5 votes

If $f(x) =\sin^{1200}(x) \ln(1+x)^{500}\arctan^{300}(x)$ how to find $f^{(2000)}(0)$?

Using the series expansion of $f(x)$ is probably the intended method: $$\sin x = x - \frac{x^3}6 + \cdots \\ \log(1+x) = x - \frac{x^2}2 + \cdots \\ \arctan x = x - \frac{x^3}3 + \cdots \\[2ex] \...
user170231's user avatar
  • 18.9k
3 votes

What is the relationship among $a, b$ and $c$ for $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-\left(ax^2+2bxy+cy^2\right)}dx dy=1$

Integrate in polar coordinates as follows \begin{align} &\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-\left(ax^2+2bxy+cy^2\right)}dx dy\\ =& \int_0^{2\pi}\int_0^\infty e^{-r^2\left(a\cos^...
Quanto's user avatar
  • 95.3k
3 votes
Accepted

Prove that a subset of $R^n$ is bounded

Note that, if $(x,y)\in S$, then $4x^2 + y^2 = 4$. Now, $y^2$ is always positive, so $4x^2 + y^2 = 4\implies4x^2 \leq 4\implies x^2 \leq 1$ so $x\in [-1,1]$ Analogously, $4x^2$ is positive, and so $4x^...
Eduardo Magalhães's user avatar
2 votes
Accepted

Line integral off by a factor of 2

From the very scratch. I can only confirm what Ninad Munshi was saying that you are asking for problems when you do vector integrals with non Cartesian unit vectors. To disentangle the notation I ...
Kurt G.'s user avatar
  • 13.2k
2 votes

Show that $ \int_0^{\pi\over 2}\frac{\sin(2nx)}{\sin^{2n+2}(x)}\frac{1}{e^{2\pi \cot x}-1}dx =(-1)^{n-1}\frac{2n-1}{4(2n+1)} $

(Assuming $n$ is a positive integer.) As you have noted $$\frac{\sin 2nx}{\sin^{2n}x}=\Im(\cot x+i)^{2n}=\frac1{2i}\left((\cot x+i)^{2n}-(\cot x-i)^{2n}\right),$$ the given integral, after the ...
metamorphy's user avatar
  • 38.8k
2 votes
Accepted

Limit of difference of two functions

It's not clear to me if this is the "reasoning" you are hoping is explained, but here goes. When you prove the existence of a limit using the $\epsilon$-$\delta$ definition, you are proving ...
Ahmad Barhoumi's user avatar
1 vote

Does a reverse triangle inequality hold for any nonnegative convex function?

Note that if $f(p,q)$ is a function with the given properties, then so is $h(p,q) = f(p,q)g(p)$ for any positive function $g$ whatsoever. But $h(p,q)+h(q,r)<h(p,r)$ is equivalent to $f(p,q)g(p)+f(q,...
Greg Martin's user avatar
  • 77.6k
1 vote
Accepted

Help to write the domain in a forma correct way

This is fine using comma as a logical "and", more formally we can state that $$D = \{(x, y) \in \mathbb{R}^2 | (x \geq 0 \land y \geq 0) \cup (x \leq 0 \land y \leq 0)\}$$ or also simply $$...
user's user avatar
  • 154k
1 vote

Calculating $\int_0^{\infty } \left(\text{Li}_2\left(-\frac{1}{x^2}\right)\right)^2 \, dx$

The following is another way to show that $$ \begin{align} \int_{0}^{\infty} \operatorname{Li}_2^2 \left(-x^{-1/a} \right) \, \mathrm dx &= a\int_{0}^{\infty} \frac{\operatorname{Li}_{2}^{2}(-u)}{...
Random Variable's user avatar

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