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I want to evaluate the below function at limit zero. I tried with a function which has double differentiation for which it is giving me a finite value. For triple differentiation it is consistently giving output as zero. I want to know if the function given below can be evaluated and if not what could be the reason.

RF[r_] := (1/r)0.200731 (-0.000343151 r SphericalBesselJ[3, 0.137662 r] - 
   0.0642014 r SphericalBesselJ[3, 0.205217 r] + 
   0.116823 r SphericalBesselJ[3, 0.269851 r] - 
   0.188628 r SphericalBesselJ[3, 0.333395 r] + 
   0.174251 r SphericalBesselJ[3, 0.3964 r] + 
   0.0371598 r SphericalBesselJ[3, 0.459094 r] - 
   0.124404 r SphericalBesselJ[3, 0.521592 r] - 
   0.0878417 r SphericalBesselJ[3, 0.583959 r] + 
   0.0341412 r SphericalBesselJ[3, 0.646234 r] + 
   0.110696 r SphericalBesselJ[3, 0.70844 r] + 
   0.0979234 r SphericalBesselJ[3, 0.770594 r] + 
   0.0284896 r SphericalBesselJ[3, 0.832709 r] - 
   0.048473 r SphericalBesselJ[3, 0.894793 r] - 
   0.0981621 r SphericalBesselJ[3, 0.956851 r] - 
   0.11019 r SphericalBesselJ[3, 1.01889 r] - 
   0.0901751 r SphericalBesselJ[3, 1.08091 r] - 
   0.0513123 r SphericalBesselJ[3, 1.14292 r] - 
   0.00632141 r SphericalBesselJ[3, 1.20491 r] + 
   0.0352401 r SphericalBesselJ[3, 1.26689 r] + 
   0.0682566 r SphericalBesselJ[3, 1.32887 r] + 
   0.0908501 r SphericalBesselJ[3, 1.39084 r] + 
   0.103587 r SphericalBesselJ[3, 1.4528 r] + 
   0.108074 r SphericalBesselJ[3, 1.51475 r] + 
   0.106441 r SphericalBesselJ[3, 1.5767 r] + 
   0.100633 r SphericalBesselJ[3, 1.63865 r] + 
   0.0923663 r SphericalBesselJ[3, 1.70059 r] + 
   0.0828932 r SphericalBesselJ[3, 1.76253 r] + 
   0.0731573 r SphericalBesselJ[3, 1.82446 r] + 
   0.0637315 r SphericalBesselJ[3, 1.88639 r] + 
   0.0549955 r SphericalBesselJ[3, 1.94832 r] + 
   0.04711 r SphericalBesselJ[3, 2.01025 r] + 
   0.0401549 r SphericalBesselJ[3, 2.07217 r] + 
   0.0340995 r SphericalBesselJ[3, 2.13409 r] + 
   0.0288995 r SphericalBesselJ[3, 2.19601 r] + 
   0.02446 r SphericalBesselJ[3, 2.25793 r] + 
   0.0207028 r SphericalBesselJ[3, 2.31985 r] + 
   0.0175275 r SphericalBesselJ[3, 2.38176 r] + 
   0.0148598 r SphericalBesselJ[3, 2.44368 r] + 
   0.0126148 r SphericalBesselJ[3, 2.50559 r] + 
   0.0107338 r SphericalBesselJ[3, 2.5675 r] + 
   0.0091515 r SphericalBesselJ[3, 2.62941 r] + 
   0.0078254 r SphericalBesselJ[3, 2.69132 r] + 
   0.00670751 r SphericalBesselJ[3, 2.75323 r] + 
   0.00576856 r SphericalBesselJ[3, 2.81514 r] + 
   0.00497396 r SphericalBesselJ[3, 2.87705 r] + 
   0.00430427 r SphericalBesselJ[3, 2.93895 r] + 
   0.00373467 r SphericalBesselJ[3, 3.00086 r] + 
   0.00325262 r SphericalBesselJ[3, 3.06276 r] + 
   0.00284023 r SphericalBesselJ[3, 3.12467 r] + 
   0.00248963 r SphericalBesselJ[3, 3.18657 r] + 
   0.00218777 r SphericalBesselJ[3, 3.24848 r] + 
   0.00192993 r SphericalBesselJ[3, 3.31038 r] + 
   0.00170643 r SphericalBesselJ[3, 3.37228 r] + 
   0.00151461 r SphericalBesselJ[3, 3.43418 r] + 
   0.00134719 r SphericalBesselJ[3, 3.49608 r] + 
   0.00120281 r SphericalBesselJ[3, 3.55798 r] + 
   0.0010759 r SphericalBesselJ[3, 3.61989 r] + 
   0.000965963 r SphericalBesselJ[3, 3.68179 r] + 
   0.000868629 r SphericalBesselJ[3, 3.74369 r] + 
   0.000783953 r SphericalBesselJ[3, 3.80558 r] + 
   0.000708442 r SphericalBesselJ[3, 3.86748 r] + 
   0.000642491 r SphericalBesselJ[3, 3.92938 r] + 
   0.000583255 r SphericalBesselJ[3, 3.99128 r] + 
   0.000531332 r SphericalBesselJ[3, 4.05318 r] + 
   0.000484364 r SphericalBesselJ[3, 4.11508 r] + 
   0.000443062 r SphericalBesselJ[3, 4.17698 r] + 
   0.000405441 r SphericalBesselJ[3, 4.23887 r] + 
   0.000372266 r SphericalBesselJ[3, 4.30077 r] + 
   0.000341841 r SphericalBesselJ[3, 4.36267 r] + 
   0.000314949 r SphericalBesselJ[3, 4.42456 r] + 
   0.00029012 r SphericalBesselJ[3, 4.48646 r] + 
   0.000268132 r SphericalBesselJ[3, 4.54836 r] + 
   0.0002477 r SphericalBesselJ[3, 4.61025 r] + 
   0.000229577 r SphericalBesselJ[3, 4.67215 r] + 
   0.00021263 r SphericalBesselJ[3, 4.73405 r] + 
   0.000197583 r SphericalBesselJ[3, 4.79594 r] + 
   0.000183424 r SphericalBesselJ[3, 4.85784 r] + 
   0.000170843 r SphericalBesselJ[3, 4.91973 r] + 
   0.000158933 r SphericalBesselJ[3, 4.98163 r] + 
   0.000148347 r SphericalBesselJ[3, 5.04352 r] + 
   0.000138266 r SphericalBesselJ[3, 5.10542 r] + 
   0.000129305 r SphericalBesselJ[3, 5.16731 r] + 
   0.00012072 r SphericalBesselJ[3, 5.22921 r] + 
   0.000113092 r SphericalBesselJ[3, 5.2911 r] + 
   0.000105741 r SphericalBesselJ[3, 5.353 r] + 
   0.0000992116 r SphericalBesselJ[3, 5.41489 r] + 
   0.0000928812 r SphericalBesselJ[3, 5.47678 r] + 
   0.0000872617 r SphericalBesselJ[3, 5.53868 r] + 
   0.0000817795 r SphericalBesselJ[3, 5.60057 r] + 
   0.0000769142 r SphericalBesselJ[3, 5.66247 r] + 
   0.0000721356 r SphericalBesselJ[3, 5.72436 r] + 
   0.0000678925 r SphericalBesselJ[3, 5.78625 r] + 
   0.0000636919 r SphericalBesselJ[3, 5.84815 r] + 
   0.0000599513 r SphericalBesselJ[3, 5.91004 r] + 
   0.0000562072 r SphericalBesselJ[3, 5.97193 r] + 
   0.000052841 r SphericalBesselJ[3, 6.03383 r] + 
   0.0000494044 r SphericalBesselJ[3, 6.09572 r] + 
   0.0000462225 r SphericalBesselJ[3, 6.15761 r] + 
   0.0000428304 r SphericalBesselJ[3, 6.21951 r] + 
   0.0000395045 r SphericalBesselJ[3, 6.2814 r]) 

