Find a real entire function $f(z)$ asymptotic to $\ln(x^2+1)$ for real $x$.

Question

Find a real entire function $f(z)$ asymptotic to $\ln(x^2 +1)$ for real $x$.

More specific I want $f(0)=0$ and $\frac{1}{2} \ln(x^2+1) < f(x) < 2 \ln(x^2+1)$.

Or prove it does not exist.

Answer 1

This is an increased elaboration on my previous answer, which is at the character count limit anyway. It is equivalent to work with $f(x) \approx \exp(x)\ln(x+1)$ where $\text{entiref}(x) = f(x)\exp(-x) \approx \ln(x+1)$. In particular, the graphs have a more natural approximately $2\pi i$ exponential appearance with this approach, making it easier to understand where the zeros are.

$$f_{\ln(x+1)} = \sum_{n=0}^{\infty} a_n x^n \approx \exp(x)\ln(x+1)$$ $$f_{\ln(x^2+1)} = \sum_{n=0}^{\infty} a_n x^{2n} \approx \exp(x^2)\ln(x^2+1)$$

So from this point forward, I work only with the ln(x+1) equation, for theoretical simplicity. The Op's requirement forces $a_0=0$, and ideally $a_1=1$, since $\ln(x+1)=x-x^2/2...$

BEGIN solution2. Here is a different, apparently numerically identical result that gives the same numerical results as the answer I posted below. Here $ \gamma \approx 0.5772156649015\;\;$ where $\gamma=$ is the euler-mascheroni constant.

$F(x) = C + \int \frac{1-\exp(-x)}{x}\;\;C=\gamma\;\;\;$ This is an asymptotic ln(x) function $F(x)$ is entire since (1-exp(-x))/x is trivially entire function, we integrate termwise, which leaves a constant C. F(z) is from https://mathoverflow.net/questions/26243/asymptotic-approximation-of-x-alpha-by-entire-functions on on mathoverflow; my observation is that numerically, $C=\gamma$.

Now, consider $F_1(x)=F(x+1)+\exp(-x)\cdot(b_0 + b_1x)\;\;\;$ For any constants b0 and b1, $F_1(x)$ is an asymptotic solution to $\ln(x+1)$ as x gets arbitrarily large. We choose $b_0+b_1 x$ such that $F_1(0)=0$ and $F_1'(x)=1$.

$b_0 = -F(x+1)\approx -0.2193839343955 \;\;\;$ asymptotic ln(x) at x=1

$b_1 = 1 - F(x+1) - \frac{d}{dx}F(x+1)\approx 0.1484955067759\;\;\;$ derivative of asymptotic ln(x)

$F_1(x) = F(x+1)+\exp(-x)\cdot(b_0 + b_1\cdot x)\;\;\; $ asymptotic ln(x+1) function

What is perhaps surprising, is that $F_1(x)$ is identical to the solution posted below to the limits of numerical accuracy, so that using the solution below setting $a_0=0, a_1=1$, $F_1(x)=\exp(-x)f(x)\;\;$ So all of the plots and Taylor series are good. END solution2

Then we define all of the other Taylor series coefficients of f(x) as the following limit which is the Cauchy integral of of $\;(\exp(x)\ln(x+1))\;$, while ignoring the fact that there is a logarithmic branch cut at the negative real axis. However, that branch discontinuity is multiplied by exp(-x) at the negative real axis, so as x gets bigger, the discontinuity becomes arbitrarily insignificant, so the equation for all of the Taylor series coefficients of f(x) converges in the limit as r goes to infinity.

