2.86295 + 3.22327 i

[tommy1729]

Consider the equation z^z = z.

This is fascinating because it makes the sequence ( power tower or tetration base z ) 
z^^n 1-periodic in n.

Maybe you think now that therefore it must be within the shell-tron.
Or maybe you think it is on its edge.

Let’s examine.

 z = z^z.

1 , -1 are solutions.

And maybe some accept 0. But imho 0^0 = 1.

These solutions ( bases) are problematic for tetration to say the least.

Please correct me if I’m wrong about that !!

But !

There are nonreal solutions as well !

And they are outside the shell-tron !

Even more surprising the derivative at their fixpoints is not a half-rotation !!

So counter-intuitive I might say.

Im not sure how many have repelling or attracting fixpoints. 
I think none are parabolic.

That is interesting and surprising.

The idea of cyclic orbits emerges.

This reminds me of the base 1.7129 i.

But this is different. 

The smallest solution is ( or its conjugate )

About

2.86295 + 3.22327 i


I am fascinated by it.



Regards 

Tommy1729

[Gottfried]

Hmmm, just to get more familiar to it....
\( z=z^z \)                           

\( 0=z^z-z \)                           

\( 0=z \cdot (z^{z-1} - 1) \)                           

\( 0= f^{\circ 2}(1) \)                           
where
.                      \( f(x)=z^x-1 \)                           


To look for zeros it might be helpful to list zeros of the real- and the imaginary parts separately first.                                

update: the Wolfram-alpha-contourplots for z^(z-1)-1 from 1+I to 10+10*I  (separate for real and for imaginary parts) give zeros on continuous lines, and an overlay seems to indicate more systematical zeros at the crossings of the lines.                                  

Some small programming in Pari/GP gave the following additional solutions  

Code:

                                           

 2.863+3.223*I
 3.727+5.318*I
 4.433+7.194*I
 5.057+8.947*I
 5.627+10.62*I
 6.160+12.23*I
 6.664+13.79*I
 7.144+15.31*I
 7.606+16.80*I
 8.051+18.27*I
 8.482+19.70*I
 8.902+21.12*I
 9.310+22.51*I
 9.710+23.89*I
...

which can easily be prolonged because the pairwise distances seem to approximate  a linear scheme.  Of course the conjugate numbers are also roots.                                                      
This does not exclude further roots, I would especially look for them in the angle indicated by the interpolation curve through the roots (and their conjugates) with the real axis                  


update 2:   I've computed the regression-line for the progression of the real parts of the roots and for the imaginary parts of the roots.                  

Code:

Re(n) = 1.207 E-12*n^13 - 1.275 E-10*n^12 + 0.000000006121*n^11 - 0.0000001769*n^10 + 0.000003433*n^9 - 0.00004727*n^8 + 0.0004760*n^7
             - 0.003558*n^6 + 0.01987*n^5 - 0.08292*n^4 + 0.2585*n^3 - 0.6192*n^2 + 1.714*n + 1.576
Im(n) = 1.340 E-12*n^13 - 1.416 E-10*n^12 + 0.000000006805*n^11 - 0.0000001968*n^10 + 0.000003824*n^9 - 0.00005274*n^8 + 0.0005321*n^7
            - 0.003990*n^6 + 0.02239*n^5 - 0.09412*n^4 + 0.2979*n^3 - 0.7428*n^2 + 3.151*n + 0.5923

est_root(n)= Re(n)+ I*Im(n)

                                   
This reproduces the given roots for n=1 to 14 perfectly and gives guesses for the next couple of roots. Such further guesses should be improved by an application of the Newton-rootfinder algorithm, of course.                            

update 3: picture based on first 41 complex roots (41=1+2*20):

                                             


update 4: contourplot of imaginary zeros (black lines) and real zeros (white lines) for z^{z-1}-1 . The smallest known 9 roots from the previous picture are dotted with red color. The image suggests there are *no more* zeros besides that on the curce determined in the previous plot, because the black and white lines have pairwise only one crossing.        

