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Let $\hat{\theta}$ be one of the position operators in cylindrical coordinates $(r,\theta,z)$. Then my question is, for what periodic functions $f$ (with period $2\pi$) is $f(\hat{\theta})$ a Hermitian operator? In case it's not clear, $f(\hat{\theta})$ is defined by a Taylor series.

The reason I ask is because of my answer here dealing with an Ehrenfest theorem relating angular displacement and angular momentum. This journal paper says that the two simplest functions which satisfy this condition are sine and cosine. First of all I'm not sure how to prove that they satisfy the condition, and second of all I want to see what other functions satisfy it.

What makes this tricky is that the adjoint of an infinite sum is not equal to the infinite sum of the adjoints.

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You're overthinking this. The function-of-an-operator $f(\hat\theta)$ can be defined via its Taylor series, but it can also be defined by its action on the angle representation: $$⟨\theta|f(\hat\theta) = f(\theta)⟨\theta|.$$ In this sense it is entirely analogous to functions of the position operator, $f(\hat x)$, in ordinary 1D quantum mechanics.

As such, the only requirement on $f$ for $f(\hat \theta)$ to be hermitian is that $f$ be real-valued (and, implicitly, that it be a well-defined function, i.e. no multivaluedness). It doesn't even need to be continuous (so e.g. $\hat \theta$ itself is hermitian); discontinuities do affect some of the operator's other properties (like, in the previous example, the derivative $f'(\hat\theta)$, which can have distribution-valued components like $\delta(\theta-\pi)$ where $f$ has a discontinuity), but not its hermiticity.

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