- #1
mancini0
- 31
- 0
Homework Statement
Hi guys, I am having difficulty with the following problem:
Show the complex sine function has real valued zeros.
Homework Equations
I know that sin(z) = (e^iz - e^-iz) /2i
The Attempt at a Solution
I have to find roots of sin(z), i.e, when the above equation = 0. First I multiplied both sides by 2i, to cancel the 2i in the denominator.
leaving e^iz -e^-iz = 0.
then e^iz = e^-iz .
So e^iz = 1/e^1z
Now I am stuck. I can break it down to trigonometric form, but I'm not sure how that helps...
cos(z) + i sin(z) = 1 / cos(z) + i sin(z)
I think I have to get the above expression into real and imaginary (u and v) parts:
cos(x+iy) +i (sin(x+iy) = 1/[cos(x+iy) + i*sin(x+iy)]
Any guidance would be much appreciated...
Thanks!