$$f(x)=\sum_{n=0}^{\infty} a_n x^n\;\;\; \exp(-x)f(x)\approx \ln(x+1)$$ By definition, for any entire function $f(x)$, we have for any value of r $$a_n = \oint x^{-n} f(x) = \int_{-\pi}^{\pi} \frac{1}{2\pi} (re^{-ix})^{-n} f(re^{ix}) )\; \mathrm{d}x\;\;$$ The conjecture is that this is an equivalent definition for $a_n$, where $f(x) \mapsto \exp(x)\ln(x+1)$ $$a_n =\lim_{r\to\infty} \int_{-\pi}^{\pi} \frac{1}{2\pi} (re^{-ix})^{-n}(\exp(re^{ix})\ln(1+re^{ix}))\; \mathrm{d}x\;\;$$

This is the equation I used to generate the f(x) Taylor series, where asymptotically $\ln(x+1)\approx f(x)\exp(-x)$. A pretty good approximation for the Taylor series coefficients is $a_n \approx \frac{\ln(n+1.5)}{n!}$, for larger values of n. Interestingly, the equation also has negative Taylor series coefficients. $a_{-1} \approx -0.3678794411714=\frac{-1}{e}$, which needs investigation. Because of this for large values of x, $f(x)\approx \frac{1}{e\cdot x} + \exp(x)\ln(x)$; this also allows approximating the location of the zeros of f(x); see below. For large negative values of x, the exp(-x) term cancels out the ln(x), leaving just $\frac{1}{e\cdot x}$

If we use the equation above for a0 and a1, we get $a_0 \approx 0.2193839343955$ instead of $a_0=0$, and we get $a_1 \approx 0.8515044932241$ instead of $a_1=1$. For large enough values x, the limit equations converge very slightly faster than using $a_0=0$ and $a_1=1$, but for both sets of equations, the error term for f(x) as x gets larger is approximately $f(x)\exp(-x) - \ln(x+1) \approx \exp(-x)\;\;$

Here is complex plane graph of $f(x)\exp(-x)$ from -40 real to +100 real, and from -10 imaginary to +100 imaginary, with grids every 10 units. You can see the zeros pretty clearly, near the imaginary axis, occurring approximately every $2\pi i$. To the right of the zeros, as $\Re(x)$ increases, the function approximates $\ln(x+1)$ increasingly well, with the error term decreasing exponentially.

Here is the nearly identical graph using $a_0=0$ and $a_1=1$.

update I added the zeros of the asymptotic approximation. The zeros of f(x) are interesting, both in and of themselves, and because I conjecture they can be approximated increasingly well by the zeros of $\;\;\exp(z)\ln(z+1) - a_{-1}/x^1 - a_{-2}/x^2 ...\;\;$ where the $a_{-1}, a_{-2}, a_{-3}$ terms are the Laurent series terms from the equation for the Taylor series approximation terms of $\exp(x)\ln(x+1)$. Since $\exp(x)\ln(x+1)$ has a branch, the Laurent series appears to be a non-convergent series, with some ideal number of terms for best convergence at a particular radius. $a_{-1}=-e^{-1}\; a_{-2}=2e^{-1}$, although I don't have a closed form equation for the general Laurent terms. Here are the first 18 zeros of f(x), from the first plot above. An eight term Laurent series expansions gives the zero approximation to 10 decimal digits accuracy for the 12th ... 18th zero below; a one term Laurent series is accurate to about 0.03 for the same zeros, using the zeros of the approximation $\;\;\exp(z)\ln(z+1) - a_{-1}/x^1 - a_{-2}/x^2 ...$

zero[ 1]= -0.3236449221346
zero[ 2]= -4.014193883364 + 6.790134224598*I
zero[ 3]= -4.782847137525 + 13.31987455364*I
zero[ 4]= -5.248507475714 + 19.72207503441*I
zero[ 5]= -5.583409659397 + 26.07792021174*I
zero[ 6]= -5.844835249373 + 32.41104823228*I
zero[ 7]= -6.059115679636 + 38.73113851434*I
zero[ 8]= -6.240572105211 + 45.04296385204*I
zero[ 9]= -6.397864071944 + 51.34917648054*I
zero[ 10] -6.536627830306 + 57.65137956884*I
zero[ 11] -6.660737008441 + 63.95060483952*I
zero[ 12] -6.772967940649 + 70.24754933585*I
zero[ 13] -6.875378096927 + 76.54270265230*I
zero[ 14] -6.969534334368 + 82.83641978994*I
zero[ 15] -7.056657240617 + 89.12896507913*I
zero[ 16] -7.137716050140 + 95.42053981217*I
zero[ 17] -7.213493140112 + 101.7113002628*I
zero[ 18] -7.284629098684 + 108.0013698029*I