   

[tommy1729]

Hmmm, just to get more familiar to it....
\( z=z^z \)                           

\( 0=z^z-z \)                           

\( 0=z \cdot (z^{z-1} - 1) \)                           

\( 0= f^{\circ 2}(1) \)                           
where
.                      \( f(x)=z^x-1 \)                           


To look for zeros it might be helpful to list zeros of the real- and the imaginary parts separately first.

 
Interesting.

I prefer to look at it differently or at least used too.

Im not aware of any kind of closed form like lambert W but I have not really tried.

Here is what I did to get 2.86295 + 3.22327 i



Let z^z = z.

Take ln on both sides.

z ln(z) = ln(z) + 2 k pi I 

Now if we let k be 1 we get

2.86295 + 3.22327 i = z

And if k = -1 we get

2.86295 - 3.22327 i

And they appear the smallest ones.
That is probably easy to prove though I did not.

I have not considered other ln branches ( on the left side ) , my first guess is they are a invalid solutions ?

——

As for your solution you auto find z = 0 as a solution which is funny since it is invalid ; 0^0 = 1 !
Well at least that is standard. Limit cases and other opinions may differ.
As an explanation we could say you divided by z = 0. If that is satisfactory is another matter.

So as you say we end up with an expression that is a double iteration in disguise. 

Consider z^v - 1 = 0

So z^v = 1

This implies v = the period of the function z^t. 

Also v = z - 1.

——

We could use Newton iterations to find 2.86295 + 3.22327 i.

Or we have a Taylor series similar to lambert W.

I assume the real part , Imaginary part , norm and argument are all transcendental.
A proof would be nice.

Recall that e^(1/e) and other typical tetration number are also not proven transcendental or even irrational !

I assume there is a way to transform your equations to mine and vice versa without first going back to z^z = z. 
Not sure.

I welcome other attempts to solve that equation.

Notice that the real part of ln ln 2.86295 + 3.22327 i is larger than 0 hence it is not on the shell-tron boundary !! 

That is remarkable considering its cyclic behavior.

I was thinking about “ its cousin base “

z^z = z

z^^( “ oo “)  = z

z^(1/z) = Q 
(z^z)^(z^-z) = S

Now

Q^z = z^(z/z) = z = z^z

So Q = z.
It easy to show S = Q = z.

Nothing special 

However we also have

T^(1/T) = z

Where T =\= z.

And T within the Shel-tron.

I have to think more about that. 

Regards

Tommy1729

[tommy1729]

Easy generalizations are the similar power tower equations

\( z_n^^{<n>}= z_n \)

Where <n> means height n ofcourse.

But I guess we should start simple.

Regards 

Tommy1729

[Gottfried]

Let z^z = z.

Take ln on both sides.

z ln(z) = ln(z) + 2 k pi I 

Now if we let k be 1 we get

2.86295 + 3.22327 i = z

And if k = -1 we get

2.86295 - 3.22327 i

And they appear the smallest ones.
That is probably easy to prove though I did not.

Yes, your ansatz is similar to my first ansatz in my linked MSE-problem asking for roots z^z^z+1 which I also rewrote initially in its logarithmic form, and did not consider the multiplicity of complex logarithms.      (see https://math.stackexchange.com/a/1415538/1714)                       

Also, the form \( f^{\circ2}(1)=0 \) directs immediately to the conjugate expression for z^z (and I''ve indeed cancelled the z-factor (which introduces the need to handle z=0 extra!) I would have liked to work directly with this, say an adapted LambertW-function or so, but hadn't a good idea.                     

I'm curious whether we find more zeros and besides the interpolation-line indicated by my 14 solutions...

update: I'm now -after the newest image in my older answer- convinced there are no further roots besides that of the rough line shown in my previous post.

[sheldonison]

update: the Wolfram-alpha-contourplots for z^(z-1)-1 from 1+I to 10+10*I  (separate for real and for imaginary parts) give zeros on continuous lines, and an overlay seems to indicate more systematical zeros at the crossings of the lines.                                  

