Hans Olo
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Are "mid-derivatives" a thing?
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3 votes

You are looking for a thing called fractional derivative and is a part of fractional calculus. In particular it is defined as $$ (D^n f) (x)=\frac{1}{\Gamma(n)} \int_0^x (x-t)^{n-1} f(t) dt $$ Note ...

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How to factorize $2x^2-3xy-2y^2+7x+6y-4=0$ without software?
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2 votes

Since the polynomial is quadratic in both $x$ and $y$, you can write it in the form: $$f(x,y) =2 x^2 - x (3 y - 7) - (2 y^2 - 6 y + 4) $$ $$= p^i F_{ij} p^j=(x-x_0,y-y_0)^i F_{ij}(x-x_0,y-y_0)^j$$ ...

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How could I calculate T such that the length of curve C is equal to 1?
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2 votes

Note that the length $S$ of a parametrized curve $s(t)$ is given by $$ S=\int_C \sqrt{x'(t)^2+y'(t)^2+z'(t)^2}dt$$ with the integration being along the curve $C$ and $s(t)=(x(t),y(t),z(t))$, given by ...

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Solution of a second order non-linear ODE
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1 votes

This is an Emden-Fowler type equation. Your specific case has a particular solution of $$y(x)=\pm\frac{2}{\sqrt{3}} x^{3/2},$$ but even with this, the original ODE doesn't seem to be solvable. As of ...

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Why does $a+b+c=3\;$ imply $\;a\sqrt{a+3} + b\sqrt{b+3} + c\sqrt{c+3} \ge 6$
1 votes

First set up a function $$f(a,b,c)=\sqrt{a+3} a+\sqrt{b+3} b+\sqrt{c+3} c.$$ Then, set $a=3-b-c$ to eliminate $a$ and get: $$f(b,c)=-\sqrt{-b-c+6} (b+c-3)+\sqrt{b+3} b+c \sqrt{c+3},$$ then find the ...

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possible typo in a math books
1 votes

The absolute value of a number is always positive (or at least zero), ie for both $y_1=2$ and $y_2=-2$ we have $|y_1|=|y_2|=2$. Conversely, $|y|=1$ means $y=\pm 1$ as expected.

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How to evaluate this integral $\int_{-\infty}^{\infty}\exp(-ay^2)dy$
1 votes

Set y=$Y/\sqrt{a}$ and see what happens ;-)

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Proof that the exponential function is the only solution to dy/dx = y, y(0) = 1
0 votes

A proof using only the definitions of the derivative and the exponential function as limits, respectively $$ y'(x)=\lim_{h\rightarrow0}\left(\frac{y(x+h)-y(x)}{h}\right),~~(1) $$ and $$ e^x =\lim_{n\...

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Show that there is a unique polynomial $\int_{x}^{x+1} b_{n}{(t)} dt = x^n$
0 votes

Taking a derivative with respect to $x$ yields $$ -n x^{(-1 + n)} - b_n(x) + b_n(1 + x)=0,$$ which is a simple difference equation with solution $$b_n(t)=-n \zeta_{1 - n}(t),$$ where $\zeta_{1 - n}(t)$...

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Proof of Wikipedia formula about Ricci curvature
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My guess is that since the covariant volume $dV=\sqrt{-g} d^n x$, where $g=|g_{ij}|$ is the determinant of the metric, is conserved we have that: $$\sqrt{-g} dV_g=dV_{Euclidian}$$ since the Euclidean ...

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Is there a close form solution for this series?
-1 votes

Playing around with the series numerically (plotting it etc) it seems it's equal to $\frac12 \sin^2(\frac{x}{2})$. I don't have a proof right now (and I'm too lazy to work one out), but it seems to ...

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