Here is the Taylor series for $f(x)\approx\exp(x)\ln(x+1)$, using the very slightly faster converging a0, a1 terms mentioned above. Using these terms, the error term at x=100 is $1.3\cdot 10^{-46}$, where as with $a_0=0$ and $a_1=1$, the error term at x=100 is $5.4\cdot 10^{-43}$. Both equations converge to $\ln(x+1)$ extremely well.

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+x^300*  1.8652589001443449088121380514937167555694076037155 E-614 }

Answer 2

As said in the comments I will write a partial "answer". I do this because its to much for a comment and I feel its not an edit to the question. I might edit the "answer" a few times.

I believe there is a real entire $f(x) = O(ln(x^2+1))$ where we use big-O notation.

Here is the assumed "proof sketch" :

$f(x) = \sum_{n=1}^\infty a_n x^{2n} = O(ln(x^2+1))$

Where $a_n$ alternates sign starting with $a_1 < 0$.

Also $|a_{2n}| > |a_{2n-1}|$.

$x > 0 \implies a_{2n-1}x^{4n-2} + a_{2n} x^{4n} < O(ln(x^2+1))$

$x > 1 \implies ln(a_{2n-1}x^{4n-2} + a_{2n} x^{4n}) < O(ln(ln(x^2+1)))$

$x > 1 \implies (4n-2) ln(x) + ln(a_{2n-1} + a_{2n} x^2) < O(ln(ln(x^2+1)))$

Now relaxing the equation :

$x > 1 \implies (4n-2) ln(x) + ln(a_{2n-1}) + ln(1 + A_n x^2) = C(ln(ln(x^2+1)))$

$x > 1 \implies ln(a_{2n-1}) = C(ln(ln(x^2+1))) - (4n-2) ln(x) - ln(1 + A_n x^2)$

Now $a_{2n-1} = exp ( min ( C(ln(ln(x^2+1))) - (4n-2) ln(x) - ln(1 + A_n x^2) )$

The minimum must occur where the derivative is $0$.

Let the minimum be attained at $x= y$ then

$ a_{2n-1} = exp( C(ln(ln(y^2+1))) - (4n-2) ln(y) - ln(1 + A_n y^2)$

SO we must compute $y$ by setting the derivative to 0 and solving for x.

I use an approximation of the derivative ;

$\frac{d}{dx} C(ln(ln(x^2+1))) - (4n-2) ln(x) - ln(1 + A_n x^2) = $ ( approx )

$-(4n-2)/x + C/(x ln(x))$

So we need to solve : $(-(4n-2) + C/ln(x))/x = 0$.

Or equivalently : $C = (4n-2) ln(x)$

$y = x = exp(C/(4n-2))$

(end proof sketch)

Numerically it seems pretty good.

I must thank tommy1729 and Sheldon (https://math.stackexchange.com/users/43626/sheldon-l)

I got the inspiration from them at the tetration forum : http://math.eretrandre.org/tetrationforum/showthread.php?tid=863

(they call this " fake function theory " )

I also wonder about least squares methods. And integral representations.

This does not constitute a proof , I know. But its too big for a comment.

EDIT :

Actually this might be a disproof ?? It seems the sequence $a_n$ might not be shrinking fast enough to give an entire function ? Or did I make a big mistake ?

But the sine function seems to suggest alternating sequences are very powerfull in approximating.

Is this the wrong way to compute it ? Did I make wrong assumptions ? A bit confused.