Some small programming in Pari/GP gave the following additional solutions  

Nice work Gottfried.  I could plot sexp for the Kneser solution for a couple of these bases with repelling fixed point b^b=b.... Kneser/fatou.gp uses the primary fixed points, which are different for these bases, but also both repelling.  These tetarion bases are outside the ShellThron region.

Code:

b^b=b; b^L1=L1; b^L2=L2;
b=~2.8630+3.2233*I;  L1=~0.23429+0.72594*I;  L2=~-0.50419-1.0820*I;  
b=~3.7273+5.3180*I;  L1=~0.13364+0.66506*I;  L2=~-0.45567-0.86504*I;
b=~4.4332+7.1938*I;  L1=~0.08774+0.62923*I;  L2=~-0.42709-0.76959*I;

Kneser tetration plot for b=~2.8630+3.2233*I from -3+3i..+8-4i; generated using fatou.gp
   
When iterating exponentials, one tends to quickly get very large numbers or very small numbers.  In this plot, very small numbers have Tet(z)=~0 which is black, and then Tet(z+1)=~1; Red.  And Tet(z+2)=~2.8630+3.2233*I; Orangish.  Since its a repelling fixed point Tet(z+n) will eventually break away from the fixed point.
Continued Kneser tetration plot from +4+3i..+15-4i; Notice how the unstable fixed point takes over.  The white regions are large in magnitude and are seeds for super-exponential growth, but those regions are also unstable since when iterating large complex numbers half the time you go from very large to very small. 

   

[Gottfried]

Code:

b^b=b; b^L1=L1; b^L2=L2;
b=~2.8630+3.2233*I;  L1=~0.23429+0.72594*I;  L2=~-0.50419-1.0820*I;  
b=~3.7273+5.3180*I;  L1=~0.13364+0.66506*I;  L2=~-0.45567-0.86504*I;
b=~4.4332+7.1938*I;  L1=~0.08774+0.62923*I;  L2=~-0.42709-0.76959*I;

Kneser tetration plot for b=~2.8630+3.2233*I from -3+3i..+8-4i; generated using fatou.gp

When iterating exponentials, one tends to quickly get very large numbers or very small numbers.  In this plot, very small numbers have Tet(z)=~0 which is black, and then Tet(z+1)=~1; Red.  And Tet(z+2)=~2.8630+3.2233*I; Orangish.  Since its a repelling fixed point Tet(z+n) will eventually break away from the fixed point.
Continued Kneser tetration plot from +4+3i..+15-4i; Notice how the unstable fixed point takes over.  The white regions are large in magnitude and are seeds for super-exponential growth, but those regions are also unstable since when iterating large complex numbers half the time you go from very large to very small. 

Hi Sheldon - thanks for your remarks! Unfortunately I seem to have been "out-of-subject" ;-) too long: I don't get the relation of your graphic with the problem of zeros of z^z - z. Could you please explain further?

[sheldonison]

Hi Sheldon - thanks for your remarks! Unfortunately I seem to have been "out-of-subject" ;-) too long: I don't get the relation of your graphic with the problem of zeros of z^z - z. Could you please explain further?

Hi Gottfried,

I only made a Kneser complex plane graph for one tetration base; B=2.8630+3.2233*I; although I observed that the next two examples also converged in fatou.gp and would allow for similar graphs.  

I also observed that these examples have the base as a different repelling fixed point from the fixed points used by Kneser's algorithm.  The complex plane plot is a pretty picture showing different approaches to near the fixed point of B.  Here is the path from the Kneser lower fixed point of -0.504189-1.082018*I iterating towards the B since sexp(2.5) gets very close to zero after sexp(1.5) gets large and negative.

