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but it is tilted about theta=artan(x) wit the upper half of the '8' removed about the midpoint and the lower part removed about the of other half. It looks like a tilted 'S' but flipped, and it's monotonically increasing or decreasing.

function/curve

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You mean $\propto$ or $\sim$? –  Hurkyl Jan 19 '13 at 20:22
    
$\int$ ? The description could use work –  Jonathan Jan 19 '13 at 20:23
    
the second one, but it reaches a maximum-->minimum or minimum->maximum at the endpoints. –  Jennifer Aneta Jan 19 '13 at 20:24
    
Are you asking about a mathematical symbol for writing, or a geometric shape for drawing? –  Hurkyl Jan 19 '13 at 20:25
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Is it possibly the logistic function? –  MJD Jan 19 '13 at 21:24

3 Answers 3

up vote 2 down vote accepted

Here's $\theta = \arctan{x}$,

enter image description here

Also note that the sigmoid function is a mathematical function having an "S" shape ("aka": a sigmoid curve). Often, the sigmoid function refers to the special case of the logistic function:

$y = \dfrac{1}{1 + e^{-x}}$

enter image description here


If you are looking for the "reflection" or "inverse" of a "tilted $S$-shaped curve:

The inverse of $\;\theta = \arctan(x)\;$ is $\;x = \tan(\theta),\;$ and when plotted you get what you might be looking for if you restrict $\theta$ to $\theta \in (-\pi/2, \pi/2)$:

$x = \tan(\theta),\; \theta\in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$

enter image description here

Somewhat similarly, what you seem to be describing (a reflected/flipped tilted "S") looks a bit like the simple cubic function: $y = x^3$, (the graph can be translated, or rotated to tilt it more.)

$y = x^3$

enter image description here

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How did this not get any thumbs-up? +1 –  Amzoti May 4 '13 at 2:19
    
$\ddot\smile$. I am at the uni. –  Babak S. Sep 28 '13 at 3:08

a sigmoid function http://en.wikipedia.org/wiki/Sigmoid_function. It has lots of applications in natural science, e.g., biological populations driven by evolution/natural selection.

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It is also called the logistic curve.

Another curve that looks much like it is the $\int_{-\infty}^x e^{-t^2} dt$.

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