Correct, @GGG. So switch the names of $j$ and $k$ and remember that you can do a finite sum in any order you want (so you can switch the order of summation).
that's $|f^9(x,y)|$, with the blue and pale yellow (tope?) regions being below $e^{-5}$ and above $e^{5}$ respectively
it appears to go to zero or infinity with doubly-exponential speed, so it's pretty sharp; i had to go that far in tolerance to get a nice-looking boundary
i don't think he claimed to have come up with it. the story i remember is that the book he learned out of taught him different tools than the standard ones for integration
No, the Leibniz rule for differentiating $\int_a^{g(x)} f(x,t)dt$ is an application of that, @PVAL. But I have a proof using Fubini in my exercises in my book.
Depends on what you know, @GGG. Thinking about writing nonsingular matrices as a product of elementary matrices. And then doing the singular case separately.
I think it works in this case as well because the integral is symmetric. We write it over two interval after applying the substitution and then add them together again.
part of the problem i have with this line of conversation, though, is that while i can talk about the anxiety, i really can't explain it. it's not something whose dynamics happen at the level of arguments and logic.
@Michael $\displaystyle \int_{0}^{1} \frac{1}{(u^2+1)(1/u^x+1)}\;{du} = \int_{0}^{1} \frac{1}{(t^2+1)(1/t^x+1)}\;{dt}$. It doesn't matter whether you denote it $u$ or $t$ or $x$ or whatever you still get the same value. That's what a dummy variable is. So we change it back to $t$ so that we can add the two integrals conveniently.
Wikipedia is being more general than you need, there.
Just know that $\Bbb R$ is an example of a partially ordered set.
(If you're curious, a partially ordered set is a set together with an order $\preceq$ satisfying several axioms that we expect an ordering to have. The "partial" bit means that it's possible for us to have neither $x\preceq y$ nor $y\preceq x$.)
In our lecture notes for our complex analysis class, we were given the following theorem:
Theorem: There exists a unique complex function $f$ such that
$f(z)$ is a single valued function $f(z) \in \mathbb{R}$ whenever $z \in \mathbb{R}$ and $f(1)=e$.
$\forall z_{1},z_{2} \in \mathb...
He defines $M$ to be the maximum value of $f$ on $[-N,N]$.
And, because of the way he chose $N$, he knows that $f$ never gets a value bigger than $M$ on the set $(-\infty,-N)\cup(N,\infty)$ (i.e. the rest of the reals).
So, what do we know?
We know that $M$ is an upper bound (because it's an upper bound both on $[-N,N]$ and $(-\infty,-N)\cup(N,\infty)$).
And we know that nothing smaller than $M$ is an upper bound, because $f$ attains the value $M$.
Thus, $M$ is the least upper bound; that is, it's the supremum.
And since we already know $f$ attains the value $M$ (for some $x$ in $[-N,N]$), we get that $M$ is the maximum, as well.
Let $\alpha(s) = (x(s),y(s))$ be the arc length parametrization of a plane, smooth, closed, convex curve, of length $L$. Let $J:(0,L)\to\mathbb{R}$ be a smooth and Bounded variation (BV introduced after comment from Dirk of possibility of blow up at 0 or L) function. Let $\Omega$ be the set of al...
Is there a continuously differentiable function $f:\mathbb{R}\to\mathbb{R}$ such that $\exists x,f'(x)=0$ and $f$ is either increasing or decreasing on both sides of $x$, yet there is no inflection at $x$?
@brody not sure it helps but: defining $g(x)=f'(x)$, that'd amount to finding $g(x)$ such that $g(0)=0$ (picking $x=0$ WLOG) and $g(x)>0$ for $0<|x|<\epsilon$ but for which $g'(0)\neq 0$
Oh actually there's nothing left to do at all. $$[1]_{2009} = [315\cdot2009-346\cdot1829]_{2009} = [-346 \cdot 1829+2009(1829)]_{2009} = [1663 \cdot 1829]_{2009}$$
But there MUST be a way to find modular inverse without using Euclidean algorithm. I'd appreciate if someone could link me to it, 'cause I can't find it.
I believe it meets your criteria @Semiclassical but its primitive (which I believe is $f(x)=\sgn(x)x^2$) changes concavity at $x=0$, but I want otherwise.
In our lecture notes for our complex analysis class, we were given the following theorem:
Theorem: There exists a unique complex function $f$ such that
$f(z)$ is a single valued function $f(z) \in \mathbb{R}$ whenever $z \in \mathbb{R}$ and $f(1)=e$.
$\forall z_{1},z_{2} \in \mathb...