Code:

 n;sexp(n) real             imag
  -10.50 -0.506879459669 -1.081547675946*I
   -9.50 -0.503719166668 -1.076547203763*I
   -8.50 -0.493149637885 -1.081945538564*I
   -7.50 -0.501967307955 -1.104327690008*I
   -6.50 -0.549186543972 -1.089827111759*I
   -5.50 -0.525023017294 -0.995149128209*I
   -4.50 -0.345334360865 -1.019099153870*I
   -3.50 -0.297562306319 -1.396335887620*I
   -2.50 -1.389515585126 -1.581602966335*I
   -1.50 -0.470176399198  0.167876843684*I
   -0.50  0.431553569486 -0.066007303590*I
    0.50  1.915502539029  0.526030319479*I
    1.50 -7.670428285966  7.221368273487*I  negative value
    2.50 -0.000000018154 -0.000000024463*I  close to zero
    3.50  0.999999994134 -0.000000051078*I  close to 1
    4.50  2.862954491212  3.223273719942*I  close to B;
    5.50  2.862955485181  3.223276200105*I
    6.50  2.862939259644  3.223286908820*I
    7.50  2.862839224324  3.223186906491*I
    8.50  2.863405711080  3.222327752266*I
    9.50  2.870363720892  3.225107746164*I  each iterations is
   10.50  2.860449530266  3.279014376166*I  less close to B

Here is another example showing superexponential growth leading to the B fixed point
n;sexp(n) real             imag
  -10.80 -0.506711953156 -1.082462980055*I
   -9.80 -0.505520576103 -1.077039610068*I
   -8.80 -0.494475720250 -1.078444858441*I
   -7.80 -0.495122081123 -1.100923128257*I
   -6.80 -0.541277048292 -1.103507639857*I
   -5.80 -0.550805490837 -1.011174310369*I
   -4.80 -0.381693338778 -0.978549052364*I
   -3.80 -0.236035997893 -1.286771958363*I
   -2.80 -1.022818800959 -1.833802406278*I
   -1.80 -0.971588970934  0.412810040788*I
   -0.80  0.166605373584 -0.036790753518*I
    0.20  1.310922364180  0.114265817726*I
    1.20  1.802771756084  5.896405085981*I
    2.20 -0.072428251453 -0.062705439536*I
    3.20  0.937447648141 -0.144360980939*I
    4.20  3.715993464216  2.438657532303*I
    5.20 26.577478568061 11.819890866600*I
    6.20  -1466498688753   3060160492434*I ** very large magnitude
    7.20  0.000000000000  0.000000000000*I ** extremely tiny magnitude
    8.20  1.000000000000  0.000000000000*I
    9.20  2.862954135717  3.223273836391*I ** extremely close to B
   10.20  2.862954135717  3.223273836391*I ...
...
A third example, from the upper Kneser fixed point; 0.23429+0.725944*I
 -10.50+0.30*I  0.292621589856  0.716091362119*I
  -9.50+0.30*I  0.229318668308  0.805625645051*I
  -8.50+0.30*I  0.140622561439  0.693906067361*I
  -7.50+0.30*I  0.289947230263  0.618938924348*I
  -6.50+0.30*I  0.370576060735  0.826423265101*I
  -5.50+0.30*I  0.042962728865  0.854091637519*I
  -4.50+0.30*I  0.146280541766  0.496511638703*I
  -3.50+0.30*I  0.537929795299  0.611163100699*I
  -2.50+0.30*I  0.290263780298  1.277238365086*I
  -1.50+0.30*I -0.267476739709  0.445564945926*I
  -0.50+0.30*I  0.423002091636  0.191529687303*I
   0.50+0.30*I  1.268662282738  0.938873534671*I
   1.50+0.30*I -2.212727431327  1.857262524570*I
   2.50+0.30*I  0.005451966339  0.006145429371*I  close to zero, but not that close
   3.50+0.30*I  1.002687766483  0.013621434548*I
   4.50+0.30*I  2.769720532534  3.261156764762*I  unstable approach to fixed point of B
   5.50+0.30*I  2.482695908749  2.666868294719*I
   6.50+0.30*I  3.792132767998 -1.130095791095*I
   7.50+0.30*I 12.925015630715 662.01740679844*I
   8.50+0.30*I  0.000000000000  0.000000000000*I  extremely close to zero now
   9.50+0.30*I  1.000000000000  0.000000000000*I
  10.50+0.30*I  2.862954135717  3.223273836391*I  extremely close to B

Here are the values for Kneser's tetration for B=2.8630+3.2233*I at the real axis

