# Tagged Questions

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

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### Determine numerical infinity for Schrodinger equation $−\psi''(z) − (iz)^ N \psi(z) = E\psi(z)$

Consider the following one dimensional Schrodinger equation within the complex plane of $z$ $$−ψ''(z) − (iz)^ N ψ(z) = Eψ(z).$$ where $N$ can be any real number, the boundary condition is $ψ(z) → 0$ ...
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### Which one is asymptotically larger?

The question is to find out which among $n^\sqrt{n}$ or $n^(log_2 n)$ is asymptotically larger? Now as a solution I read somewhere that if we take log on both sides and then compute which one is ...
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### Prove that the value of the constant $C$ must be $1$

After proving the prime number theorem in class, our professor directs us to a remark by Lagrange that for large values of $x$, $\pi(x)$ is approximately equal to $$\frac{x}{\log x - B}.$$ (This is ...
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### Asymptotic distribution of zero-drift Geometric Brownian Motion as $t \to \infty$

If we fix the drift at $\mu = 0$, then my geometric brownian motion will have stationary mean, but it seems that the variance will grow without bound. What does the limiting distribution look like for ...
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### $\lim_{n\to\infty}\frac{f(n)}{h(n)}$ if $f\in o(g)$ and $g\in O(h)$

I would appreciate it if anybody could check my attempted solution to this question. I'm guessing since the question says 'values' rather than 'value' that I haven't finished the question. So if you ...
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### The behaviour of $\sum k^a 2^k$.

Let $a$ be a real number, I want to find a simple equivalent (or if it is possible, an asymptotic expansion) of $$\sum_{k=1}^n k^a 2^k.$$ I believe that the sum is $\sim n^a 2^{n+1}$ (tried many ...
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### Proof involving the Big-O notation

I am stuck on a proof question involving the big-O notation: Prove that if $f(x)$ is $O(x^3)$ then $f(x+x^2)$ is also $O(x^3)$. I am stuck because $f(x)$ can be any arbitrary polynomial. I started ...
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### Graph with both slant and horizontal asymptotes

Is there such a graph? A graph that increases at a decreasing rate with the graph approaching a slant asymptote as x decreases to negative infinity while the graph approaching a horizontal asymptote ...
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