That's true oay, thanks!
Maybe instead of
"-It has an upper bound of 1"
I should have written
-it's limit at +-Infinity = 1
Not sure if even that makes it watertight. But anyway, HS-Scientist's suggestion is more suitable for what I need ;)
Thanks HS! Impressively quick response.
That looks very promising actually. I might modify it with an extra parameter:
f(x)=1-ce^(-(bx)^2)
Then, by changing the value of b I can change the rate at which it approaches the upper bound.
Anyway, looks very good and helpful!
PS Sorry about using...
Hi everyone,
I'm trying to find a function of single variable f(x) with the following properties:
-It is symmetric around zero
-It is differentiable everywhere
-f'(x)≥0 for all x>0
-f'(x)=0 when x=0
-f'(x)≤0 for all x<0
(I think these last two actually follow from the first three?)...
Thanks tiny-tim!
I thought that might be the case. So many other series seem to have been named that I just thought it was odd!
No, I can't prove the identity you gave in your post... yet! But I will try.
Hi, a naive question here, but I was wondering if the series
\sum^{∞}_{k=0} k a^{k}
has a particular name? As in 'geometric series' for \sum^{∞}_{k=0} a^{k} ?
And what about the more general \sum^{∞}_{k=0} k^{n} a^{k} ?
As a related question, you seem to be able to get the formula for the...
This is actually related to a post I made earlier in the differential equations forums, but I've since realized that solving the equations themselves is not necessarily the best way to get where I want to go. Perhaps it's better suited to this forum, since it is an integration problem that I...
Thanks again micromass.
That solves my problem (or actually shows that it is unsolvable), and makes me appreciate Mathematica a bit more!
Cheers,
zeroseven
Thanks again, I always wondered how we could be sure that the integral of exp(-x^2) doesn't exist!
So does this mean that when Mathematica works on an integral, it's actually running a rigorous mathematical proof on whether that integral exists or not? Does it use the Risch algorithm that you...
Thanks, that's helpful and very interesting.
I have no idea how I would even begin to go about proving that a solution doesn't exist for an integral! Is this something that can only be done by computer algorithms?
I've been trying to figure out how to integrate 1/(x+ln(x)) but am not getting anywhere. Mathematica can't do it, and I haven't found it in lists of integrals.
Does anyone know if this integral exists in closed form?
Same goes for (x+ln(x))/(1+x+ln(x))
Thanks!
zeroseven
Another update: Lotka-Volterra equations was a great tip. They are almost identical in form to my equations, and cannot be solved with elementary functions, which convinces me that my equations can't be either. Lambert's W works in both cases, though.
Anyone interested, have a look here...
Interestingly (and frustratingly) I am unable to obtain the elementary function solution even for the special case a=b if I do it by eliminating dt. If I use the method I described in the first post, I can get a fairly simple elementary function for the integral that I need. But if I start by...
First, thanks for the replies everyone!
Second, I need to apologize.. Seems I made a small type in my first post ... the equations should be
dx/dt=-ax-cxy
dy/dt=-by-cxy
(so bx in the first post should be by)
I'm embarrassed about this happening in my very first post on the forums...
Just wanted to add a bit more detail about what I need to do:
The end result that I want is the integral
\int^{∞}_{0}cx(t)y(t)dt
So again, if it is possible to obtain this without having analytical solutions for x(t) and y(t) separately, that is fine. I don't even need x(t)y(t) like I...
Hi everyone, new member zeroseven here. First, I want to say that it's great to have a forum like this! Looking forward to participating in the discussion.
Anyway, I need to solve a pair of differential equations for an initial value problem, but am not sure if an analytical solution exists. I...