# Tagged Questions

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### How find this value of $A$?

Question： Let $z\in C$ Find this value $A$,such $$\lim_{k\to +\infty}\left(k-\dfrac{W_{k^2}(z)}{W_{k}(z)}\right)= A\cdot i$$ where $i^2=-1$,and $w_{k}(z)$ is Lambert $W$ function:see ...
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### Is $y(x)=\frac{1}{2}M\left[1-\cos\left(\frac{\pi}{M}x\right)\right]$ an integer

Let ${\rm y}\left(x\right) = \frac{1}{2}M\left[1-\cos\left(\frac{\pi}{M}x\right)\right]$. Is ${\rm y}\left(x\right)$ an integer for each $x = 1,2,\ldots,M$ when $M\to\infty$ ?.
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### Limit of the integral, integral of the limit (no dominated convergence)

I'm working on the integral: $\lim_{x \rightarrow 0} \int_0^T \frac{1}{\sqrt{\pi t}} \exp \left(-\frac{x^2}{t}-t\right) \left(1-\frac{t}{T} \right) \mbox{d}t$ I'ld like to inverse the integral and ...
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### Evaluating limit involving hypergeometric function $\lim_{z\rightarrow1^-}{}_2F_1(-a,-b,-(a+b),z)$

The following expression involving a hypergeometric function popped out as a solution to an integral that I've been working on (via Gradshteyn and Ryzhik): ...
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### Error Function limit

$$\prod_{n=1}^{\infty}{\frac{2}{\sqrt{\pi}}\int_0^n e^{-x^{2}} \mathrm{d}x} \approx 0.83874$$ Is it a known constant? I couldn't find anything about it. Do you know ways to calculate the value ...
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### Find $\lim_{x\to \infty} \ln(\exp(\operatorname{LmW}(x))+1)(\exp(\operatorname{LmW}(x))+1) - x - \ln(x)$

Find $\lim_{x\to \infty} \ln(e^{\operatorname{LambertW}(x)}+1)(e^{\operatorname{LambertW}(x)}+1) - x - \ln(x)$ Where the $LambertW$ function is defined here : http://en.wikipedia.org/wiki/Lambert_W ...
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### How to prove asymptotic limit of an incomplete Gamma function

How can I prove: $$\lim_{z\to\infty}{\Gamma(z, x)\over\Gamma(z)} = 1$$ ? Here $\Gamma(z, x)$ is the upper incomplete gamma function and $\Gamma(z)$ is the gamma function. This must be something ...
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### Limit of an integral containing a product

I'm stuck on the following problem: given the integral: $$I(N)=\int \prod_{k=1}^N \left(k-\frac{k}{x}\right) \, dx$$ calculate the following limit: $$I_{\infty}(N)=\lim_{x\to\infty}I(N)$$ I know that ...
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### Show $\lim_{x\to0}\frac{\Gamma(x)}{\psi(x)}=-1$

How to show that $$\lim_{x\to0}\frac{\Gamma(x)}{\psi(x)}=-1$$ where $\psi(x)$ is the digamma function.
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### Compute $\lim_{x\to\infty} \frac{{(x!)}^{\frac{1}{x}-1} (x\Gamma(x+1) \psi^{(0)}(x+1)-x! \log(x!))}{x^2}$

What's the strategy one may use when facing a limit like this one? I think it's more important to know the possible ways to go than the answer itself. It's a problem that came to my mind again when I ...
Is there any simple way of computing the following sum? $$\sum_{k=1}^\infty \frac1{k\space k!}$$