Unboundness and periodicity for complex trig functions

[ppy]

Hi
I just found out that cos(z) and sin(z) are unbounded and tend to ∞ which I find strange ! But the part I'm struggling with is that I can't reconcile that fact with the fact that they both have a period of 2pi. Surely that means that each value in the range 0-2pi is repeated in the range 2pi-4pi and so on ?
Thanks

 

[Mandelbroth]

Hi
I just found out that cos(z) and sin(z) are unbounded and tend to ∞ which I find strange ! But the part I'm struggling with is that I can't reconcile that fact with the fact that they both have a period of 2pi. Surely that means that each value in the range 0-2pi is repeated in the range 2pi-4pi and so on ?
Thanks

Since we know that, for example, $\sin(z)=\frac{e^{iz}-e^{-iz}}{2i}$, if we set $z=ix$, then $\sin(ix)=\frac{e^{-x}-e^{x}}{2i}$.

For values $x_0\in(0,2\pi)$, is there a corresponding $x_1\in(2\pi,4\pi)$ for which $\sin(ix_0)=\sin(ix_1)$? What does this imply about the periodicity of the sine function as the imaginary part of $z$ gets larger while the real part stays the same?

 

[ppy]

I think I might be getting there ! The period is 2pi but there is no imaginary period ? So this means sin z repeats for every real value of 2pi but never repeats for imaginary values ?

 

[Mandelbroth]

I think I might be getting there ! The period is 2pi but there is no imaginary period ? So this means sin z repeats for every real value of 2pi but never repeats for imaginary values ?

Essentially, yes.

 

[mathman]

For pure imaginary arguments, sin and cos are essentially sinh and cosh for real arguments.