sin(x)cosh(y) = 0 = cos(x)sinh(y)
$If \: sinh(y)=0, then \: cosh(y)=1 \: (cosh^2 - sinh^2=1) $
$ \therefore sin(x) = 0, \therefore x = n\pi, n=0,\pm1,\pm2 ... $
$ y = arcsinh(0) = 0 $
$ \therefore $ 1st root is $ (0,n\pi) $
B) I also would like to do it using exponents:
$ sin z = \frac{1}{2i}\left(e^{z}-e^{-z}\right) $
$ \therefore \left(e^{z}-e^{-z}\right) = 0 $
$ \therefore e^{z} = e^{-z}, \: \therefore x+iy = -(x+iy)$
This just gives x = 0 = y which seems trivial?
Thanks in advance.
A) sin z = sin(x+iy) = sin(x)cosh(y) + i cos(x)sinh(y)
Roots are when sin(x)cosh(y) = 0 = cos(x)sinh(y)
$If \: sinh(y)=0, then \: cosh(y)=1 \: (cosh^2 - sinh^2=1) $
$ \therefore sin(x) = 0, \therefore x = n\pi, n=0,\pm1,\pm2 ... $
$ y = arcsinh(0) = 0 $
$ \therefore $ 1st root is $ (0,n\pi) $
B) I also would like to do it using exponents:
$ sin z = \frac{1}{2i}\left(e^{z}-e^{-z}\right) $
$ \therefore \left(e^{z}-e^{-z}\right) = 0 $
$ \therefore e^{z} = e^{-z}, \: \therefore x+iy = -(x+iy)$
This just gives x = 0 = y which seems trivial?
Thanks in advance.