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4 votes

Formula for bump function

If you only require the first order derivative to be continuous, then you can take $$ f(x) = \begin{cases} \cos^2\frac{\pi x}{2}&\text{when $-1\leq x\leq 1$}\\ 0,&\text{otherwise} \end{cases} $...
md2perpe's user avatar
  • 27.3k
4 votes

Formula for bump function

I have a stricter version than what is being asked for. I have had a very similar question in the past nag at me but never figured it out. What I had been interested in was more restrictive than OP ...
Cameron Williams's user avatar
1 vote

Find $\displaystyle\lim_{n \to \infty} \int_0^\infty \frac{1+\frac{x}{\sqrt{n}}e^{-x/n}}{(x+1)^2} \, dx$

Since there are different scales at play ($\sqrt{n}$ and $n$) in this kinds of situations is often useful to split the integral in intervals that grow as $n\rightarrow\infty$ and/or use the following ...
Mittens's user avatar
  • 40.7k
1 vote

Laplace transform of $\sin(\omega t)$

There are a few ways to find $\int \sin(\omega t) e^{-s t} dt$. One way is via integration by parts, and another is by using Euler's identity $e^{i x} = \cos x + i \sin x$ to relate the integral to ...
ConMan's user avatar
  • 25.6k
1 vote

Integral of Thomae's function

Note that for every partition $P$ of $[0,1]$, $L(f,P)=0$. Thus,$\int_0^1f=L(f)$=sup$\{L(f,P):P\; \text{is partition of }[0,1]\}$=sup$\{0\}=0$
suraj tidke's user avatar

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