1. ## e^(x^n) functions

After being inspired by my calculus textbook to try and solve the yet to be solved antiderivative of e^(-x^2) function, I came to realize that the family of functions e^(x^n)has no easy way, if any way of being solved.

So before spending a lot of time on this, is there any way to solve the family of the functions e^(x^n)?

Thanks!

How are these put into LATEX?

2. Originally Posted by Truthbetold
After being inspired by my calculus textbook to try and solve the yet to be solved antiderivative of e^(-x^2) function, I came to realize that the family of functions e^(x^n)has no easy way, if any way of being solved.

So before spending a lot of time on this, is there any way to solve the family of the functions e^(x^n)?

Thanks!

How are these put into LATEX?
It cannot be done by any ordinary terms, but using the fact that $\displaystyle e^{x^2}$ is analytic $\displaystyle \forall{x}\in\mathbb{R}$

we can input the power series to et

$\displaystyle \int\sum_{n=0}^{\infty}\frac{x^{2n}}{n!}dx=\sum_{n =0}^{\infty}\frac{x^{2n+1}}{(2n+1)n!}+C$

So then for

$\displaystyle \int{e^{x^{\psi}}}dx$

we would have

$\displaystyle \int\sum_{n=0}^{\infty}\frac{x^{\psi{n}}}{n!}dx=\s um_{n=0}^{\infty}\frac{x^{\psi{n}+1}}{(\psi{n}+1)n !}+C$

3. Originally Posted by Truthbetold
After being inspired by my calculus textbook to try and solve the yet to be solved antiderivative of e^(-x^2) function, I came to realize that the family of functions e^(x^n)has no easy way, if any way of being solved.

So before spending a lot of time on this, is there any way to solve the family of the functions e^(x^n)?

Thanks!

How are these put into LATEX?
Obviously n = 1 can be done. n = 2 is the classic example of when it can't be done. In fact, it can't be done for n > 1.

Are you restricting n to be a positive integer? It can be done (using substitution and then integration by parts) when n = 1/2. However, I can't say off-hand whether it can be done for all values 0 < n < 1.

There's a theorem you can apply to test whether an integral is doable. Try reading this: Integration of Nonelementary Functions to get some background.

(Of course, all integrals can be done if you define an appropriate function. Eg. $\displaystyle \int e^{x^2} \, dx$ can be done using the Error Function).

4. Originally Posted by mr fantastic
Obviously n = 1 can be done. n = 2 is the classic example of when it can't be done. In fact, it can't be done for n > 1.

How come it cannot be done? Is there just no way by using the quotient rule to get only e^(x^2)?

I find that difficult to believe.
I read that Integration of Nonelementary Functions.
I'm not sure what to think about it. I certainly don't want to believe it.
I'm still going to work on this anyway.

MathStud26,
I am really a neophyte in Calculus (as in high school AP Calculus AB).
I've heard of power series, but what you did is beyond my limited knowledge.

5. Originally Posted by Truthbetold
How come it cannot be done? Is there just no way by using the quotient rule to get only e^(x^2)?

I find that difficult to believe.
I read that Integration of Nonelementary Functions.
I'm not sure what to think about it. I certainly don't want to believe it.
I'm still going to work on this anyway.

MathStud26,
I am really a neophyte in Calculus (as in high school AP Calculus AB).
I've heard of power series, but what you did is beyond my limited knowledge.
I'll bet anything you want that not you (or anyone) will succeed in getting an answer in terms of a finite number of elementary functions. Show me what you got ......

6. Originally Posted by Truthbetold
How come it cannot be done? Is there just no way by using the quotient rule to get only e^(x^2)?
Certainly not: you can find an antiderivative, but there is no elementary function which has that derivative. And by elementary, all I mean is a function composed of a finite number of additions, subtractions, multiplications, divisions, root extractions, and exponential and logarithmic functions. You can happily find another function that will give you that derivative, but you won't be able to write it using the standard algebraic expressions that you are used to. However, if you insist, Mathematica gives this evaluation:

$\displaystyle \int e^{x^2}\,dx = \frac12\sqrt{\pi}\,{\rm erfi}(x)$

where $\displaystyle \rm erfi$ is the imaginary error function (and again, I should emphasize that this function is not elementary--it is unlike any of the other functions you have likely encountered before).

Originally Posted by Truthbetold
I certainly don't want to believe it.
I'm still going to work on this anyway.
Mathematics isn't about what you want to believe; for that you should turn to religion. But by all means, investigate the problem some more and you may still learn some interesting things. Just remember that once a statement in mathematics is soundly proven (as this has been), your "feelings" make no difference in the matter.

Originally Posted by Truthbetold
MathStud26,
I am really a neophyte in Calculus (as in high school AP Calculus AB).
I've heard of power series, but what you did is beyond my limited knowledge.
AB probably won't go that far, but when you get a little higher up (as you surely will in college), you will become familiar with the use of such series representations.