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I need to find an expression for $n$th derivative of $f(x) = e^{x^2}$. Really need help.

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Can you find the first few derivatives yourself? –  Ben Millwood Sep 10 '12 at 17:43
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Nevertheless, you'll find people much more willing to help you if you show that you've done some work yourself - besides, someone else might be able to see the pattern. –  Ben Millwood Sep 10 '12 at 17:57
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try taking derivative of taylor expansion of $e^{x^2}$ –  Santosh Linkha Sep 10 '12 at 18:05
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More generally, $$ \frac{d^n}{dx^n}e^{x^p} = \left(\sum_{k \in \mathbb{Z}}a_{n,k;p}x^{kp-n} \right) e^{x^p} $$ where the coefficients $a_{n,k;p}$ form the exponential Riordan array $A_p = [1, (1+x)^p - 1]$. This can be proved using some of the techniques from math.stackexchange.com/questions/18284 and is given as an exercise (although not in those terms) in Comtet's Advanced Combinatorics. –  Peter Taylor Sep 10 '12 at 19:52
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7 Answers

up vote 20 down vote accepted

Here is a less elementary solution.

We consider the exponential generating function

$$G(t,x) = \sum_{n=0}^{\infty} \frac{t^n}{n!} \frac{d^n}{dx^n}e^{x^2}.$$

Then we can identify this series as the McLaurin series of $e^{(x+t)^2}$ near $t = 0$. Thus we must have

$$G(t,x) = e^{(x+t)^2} = e^{x^2} e^{2xt} e^{t^2} = e^{x^2}\left( \sum_{k=0}^{\infty} \frac{(2x)^k}{k!} t^k \right)\left( \sum_{l=0}^{\infty} \frac{1}{l!} t^{2l} \right). $$

Expanding this series and comparing, we have

$$ \frac{d^n}{dx^n} e^{x^2} = \left( \sum_{j=0}^{\lfloor n/2 \rfloor} \frac{n!}{j!(n-2j)!}(2x)^{n-2j} \right) e^{x^2}. $$

Here is an example:

enter image description here

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I love this, particularly because my intuition was that a closed form expression using only simple operations like factorial would not be possible. –  alex.jordan Sep 10 '12 at 18:46
    
It would be even simpler to use the Hermite polynomials for this, no? –  J. M. Sep 12 '12 at 2:22
    
@J.M., Unfortunately I know not much about the Hermite polynomials. I have heard of these polynomials, and also know that they are closely related to the Gaussian-weighted $L^2$ space, but the important thing is that I even do not know the exact definition or a representation formula for this nice family. That's why I just made a rather elementary approach. –  sos440 Sep 12 '12 at 2:37
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In that case, you'll want to see the Rodrigues formula for the Hermite polynomials. Notice anything? ;) –  J. M. Sep 12 '12 at 2:42
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@J.M., Now it's also clear to me! We may write $$\frac{d^n}{dx^n}e^{x^2} = i^{-n}H_n(ix)e^{x^2}.$$ –  sos440 Sep 12 '12 at 2:46
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I'm not sure if this is what you are after, but there is a recursion that you can establish. Each derivative will be the product of a polynomial in $x$ with $e^{x^2}$. If $f^{(n)}(x)=p_n(x)e^{x^2}$, then we have $$p_0(x)=1$$ and $$p_n(x)=2x\,p_{n-1}(x)+p^\prime_{n-1}(x)$$

This recursion has the following associated infinite matrix:

$$M=\begin{bmatrix} 0 & 1 & 0 & 0 & \cdots\\ 2 & 0 & 2 & 0 & \cdots\\ 0 & 2 & 0 & 3 & \cdots\\ 0 & 0 & 2 & 0 & \cdots\\ \vdots & \vdots & \vdots & \vdots & \ddots\\ \end{bmatrix}$$

viewed as a transformation on the vector space of polynomials in $x$ with basis $\left\{1,x,x^2,x^3,\ldots\right\}$. The first column of $M^n$ gives you the $n$th derivative of $f$. If there is any hope to solve the recursion explicitly (which I'm not optimisitic for) then I'd recommend further study of this matrix. Maybe it can be diagonalized, and that would give way to a formula for $f^{(n)}$.

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I was hoping for some kind ofpolinomial in x with factorials , :) –  Mykolas Sep 10 '12 at 17:59
    
Moreove, $p_n(x) = x g_n(x^2)$ for $n$ odd and $p_n(x)=h_n(x^2)$ for $n$ even. –  lhf Sep 10 '12 at 18:08
    
Try to find more patterns with WA. –  lhf Sep 10 '12 at 18:09
    
\prime works nice for things like $p^\prime$. –  Pedro Tamaroff May 20 '13 at 23:06
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Another approach is to write:$$f(x+y)=\sum_{n=0}^\infty \frac{f^{(n)}(x)}{n!} y^n$$

Now $f(x+y)=f(x)f(y)e^{2xy}$. So if we let $g_n(x)=f^{(n)}(x)/f(x)$m then we see that:

$$f(y)e^{2xy} = \sum_{n=0}^\infty \frac{g_n(x)}{n!}y^n$$

But $$f(y)=\sum_{k=0}^\infty \frac{1}{k!} y^{2k}$$ and $$e^{2xy} = \sum_{m=0}^\infty \frac{(2x)^m}{m!}y^m$$

