# Questions tagged [asymptotics]

Questions involving asymptotic analysis, including growth of functions, Big-O notation, Big-Omega and Big-Theta notations.

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### The Complexity of “The Baby Shark Song”.

This question is just for fun. I hope it's received in the same goofy spirit in which I wrote it. I just had the pleasure of reading Knuth's "The Complexity of Songs" and I thought it'd be hilarious ...
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### Eigenvalue problem for $−\psi''(x) − (ix)^ N \psi(x) = E\psi(x)$ in complex plane

To find the eigenvalue $E$ in the complex plane of $x$ for one dimensional Schrodinger equation $$−\psi''(x) − (ix)^ N \psi(x) = E\psi(x).$$ where $N$ can be any real number, the boundary condition ...
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### Asymptotic related to the infinite product of sine

The amount is somewhat complicated ($x$ is a constant): $$S_n=\sum_{k=1}^n\ln\left(1-\frac{\sin^2\big(x/(2n+1)\big)}{\sin^2\big(k\pi/(2n+1)\big)}\right)\tag{*}$$ I want to enrich my handy powerful ...
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### Upper bound for the widest matrix with no two subsets of columns with the same vector sum

Over at PPCG there is an ongoing contest going on to find the largest matrix without a certain property, called property $X$. The description is as follows (copied from the question). A circulant ...
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### Does the “prime ant” ever backtrack?

A few mathematical questions have come up from the question "The prime ant 🐜" on the Programming Puzzles & Code Golf Stack Exchange. Here is how the prime ant is defined: Initially, we have ...
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### Strong upper bound for consecutive p-smooth numbers

Størmer's Theorem provides a method to find all consecutive p-smooth numbers. In Lehmer's paper On a problem of Størmer, a very weak upper bound is given (Theorem 7). M. F. Hasler observes at A002072 ...
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### Fractional part summation and asymptotic expansion error

Let us consider the sum $$\displaystyle T_K=\sum_{n \geq \sqrt{K}}^{{K}} \left\{ \sqrt {n^2-K} \right\}$$ where $K$ is a positive integer and where $\{ \}$ indicates the fractional part. Its ...