The integration of e^(-x^2) (1 Viewer)

[Drsoccerball]

I found the taylor series










Did i integrate it or not ?

 

[Carrotsticks]

Yes you did. It is not a closed form as what most people expect, but it is an integral.

 

[leehuan]

So all of these integrals such as can be found by using their Taylor series?

 

[Paradoxica]

Not in closed form, but yes, the analytical integral of any infinitely differentiable, taylor approximable function can have it's integral expressed.

 

[Kaido]

Not in closed form, but yes, the analytical integral of any infinitely differentiable, taylor approximable function can have it's integral expressed.

LOL

 

[Drsoccerball]

So all of these integrals such as can be found by using their Taylor series?

Yes it can

 

[braintic]

I wonder how many people understand the Taylor's series beyond the application of the formula?

 

[Drsoccerball]

I wonder how many people understand the Taylor's series beyond the application of the formula?

Isn't it just finding an approximation for a graph by finding its derivatives or some crap like that forgot it already

 

[Paradoxica]

I wonder how many people understand the Taylor's series beyond the application of the formula?

I understand it in computing aspects... e.g. our school calculators use Taylor Series to approximate until a given level of accuracy is reached, at which point the algorithm halts. Used for transcendental functions and their inverses.

 

[braintic]

Can people see how the constant/linear terms in the Taylor expansion gives the equation of the tangent at the origin?

For sinx it is y=x.
For cosx it is y=1.
For e^x it is y=1+x.

 

[Physicklad]

Can people see how the constant/linear terms in the Taylor expansion give the equation of the tangent at the origin?

also the basis for the many function approximations close to the origin (or any other 'centre point'), solving differential equations, proof of the fact that all functions with defined infinite derivatives can be written as an infinite polyomial...
and deriving euler's formula!! this should be enough motivation for the whole formulation of maclaurin taylor series! this shit is truly amazing

 

[Paradoxica]

Can people see how the constant/linear terms in the Taylor expansion give the equation of the tangent at the origin?

For sinx it is y=x.
For cosx it is y=1.
For e^x it is y=1+x.

Well, by definition, the taylor expansion approximates the function around that point, and the first few terms are defined by the first few derivatives computed at that point, so the constant/linear terms are tangential to that point by definition...

 

[braintic]

also the basis for the many function approximations close to the origin (or any other 'centre point'), solving differential equations, proof of the fact that all functions with defined infinite derivatives can be written as an infinite polyomial...
and deriving euler's formula!! this should be enough motivation for the whole formulation of maclaurin taylor series! this shit is truly amazing

And ... a derivation of E=mc^2

 

[Physicklad]

I found the taylor series










Did i integrate it or not ?

and the improper integral from -infinity to positive infinity is (pi)^1/2.
is there a way to use drsoccerball's genius to find an approximation for pi?? subbing infinities into that sexy polynomial dont look so good at this moment...
anybody got ideas??

 

[Drsoccerball]

and the improper integral from -infinity to positive infinity is (pi)^1/2.
is there a way to use drsoccerball's genius to find an approximation for pi?? subbing infinities into that sexy polynomial dont look so good at this moment...
anybody got ideas??

I don't think its possible unless you find a taylor series for then maybe it would work but then youre left with an infinite sum

 

[Paradoxica]

and the improper integral from -infinity to positive infinity is (pi)^1/2.
is there a way to use drsoccerball's genius to find an approximation for pi?? subbing infinities into that sexy polynomial dont look so good at this moment...
anybody got ideas??

Last year's BOS trial had a volumes question which trapped the square of the improper integral between two limiting expressions. Perhaps ask Carrotsticks to come up with something similar.

 

[braintic]

and the improper integral from -infinity to positive infinity is (pi)^1/2.
is there a way to use drsoccerball's genius to find an approximation for pi?? subbing infinities into that sexy polynomial dont look so good at this moment...
anybody got ideas??

You don't need to sub in -inf to inf.
Integrating from -3 to 3 gives 99.998% of the total value.
The number of terms you add has a much bigger effect on the estimate than the size of the limits.

 

[Drsoccerball]

I also saw the integration of e^(-x^2) through the use of two integrals ? Can someone explain wot dat is

 

[Paradoxica]

I also saw the integration of e^(-x^2) through the use of two integrals ? Can someone explain wot dat is

It's basically the same as the question from last year's BOS. Converting it into a (polar) volume integral by exchanging variables and then taking the limits. Trapping the squared value between two bounds which become identical. Taking the square root of both sides.

 

[braintic]

.

Hmmm ... my quote looks kind of stupid out of context.