Talk:Entire function/Archive 1

Archive 1

Trigonometric and hyperbolic functions entire?

I'm not a mathematician, but how can it be true that the trigonometric and hyperbolic functions are entire? For example, the tangent function isn't even defined at π/2, so how can it be holomorphic there? —Caesura^(t) 02:20, 1 March 2006 (UTC)

Whoever wrote that meant sine, cosine, sinh, and cosh, but I remved that thing altogether. Thanks. Oleg Alexandrov (talk) 03:08, 1 March 2006 (UTC)

I'm a layman, and have difficulties tying together the following two concepts:

1. Liouville's theorem which states that if an entire function f is bounded, then f is constant
2. Trigonometric functions like sine and cosine are entire.

Isn't sine a bounded entire function that is not constant, in contradiction to Liouville's theorem ? —Preceding unsigned comment added by 213.224.83.33 (talk) 17:54, 4 September 2008 (UTC)

You're thinking of real variables. While it is true that sinx is bounded for x real, this is no longer the case if x is complex. Indeed, consider for instance

$\sin(it)={\frac {e^{i(it)}-e^{-i(it)}}{2i}}={\frac {e^{-t}-e^{t}}{2i}}$

which is unbounded. siℓℓy rabbit (talk) 19:22, 4 September 2008 (UTC)