Mathematical function in the form $f(\cos \theta)$

Question

I was reading about Chebyshev polynomials and came across this interesting notation that defines a Chebyshev polynomial of the first kind:

$$ F_n(\cos \theta) = \cos (n\theta) $$

What threw me was $\cos \theta $ on the left hand side of the equation. I've only ever seen the syntax $f(x)$, $f(x, y)$ e.t.c before which reads "$f$ is a function of $x$" or "$f$ is a function of $x$ and $y$". I was just expecting it to be $F_n(\theta)$.

What does it mean to include "modifiers" inside the "function of" syntax? e.g. what does it mean if I wrote $f(2x) = 4x + 1$?

I then came across https://www3.nd.edu/~zxu2/acms40390F11/sec8-3.pdf which defines the Chebyshev polynomial as:

$$ F_n(\theta) = \cos (n\arccos \theta) $$

This form makes more sense to me. And if you replace $\theta$ with $\cos \theta$ you arrive back at the original definition. So is the initial form just a way of writing it as to avoid an inverse cosine? It seems like a weird thing to do just to avoid it. I can intuitively read this later form as "f is a function of theta", but don't get the same intuition when reading the first form as "f is a function of cosine theta". Is this style common in mathematics?

Answer 1

I think you have slightly rewritten both of those sources.

For things like the first, I most often see: $$F_n(\cos \theta) = \cos n\theta = \cos(n \theta),$$ parentheses added for readability. The $n$ multiplier is inside the $\cos$.

This is slightly unusual syntax, but is not that uncommon. It's an implicit definition of an auxiliary variable $\theta$ in terms of the input, which we may as well call $x$. It's well-defined because the Chebyshev polynomials are only meant to be used on $[-1, 1]$. With $x = \cos \theta$ it seems like the implicit mapping leaves not only the sign of $\theta$ undefined, but also any addition of multiples of $2\pi$, however any choice of $\theta$ gives the same result once you take the cosine of any integer multiple of $\theta$, so the overall function is well defined.

For the second, I usually see instead:

$$F_n(x) = \cos (n\arccos x).$$ This is just fixing a specific standard $\theta$, but they really do mean the same thing.

What does it mean to include "modifiers" inside the "function of" syntax? e.g. what does it mean if I wrote $f(2x)=4x+1$

Because $2x$ is an invertible function, it means exactly what you might expect -- the same as $f(x) = 2x + 1$.