EDIT 2 :

Tommy suggested finding an asymptotic to $f(x) = ln(x^2+1) exp(x^2)$. The solution would then be $f(x) exp(-x^2)$. That might work.

Answer 3

The existence of such function is guaranteed by Carleman's theorem, for any two continuous real valued functions $f<g$, there exists a real entire function in between f(x) and g(x) for all reals. see: math overflow link

For numeric results, I used the Op's answer, and Tommy's suggestion, to generate the asymptotic function $f(x) \approx \ln(x^2+1)\exp(x^2$, and then the desired asymptotic function is $f(x)\exp(-x^2)$.

There were a lot of computational difficulties for generating this asymptotic result, both in generating $f(x)$ itself and even more so, in multiplying the Taylor series of f(x) by exp(-x^2). Here is the Taylor series for $f(x) \approx \ln(x^2+1)\exp(x^2)$, which gives results accurate to 35 decimal digits if x<=10, for $f(x)\exp(-x^2)-\ln(x^2+1)$. The asymptotic function's precision and error terms improve exponentially as x gets larger, for the infinite entire asymptotic Taylor series.

{f=
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+x^180*  3.0398035416946135636028990121365441 E-138
+x^182*  3.3484840025418832038484151939256019 E-140
+x^184*  3.6483022558183007101795318927317532 E-142
+x^186*  3.9321032157621411141379422190462544 E-144
+x^188*  4.1927701514859389140909389981673311 E-146
+x^190*  4.4235272669923616137958106471637279 E-148
+x^192*  4.6182377898763770355728498617515901 E-150
+x^194*  4.7716789320345269781101284578089527 E-152
+x^196*  4.8797763202677976199850319301888485 E-154
+x^198*  4.9397831542631726529459190348494518 E-156
+x^200*  4.9503931971143089329866743631577717 E-158
+x^202*  4.9117813862635127248462260315296755 E-160
+x^204*  4.8255709416189904342543469753027592 E-162
+x^206*  4.6947308858382379843202270819872900 E-164
+x^208*  4.5234124441475715033896158230559723 E-166
+x^210*  4.3167364886479819276205447338170893 E-168
+x^212*  4.0805467669288668471298023671114476 E-170
+x^214*  3.8211449612773525249192372118017444 E-172
+x^216*  3.5450236453885623527062371927933348 E-174
+x^218*  3.2586120424601915408821719193404432 E-176
+x^220*  2.9680473428374280896879078486321870 E-178
+x^222*  2.6789814810639341414957466989183588 E-180
+x^224*  2.3964300066679809069756609257598127 E-182
+x^226*  2.1246663171265192034320134663780201 E-184
+x^228*  1.8671613336865434920788141519694894 E-186
+x^230*  1.6265659181520794885171318855735103 E-188
+x^232*  1.4047311124567213777863504355051806 E-190
+x^234*  1.2027597222569246629766702724383645 E-192
+x^236*  1.0210818811257405412205760707438274 E-194
+x^238*  8.5954698330499079592176872001434089 E-197
+x^240*  7.1752467396949083627525669755611112 E-199
+x^242*  5.9400831951367278977066109854343018 E-201
+x^244*  4.8771541485230735320907995716303265 E-203
+x^246*  3.9718058841925562252766251048314798 E-205
+x^248*  3.2083811838309789218291913075664471 E-207
+x^250*  2.5709207542651528481611339463608225 E-209