So:

$$f(y)e^{2xy} = \sum_{n=0}^\infty \frac{y^n}{n!} \sum_{m+2k=n} \frac{(2x)^m n!}{m!k!}$$

which gives us $$g_n(x)=\sum_{m+2k=n} \frac{(2x)^m n!}{m!k!}$$

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Well, I see @sos440 beat me to this answer, but since my answer is slightly organized differently, I'll leave it as is. –  Thomas Andrews Sep 10 '12 at 19:27
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I love to see questions like this in patterns/scheme. After the already given complete answers (especially the comment of Peter Taylor contains a very nice formal statement) here one more version, but which focuses on the visibility of the pattern in the coefficients. In Pari/GP I simply called

   (exp(x^2) )        / exp(x^2)   
   (exp(x^2) ')       / exp(x^2)   
   (exp(x^2) '' )     / exp(x^2)   
   (exp(x^2) ''' )    / exp(x^2)   

with so many apostrophes as needed for the n'th derivative. In the result

  1 + O(x^34)
  2*x + O(x^33)
  2 + 4*x^2 + O(x^32)
  12*x + 8*x^3 + O(x^31)
  12 + 48*x^2 + 16*x^4 + O(x^30)
  120*x + 160*x^3 + 32*x^5 + O(x^29)
  120 + 720*x^2 + 480*x^4 + 64*x^6 + O(x^28)
  1680*x + 3360*x^3 + 1344*x^5 + 128*x^7 + O(x^27)
  1680 + 13440*x^2 + 13440*x^4 + 3584*x^6 + 256*x^8 + O(x^26)

I find the pattern:

                                                   1      = exp(x^2)/exp(x^2)
                                                   2*x    = exp(x^2)'/exp(x^2)
                                    2* 1* 1      + 4*x^2  = exp(x^2)''/exp(x^2)
                                    2* 2* 3*x    + 8*x^3  = ...
                     3*4* 1* 1     +2* 4* 6*x^2 + 16*x^4  = ...
                     3*4* 2* 5*x   +2* 8*10*x^3 + 32*x^5  = ...
    4*5*6* 1* 1    + 3*4* 4*15*x^2 +2*16*15*x^4 + 64*x^6  = ...
    4*5*6* 2* 7*x +  3*4* 8*35*x^3 +2*32*21*x^5 +128*x^7  = ...

After that I find it obvious how this continues (formal description&proof are in the other answers above) and leave that schematic description just for the accidental later reader here.

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The values of the derivatives all multiplied by a factor of $e^{x^2}$ are

$2x$

$2 + 4x^2$

$8x + 4x + 8x^2 = 12x + 8x^2$

$12 + 16x + 24x^2 + 16x^3 $

The if this polynomial progression is called $Q(n)$ the general pattern is:

$Q(n+1) = Q'(n) + 2xQ(n)$ where $Q(0) = 1$

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Related problems (I), (II), (III), (IV), (V), (6), (7). I already posted a solution on this website related to this one.

Here is the first formula, which gives only the $n$th derivative of integer order of ${\rm e}^{x^2}$

$$ {{\rm e}^{{x}^{2}}}\sum _{s=0}^{n} \left( {x}^{2\,s-n}\sum _{k=0 }^{n}{2}^{k+s} \left[\matrix{n\\k+s}\right] \left\{\matrix{k+s\\s}\right\} \right)\,, $$

where $\left[\matrix{n\\k+s}\right]$ and $\left\{\matrix{k+s\\s}\right\}$ are the Stirling numbers of the first kind and the second kind respectively.

The second formula is more general. It is a unified formula for the $n$th derivative and the $n$th anti derivative of real orders (including integer orders) of ${\rm e}^{x^2}$ in terms of the Meijer $G$-function,

$$ \left( -1 \right)^{\frac{n}{2}}{2}^{n} G^{1, 2}_{2, 3}\left(-{x}^{2}\, \Big\vert\,^{-\frac{n}{2}, -\frac{n}{2}+\frac{1}{2}}_{-\frac{n}{2}, \frac{1}{2}, 0}\right)\,.$$

Note that,

(i) if $n > 0$, then the formula gives derivatives of order $n$ ($n$ can be integer or real ).

(ii) if $ n<0 $, then the formula gives anti-derivatives of order $n$ ($n$ can be integer or real).

(iii) if $n=0$, then it gives the original function.

See here for the details.

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The derivatives at $0$ may be found simply using the Taylor expansion: $$e^x = \sum_{k=0}^\infty {\frac{x^k}{k!}}$$ So that: $$e^{x^2} = \sum_{k=0}^\infty {\frac{x^{2k}}{k!}} = \sum_{k=0}^\infty {\frac{x^{k}}{k!}f^{(k)}(0)}$$ Leading to: $$f^{(2n)}(0) = {(2n)! \over n!} , \ f^{(2n+1)}(0) = 0$$ This means that for $f^{(n)}(x) = p_n(x) e^{x^2} = (\sum a^{(n)}_k x^k) e^{x^2}$:

$$a^{(2n)}_0 = {(2n)! \over n!}, \ \ a^{(2n + 1)}_0 = 0$$ And using alex's recursion: $$a^{(2n - 1)}_1 = {(2n)! \over n!}$$

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Ty nbubis, just I didn't understand this line: –  Mykolas Sep 10 '12 at 19:21
    
$$e^{x^2} = \sum_{k=0}^\infty {\frac{x^{2k}}{k!}} = \sum_{k=0}^\infty {\frac{x^{k}}{k!}f^{(k)}(0)}$$ –  Mykolas Sep 10 '12 at 19:21
    
i mean just the last part of it $f^{(k)}(0)$ –  Mykolas Sep 10 '12 at 19:24
    
@Mykolas - That's just the definition of a taylor series. –  nbubis Sep 10 '12 at 20:25
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