how do i find F(g(x)) given f(x), g(x), or compute it?

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how do i find F(g(x)) given f(x), g(x), or compute it?

Postby >-) » Tue Apr 08, 2014 12:08 am UTC

I have the function

Code: Select all

       [ + inf
E(v) = | tanh(arctanh(M)v) * e^[-(v-1)^2/(2S^2)]/(sqrt(2*pi)S) dv
       ] -inf


I didn't bother finding the antiderivative -- wolfram alpha couldn't figure anything out even with extended computation time and it doesn't look feasible.

Now i have the expression Q/R (where Q is constant) that i want to plug into E. However, if i replace all the v with Q/R, i'm left with dv. No numerical integrators i've found will take a function as the thingy behind d and i don't even know if it makes sense to do so, so i'm forced to solve it somehow.

i tried change of variables:
u = Q/R, du/dv = 1 because du = dv. Not helpful.

According to wolfram alpha the integral of f(g(x)) is indeterminite, so that doesn't help.

So how can i plug an expression into a integral?

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Re: how do i find F(g(x)) given f(x), g(x), or compute it?

Postby Schrollini » Tue Apr 08, 2014 2:17 am UTC

>-) wrote:I have the function

Code: Select all

       [ + inf
E(v) = | tanh(arctanh(M)v) * e^[-(v-1)^2/(2S^2)]/(sqrt(2*pi)S) dv
       ] -inf


No, you don't. The thing on the right-hand side depends on M and S. But v is a dummy variable, being integrated over, so this thing is not a function of v. What you really have is

E(M,S) = ∫-∞tanh (tanh-1(M) v) exp [-(v-1)2/2S2] / (√(2π)S) dv

You can plug whatever you like into M and S, but v just isn't an argument for you to adjust.
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Re: how do i find F(g(x)) given f(x), g(x), or compute it?

Postby >-) » Tue Apr 08, 2014 3:46 am UTC

sorry, i forgot to mention, both S and M are constants, v is the only variable.

I see what you mean when you say "You can plug whatever you like into M and S, but v just isn't an argument for you to adjust.", but i'm confused when you write E as a function of M and S, but not v. Surely, it is a function of v?

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Re: how do i find F(g(x)) given f(x), g(x), or compute it?

Postby ConMan » Tue Apr 08, 2014 5:09 am UTC

>-) wrote:sorry, i forgot to mention, both S and M are constants, v is the only variable.

I see what you mean when you say "You can plug whatever you like into M and S, but v just isn't an argument for you to adjust.", but i'm confused when you write E as a function of M and S, but not v. Surely, it is a function of v?

No, it isn't. The integral is over the dummy variable v, which means that outside of the integral v doesn't exist. It's like how ∫0x (z^2 dz) = 1/3 x^3 is a function of x (which is a limit of the integral and hence a parameter of the whole thing) and not z, or more appropriately how ∫01 (az^2 dz) = a/3 is a function of the free variable a, but not of z.
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Re: how do i find F(g(x)) given f(x), g(x), or compute it?

Postby skullturf » Tue Apr 08, 2014 5:41 pm UTC

Or how Sum(n,n=1..4) is 1+2+3+4=10 and is not a function of n.

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Re: how do i find F(g(x)) given f(x), g(x), or compute it?

Postby >-) » Tue Apr 08, 2014 6:43 pm UTC

yeah i see it now, that solves the question, thanks!


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