+x^252*  2.0437328270221585277476758986377295 E-211
+x^254*  1.6118318268018194488610632610936588 E-213
+x^256*  1.2612539851622949493154153063528759 E-215
+x^258*  9.7926270498507619328506410212243514 E-218
+x^260*  7.5445948936227155202620182841491190 E-220
+x^262*  5.7681758076228812615679592722215705 E-222
+x^264*  4.3765542601429201889262169463408459 E-224
+x^266*  3.2956602838905411738805382291241816 E-226
+x^268*  2.4631649944977078860504830835597113 E-228
+x^270*  1.8272997644208747788789532851884910 E-230
+x^272*  1.3455977312437524650106651013022332 E-232
+x^274*  9.8363374946541029880070340468144678 E-235
+x^276*  7.1381809615231028869961088518181511 E-237
+x^278*  5.1428106729307704472070264913462643 E-239
+x^280*  3.6787047629802064484011221342722680 E-241
+x^282*  2.6127207267821731473382386378136625 E-243
+x^284*  1.8425387570385804039507331118314404 E-245
+x^286*  1.2902904340009841445214910884150782 E-247
+x^288*  8.9727763846206630162980372068308648 E-250
+x^290*  6.1966326997131589119294793979444780 E-252
+x^292*  4.2500588021933414188786367446496555 E-254
+x^294*  2.8951085088163651685307592608650410 E-256
+x^296*  1.9587799296431179662082428708715490 E-258
+x^298*  1.3163677559370068573836516830365380 E-260
+x^300*  8.7873767778245600528981456643467563 E-263
+x^302*  5.8270809589890229808612834844831363 E-265
+x^304*  3.8385902960181587756238432788550292 E-267
+x^306*  2.5121191692848013794533790420901500 E-269
+x^308*  1.6333343564851485478337357735030594 E-271
+x^310*  1.0551026532424685686756552712554848 E-273
+x^312*  6.7720043527468723160507297423038065 E-276
+x^314*  4.3187746458416666065524473697031941 E-278
+x^316*  2.7367954321478776213957675264172161 E-280
+x^318*  1.7233760862141576032030570595919383 E-282
+x^320*  1.0784275050796011077982183929187538 E-284
+x^322*  6.7064398959285544549913919978797548 E-287
+x^324*  4.1447669204107379871157734111016636 E-289
+x^326*  2.5458439895971144620924646826388123 E-291
+x^328*  1.5541876515984266195742476223477542 E-293
+x^330*  9.4304260565984170069476612148378349 E-296
+x^332*  5.6876304211685272334771942063474693 E-298
+x^334*  3.4097255681881316920313209049478022 E-300
+x^336*  2.0319410656792319220244794928747544 E-302
+x^338*  1.2037099398982330855716119487558144 E-304
+x^340*  7.0887048796743225068464281718891817 E-307
+x^342*  4.1501264333709663750357784755238772 E-309
+x^344*  2.4155722872164402402366408421263890 E-311
+x^346*  1.3978409273259589099340096795576327 E-313
+x^348*  8.0424620773383190885938430521414217 E-316
+x^350*  4.6007457555223156980844993425732100 E-318
+x^352*  2.6169149686347582147940021505008903 E-320
+x^354*  1.4800868978782196236759264628558889 E-322
+x^356*  8.3240550670500646398165455766546783 E-325
+x^358*  4.6552879444777486212573378176444272 E-327
+x^360*  2.5890212733452180092493549562050644 E-329
+x^362*  1.4319095881582849186553262993046333 E-331
+x^364*  7.8758924446023022551600836551281423 E-334
+x^366*  4.3082540491380464458304335127991022 E-336
+x^368*  2.3438681934420224431366294599560156 E-338
