What does it mean for a function to be C^infinity?

Question

anonymous · Answer

The answer will always be that number to the square root of infinity.

anonymous · Answer

$$\mathbb C^\infty$$Like this?

anonymous · Answer

Why are all polynomials C^infinity?
I mean, if we consider the polynomial f(x)=x^2...how does that satisfy the definition.
ps. badreferences u are correct

anonymous · Answer

That means it's infinite dimensional complex. Are you in a modern algebra class?

anonymous · Answer

ok whats$$C^\infty(R)$$
cause thats what im getting at

anonymous · Answer

I'm a little hazy, but isn't that the Cauchy-Reimmann surface?

anonymous · Answer

Wait, no, nevermind, ignore me.

anonymous · Answer

What class is this?

anonymous · Answer

higher linear algebra

anonymous · Answer

its not very specific, but its a linear map !!

helder_edwin · Answer

$C^\infty$ means that it can be differentiated infinitely-many times.

anonymous · Answer

Exactly, amen

anonymous · Answer

$\mathbb C^\infty$ means something else.

anonymous · Answer

$$C^\infty(R)->C^\infty(R)$$

anonymous · Answer

$C$ or $\mathbb C$?

anonymous · Answer

single C.
I see. Now, wolfram says that all polynomials are C^infty.
Can someone explain to be how f(x) = ax^2 +bx+c is C^infty? considering that it is a polynomial?

anonymous · Answer

Ah okay.

$C(R)$ means the complex conjugate of $R$. This is the sans-serif $C$ as opposed to the shell $\mathbb C$.

anonymous · Answer

At least, in particle physics. Maybe it has another meaning in higher linear algebra. Apologies.

Infinity complex conjugates doesn't sound right.

anonymous · Answer

@TuringTest Hehe, you're better at math than me.

anonymous · Answer

@Zarkon You too get in here.

anonymous · Answer

All good guys ! http://sci4um.com/about19318.html So f(x)=0 is C^infinity :) im happy now ! thanks guys !

anonymous · Answer

Manifold calculus? Hard stuff to learn in linear algebra.

turingtest · Answer

I think it's what @helder_edwin said. Any polynomial can be differentiated to 0 by taking n+1 derivatives where n is the order of the polynomial. The derivative of 0 is 0, which can, of course, be differentiated ad infinitum. Hence all (at least finite-degree, I don't know about otherwise) polynomials are infinitely differentiable.

anonymous · Answer

Yes, @extremity found it. Why do they use the same symbol as the one you use when you learn about symmetry in physics?

anonymous · Answer

Two completely different things.