Series Approximation for y with Derivative of Floor Function

[lewis198]

I was wondering what series approximation I can use to approximate y:

y=(1-(dx/dy)^2)^1/2

when dx/dy is not trigonometric, and contains the derivative of a floor function

 

[dodo]

Hi,
the floor function is notoriously discontinuous. What do you mean by its derivative?

 

[sadhu]

y has properties of sin fuction i guess we solve x in terms of y

x=1/2*(abs(y)(1-y^2)^0.5 +inverse sin(abs(y)))

you can also get it solved for y

 

[lewis198]

okay well an approximation

 

[sadhu]

what do you meant by that

 

[epenguin]

You can transform that into dx/dy = sqrt(1 - y^2) and integrate wrt y to obtain

x = -.5 {y*srt(y^2 - 1) - ln ABS(y + sqrt(y^2 -1)} + a constant

give or take a + or -

which does not mean the solution of sadhu is not right too.

Did this d.e. emerge from any 'real' problem?

 

[olgranpappy]

Hi,
the floor function is notoriously discontinuous. What do you mean by its derivative?

how about \sum_n \delta(x-n).

 

[Gib Z]

how about \sum_n \delta(x-n).

I don't exactly know what you mean by that :( But I doubt that's a derivative. Differentiability implies continuity. The floor function, being discontinuous at an infinite number of points, is not differentiable.

 

[HallsofIvy]

That uses the delta "function". It's not a function in the true sense, but a distribution or "generalized function". And, of course, the differentiation is in the sense of distributions. Distributions do not have to be continuous in order to be differentiable. (In fact, I am not sure that "continuous" is defined for distributions!)

 

[olgranpappy]

I don't exactly know what you mean by that :( But I doubt that's a derivative.

\delta is Dirac's delta function and the n are positive integers. You can, of course, check for yourself that
\int_0^x dy \sum_n \delta(y-n)=Floor[y]

Differentiability implies continuity.

Lah dee dah, lah dee dah, I'm not worried about that...

The floor function, being discontinuous at an infinite number of points, is not differentiable.

But, if it's not differentiable then the derivative doesn't exist. So how did I just write the derivative down if it doesn't exist?... I'd rather not let the fact that something doesn't exist stop me from using it to solve a problem.

Mathematicians are so cute...

 

[Gib Z]

I was never stopping you from solving the problem. Just don't call it a "derivative". Call it something else. You didn't write down the derivative, you wrote down something that satisfies a nice equation in this same manner a derivative does for continuous functions. That doesn't mean it is a derivative.

Physicists are just sloppy. There is nothing in mathematics that stops the physics being done, but physicists are too lazy to justify their mathematics. Every working physicist I have asked admits that their sloppy mathematics leads them to errors, why not correct the problem?

 

[olgranpappy]

nah.

 

[morphism]

But, if it's not differentiable then the derivative doesn't exist. So how did I just write the derivative down if it doesn't exist?

It's simple: you didn't.

 

[olgranpappy]

oh, girls, stop being so silly.