+x^370*  1.2682599601734130141997140486447331 E-340
+x^372*  6.8255761585313529030023214624502090 E-343
+x^374*  3.6537504179850186529808586971817157 E-345
+x^376*  1.9454470640383043345151522063744741 E-347
+x^378*  1.0303701228040713220056802251251407 E-349
+x^380*  5.4284092933935871432575826027630447 E-352
+x^382*  2.8449161093678520514495836019835479 E-354
+x^384*  1.4831867268736039429035720562759859 E-356
+x^386*  7.6924289606812687736387887303841743 E-359
+x^388*  3.9690278432912187579900658177233798 E-361
+x^390*  2.0373674015999579024215378976211578 E-363
+x^392*  1.0404724062589859131137824355131333 E-365
+x^394*  5.2866324949365548094078654832688881 E-368
+x^396*  2.6725524292132869175146706128752359 E-370
+x^398*  1.3442591789564729820372553297261899 E-372
+x^400*  6.7276043932161689622055463797647739 E-375
+x^402*  3.3501898805653338209475704099873953 E-377
+x^404*  1.6600482741408619782632712400525639 E-379
+x^406*  8.1851190380409222601743302375742820 E-382
+x^408*  4.0159919520697047756962795777863189 E-384
+x^410*  1.9608062398059577710626370130100980 E-386
+x^412*  9.5271038245275776436955979522559012 E-389
+x^414*  4.6066131783811398794367271936266436 E-391
+x^416*  2.2167023535320304489442025975165265 E-393
+x^418*  1.0615681256049270688335788443041634 E-395
+x^420*  5.0595649367762156018326402207801767 E-398
+x^422*  2.4000106325239728954321198912425041 E-400
+x^424*  1.1330722982063571446658985926135453 E-402
+x^426*  5.3242228851643212061132930170268939 E-405
+x^428*  2.4901102609759374945796280980499327 E-407
+x^430*  1.1591888011476562915623901579261002 E-409
+x^432*  5.3712138150299921355619109397593612 E-412
+x^434*  2.4773233699268207877862835462388887 E-414
+x^436*  1.1373501682539365954984102549761850 E-416
+x^438*  5.1977584052351058915934485956926015 E-419
+x^440*  2.3645987423128675227958881955755498 E-421
+x^442*  1.0708466527742741848235811063637934 E-423
+x^444*  4.8276356812403859567097281296510472 E-426
+x^446*  2.1666457058982492763117755485891586 E-428
+x^448*  9.6804661647641518073467320429143415 E-431
+x^450*  4.3059429864857946197237910230468715 E-433
+x^452*  1.9068323395955325335591466711305134 E-435
+x^454*  8.4069312021948624849915116107499447 E-438
+x^456*  3.6902150407655958254380893563093443 E-440
+x^458*  1.6127365347504055955069556099905408 E-442
+x^460*  7.0174777108830590245620859325682571 E-445
+x^462*  3.0402741961576603620194757027175907 E-447
+x^464*  1.3114952288328110327887496287280118 E-449
+x^466*  5.6331460547306824417499990924856986 E-452
+x^468*  2.4092044658830599569385868148282707 E-454
+x^470*  1.0259887676107284313384139546012225 E-456
+x^472*  4.3507660565262587322577700129128264 E-459
+x^474*  1.8371763626014710809425480702452461 E-461
+x^476*  7.7251283884274221825716125421795202 E-464
+x^478*  3.2347296144298196095574184027015663 E-466
+x^480*  1.3488240565723773570312053678447237 E-468
+x^482*  5.6009954020913014783199103742920378 E-471
+x^484*  2.3161955123320472025791616310545782 E-473