This is a wavefunction which includes spherical bessel functions

urF[r] := Abs[D[D[D[RF[r], r], r], r]]^2

I want to differentiate it three times and then evaluate at limit zero

DRF = Limit[urF[r], r -> 0, GenerateConditions -> False]

I am trying to evaluate the function at limit zero but it is giving me output as 0. I want a value even if it is very small. I tried using

$MinPrecision = 20; $MaxPrecision = 30;

But it is still not working. Any suggestions on how to overcome this. Thank you in advance

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  • $\begingroup$ what happens if you change urF[r] :=... to urF[r_] :=... and change urF8[r] to urF[r] as there is no urF8[r] anywhere in your code. $\endgroup$
    – Nasser
    Mar 12 at 6:22
  • $\begingroup$ And you do not have to write D[D[D[RF[r], r], r], r] you can just write D[RF[r],{r,3}] $\endgroup$
    – Nasser
    Mar 12 at 6:25
  • $\begingroup$ Yeah its urF[r] only. I will correct it in the original question. I was trying to see if it is differentiable that is why i wrote it like D[D[D[RF[r], r], r], r]. $\endgroup$ Mar 12 at 6:43
  • 1
    $\begingroup$ if you remove Abs I get 0.0111299 as limit. Why did you add Abs ? $\endgroup$
    – Nasser
    Mar 12 at 6:45
  • $\begingroup$ I wanted the function to have a positive value. I will check it once without absolute and see if it works for all of my other function and let you know. Thank you for your help good sir. $\endgroup$ Mar 12 at 6:57

1 Answer 1

3
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After sometime, it does give answer with Abs also.

using V 14 on windows:

RF[r_] := (* see original post*)
urF[r_] := Abs[D[RF[r], {r, 3}]]^2
Limit[urF[r], r -> 0]

Mathematica graphics

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