A reoccurring statement in condensed matter physics is vaguely the following: Let $f:\mathbb{R}^n\times\mathbb{R}\rightarrow\mathbb{C}$ be some function. $\mathbb{R}^n$ are the spatial coordinates and $\mathbb{R}$ represents temperature. This function is shown to behave very differently at $T\rightarrow 0$ and $T\rightarrow \infty$. For example, it may hava a finite value for low $T$ but is identically $0$ at large enough $T$. Another example is that $f \sim \mid{r}\mid ^k$ for low $T$ and $f \sim e^{-m\mid r\mid}$ for large enough $T$. By $\sim$ I mean that the function converges to these functions at these limits.
Now, it is claimed that these types of functions can not be analytic with respect to $T$ for all $T$: There is a point in which the behavior is singular. My questions are the following:
- How does one prove this for the cases above?
- What type of arguments and from which fields are used for these types of problems? Particular theorems will be very welcomed
- A proof for similar cases will also be a good answer, as my goal is understanding the formal arguments