+x^486*  9.5387778001852536493119022676929455 E-476
+x^488*  3.9122368065366132896010583186772364 E-478
+x^490*  1.5980108789305212015372959236632205 E-480
+x^492*  6.5007546043738560791390486017894067 E-483
+x^494*  2.6338100282375868123612084089514439 E-485
+x^496*  1.0627934950764545348587435908647804 E-487
+x^498*  4.2713402573825864197269787671449632 E-490
+x^500*  1.7097684889342256728475583262169178 E-492
+x^502*  6.8167170690893784857460923010713081 E-495
+x^504*  2.7069794280609758907801550453849263 E-497
+x^506*  1.0707133556414323052704577339192125 E-499
+x^508*  4.2183912744938588321093927922482137 E-502
+x^510*  1.6554368545070885644250347049640618 E-504
+x^512*  6.4710862686884100831842289143927412 E-507
+x^514*  2.5196905612142926143864290147382080 E-509
+x^516*  9.7730307507476447488074147080763105 E-512
+x^518*  3.7759818172532326284300013425827017 E-514
+x^520*  1.4533010325546245227702971437250612 E-516
+x^522*  5.5720212566576728141525208361883109 E-519
+x^524*  2.1281774067436817198358112855254375 E-521
+x^526*  8.0974283036088274826054118150946671 E-524
+x^528*  3.0692827850287978024970428231289811 E-526
+x^530*  1.1590000294229090885478486724562258 E-528
+x^532*  4.3600647297857883309324682001259171 E-531
+x^534*  1.6340732544631760993127019842572065 E-533
+x^536*  6.1013410862961239302394223399514090 E-536
+x^538*  2.2696575831980264118461130365373784 E-538
+x^540*  8.4116781099102540484574194196715117 E-541
+x^542*  3.1059766267867071243511144656800518 E-543
+x^544*  1.1426490927215579200366990108987664 E-545
+x^546*  4.1882501517153093051205893495243755 E-548
+x^548*  1.5295483910899009974549524278253060 E-550
+x^550*  5.5655808938547381233123139109469887 E-553
+x^552*  2.0178096700795144737589310032739830 E-555
+x^554*  7.2891691901754626842544881489312288 E-558
+x^556*  2.6236727486199870277808066484473897 E-560
+x^558*  9.4098072723088711513345530197835567 E-563
+x^560*  3.3627672674347540093336157394946285 E-565
+x^562*  1.1974667459520276815693244749893372 E-567
+x^564*  4.2489946917606168696332967261626913 E-570
+x^566*  1.5023477183994287175822262696270016 E-572
+x^568*  5.2932415495228858569415743927490860 E-575
+x^570*  1.8584262661552812916649197258771482 E-577
+x^572*  6.5019957821473725533639805524051144 E-580
+x^574*  2.2668932860841137891509077343544729 E-582
+x^576*  7.8759653025974339712690776027586565 E-585
+x^578*  2.7269054608787705338470784012530895 E-587
+x^580*  9.4088197056789576557674817839914784 E-590
+x^582*  3.2352225499222988825731449809832761 E-592
+x^584*  1.1086188574907933487350934313471123 E-594
+x^586*  3.7859471734049936555531848945534595 E-597
+x^588*  1.2885049527100059791377750041493628 E-599
+x^590*  4.3704073815830744250845512499365575 E-602
+x^592*  1.4773623750012737580279726250722068 E-604
+x^594*  4.9772149455130772543535991336437389 E-607
+x^596*  1.6711865764110440014395419009948120 E-609
+x^598*  5.5925201598622233712757741494808079 E-612
+x^600*  1.8652589001443449088121380514937168 E-614; }

It turns out that you lose a lot of precision in that multiplication by exp(-x^2), and you need an extra large amount of precision in the entire function to generate accurate results at the real axis, where the derivatives are cancelling. I generated a 300 term (x^600) asymptotic for the entire function, in pari-gp using 134 decimal digits precision and a lot of tricks, which gives very good results if $|x|<=9$, and has its best accuracy of about 35 decimal digits for values near x=+9. Updated to include the first 125 terms of entiref (x^250), for Antonio Vargas.

 {entiref=
 +x^ 2*  1.000000000000000000000000000000000000000000000000000000000000000000
 +x^ 4* -0.3903080328022398631614181122699925379762414278275504252183985648850
 +x^ 6*  0.1375788748110124682424374847928600715147226065003467931915901836882
 +x^ 8* -0.03863728761483880499259264298678193713104655257223268868775160299498
 +x^10*  0.008747481601094830079665989070608193777827806357836361521383464628533
 +x^12* -0.001641193268278966279231353709768925284150022492905004523264592289770
 +x^14*  0.0002616639105782442949733463357922654190354746495134737600001108244232
 +x^16* -0.00003618561358737824400665656054216919436248417993097722079578770229568
 +x^18*  0.000004412510097357058845738729777072243729802120876650229561000776190015
 +x^20* -0.0000004808126873165497443393515090015363089752813947730516960136053584884
 +x^22*  0.00000004733084424489864100234363177603646208282049355730062960571981692851
 +x^24* -0.000000004247428934267281201811125771560623360590507330764270562145091149168
 +x^26*  0.0000000003501335594761328997744701771923262297783910556272608750113648130919
 +x^28* -2.668630243903328175021565547870923863178634487323350804813312708238 E-11
 +x^30*  1.891120738388056412638527127164007962138666819660099591006583816498 E-12
 +x^32* -1.252097029553170339693678466000739519438205823046719982007820130696 E-13
 +x^34*  7.778504706304949680843875697095710856189980020607659383183493741077 E-15
 +x^36* -4.551238453526195314455658434004708125426827807197877960382484343355 E-16
 +x^38*  2.516481981763348464137143471704196452796661296427446064374946665221 E-17
 +x^40* -1.318838800154599762565175454315150405570382986212012474812519462672 E-18
 +x^42*  6.568953128927681260495586323443395623814643892064506599693955639215 E-20
 +x^44* -3.117227042071800769320929752367637475934420891851486694342979093797 E-21
 +x^46*  1.412450547402984869528281506638892938809787199205571323311490434722 E-22
 +x^48* -6.123381116921861262956124159360437736265928988389993129732737773670 E-24
 +x^50*  2.544659298608104031544788552184204651343521455381910179877458046177 E-25
 +x^52* -1.015384688950120000418048348842714390022585963928704523014715143804 E-26
 +x^54*  3.896540198116138343915535799437739159761006073575164646517770124122 E-28
 +x^56* -1.440155194951485402851719427157641346171222033695983959825656994082 E-29
 +x^58*  5.133451976996463967204245633684011099607434720643460675804128943736 E-31
 +x^60* -1.766963080358872061808820431113781079212151015127724863334759364894 E-32
 +x^62*  5.879957458426308562941552764240728133869531432107574746089012733808 E-34
 +x^64* -1.893770931532020419347879666434363020192223533099907670187271416789 E-35
 +x^66*  5.909286356660390400719055048074241546421263606696828768530587540150 E-37
 +x^68* -1.788205488532361192026140779288595376895462864177359532326866558905 E-38
 +x^70*  5.252549951221843928744208387464144058077763922035517888655816841696 E-40
 +x^72* -1.498877612875142166111638892660063056650438790686374124478266494435 E-41
 +x^74*  4.158697751492591570701921665524932295810396287244992772370050923038 E-43
 +x^76* -1.122733248293364879313692096444812328194682894662060194420057474181 E-44
 +x^78*  2.951474664352473043243621851555023598586562525517034340782506655576 E-46
 +x^80* -7.560381699931795105957872349521925165356917620746234113693051018497 E-48
 +x^82*  1.888315095977826644356334460543477687577192792001237225453836530092 E-49
 +x^84* -4.601519242339140903411470798077886447301646630443349834596685081384 E-51
 +x^86*  1.094664590132602738233248736166351745407731167142735995520989816639 E-52
 +x^88* -2.543659239935988987323687867118984800714268535442454864259414800107 E-54
 +x^90*  5.776550852534097828063545564188677119116262766430314568849842566158 E-56
 +x^92* -1.282724525873545450951998309430456675494571718070986174948894664498 E-57
 +x^94*  2.786550193951782813731959474458748346686496730200934398002234067872 E-59
 +x^96* -5.924796219155865282227842097252363371054756731897868316742816486519 E-61
 +x^98*  1.233527598068494418478206898158068983739087998551787680776846585950 E-62
 +x^100* -2.515828404089953602462783677019212534380186712868882793528751556938 E-64
 +x^102*  5.028634635634635008744667900948918019719316369147515442777126677752 E-66
 +x^104* -9.854377284524946586809705153062349158016923826451027438638680704156 E-68
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