Code:

n;sexp(n)  real            imag
 -1.95  -1.68863033100   0.80749681432*I
 -1.90  -1.33183739912   0.60771306027*I
 -1.85  -1.12187088481   0.49293340464*I
 -1.80  -0.97158897093   0.41281004079*I
 -1.75  -0.85368869801   0.35158089365*I
 -1.70  -0.75601053277   0.30223718788*I
 -1.65  -0.67206743302   0.26104970510*I
 -1.60  -0.59798462846   0.22580024594*I
 -1.55  -0.53125894475   0.19506474235*I
 -1.50  -0.47017639920   0.16787684368*I
 -1.45  -0.41350887904   0.14355282012*I
 -1.40  -0.36034335168   0.12159308516*I
 -1.35  -0.30997964615   0.10162337348*I
 -1.30  -0.26186618415   0.08335787988*I
 -1.25  -0.21555788662   0.06657524687*I
 -1.20  -0.17068763255   0.05110242206*I
 -1.15  -0.12694631651   0.03680352807*I
 -1.10  -0.08406853793   0.02357203679*I
 -1.05  -0.04182208027   0.01132519101*I
 -1.00  -0.00000000000   0.00000000000*I
 -0.95   0.04158545142  -0.01044962968*I
 -0.90   0.08310759688  -0.02005491968*I
 -0.85   0.12472915581  -0.02883359692*I
 -0.80   0.16660537358  -0.03679075352*I
 -0.75   0.20888664016  -0.04391927640*I
 -0.70   0.25172073041  -0.05019990162*I
 -0.65   0.29525476327  -0.05560092452*I
 -0.60   0.33963695198  -0.06007757813*I
 -0.55   0.38501819829  -0.06357107630*I
 -0.50   0.43155356949  -0.06600730359*I
 -0.45   0.47940368417  -0.06729512052*I
 -0.40   0.52873602250  -0.06732423809*I
 -0.35   0.57972616535  -0.06596260063*I
 -0.30   0.63255895544  -0.06305319831*I
 -0.25   0.68742955963  -0.05841021138*I
 -0.20   0.74454439500  -0.05181436430*I
 -0.15   0.80412185886  -0.04300734123*I
 -0.10   0.86639277455  -0.03168508107*I
 -0.05   0.93160042577  -0.01748973207*I
 -0.00   1.00000000000  -0.00000000000*I
  0.05   1.07185719071   0.02128042978*I
  0.10   1.14744561154   0.04693677824*I
  0.15   1.22704254441   0.07765981927*I
  0.20   1.31092236418   0.11426581773*I
  0.25   1.39934673899   0.15772037159*I
  0.30   1.49255037217   0.20916684954*I
  0.35   1.59072059952   0.26996017715*I
  0.40   1.69396854036   0.34170674594*I
  0.45   1.80228866766   0.42631115941*I
  0.50   1.91550253903   0.52603031948*I
  0.55   2.03318092557   0.64353488562*I
  0.60   2.15453658025   0.78197722461*I
  0.65   2.27827728009   0.94506332647*I
  0.70   2.40240544403   1.13712333668*I
  0.75   2.52394651318   1.36317064654*I
  0.80   2.63858347989   1.62893181732*I
  0.85   2.74016991978   1.94081738839*I
  0.90   2.82008979925   2.30578450717*I
  0.95   2.86643179940   2.73101309925*I
  1.00   2.86295413572   3.22327383639*I
  1.05   2.78784271893   3.78780391346*I
  1.10   2.61232867553   4.42642243211*I
  1.15   2.29936076494   5.13451386397*I
  1.20   1.80277175608   5.89640508598*I
  1.25   1.06780346359   6.67861653316*I
  1.30   0.03454216886   7.42061697158*I
  1.35  -1.35319024000   8.02333254654*I
  1.40  -3.12976426085   8.33725374335*I
  1.45  -5.27598852788   8.15529704969*I
  1.50  -7.67042828597   7.22136827349*I
  1.55 -10.02727185109   5.27358377649*I
  1.60 -11.84180728392   2.14738517285*I
  1.65 -12.39732407235  -2.04413877023*I
  1.70 -10.92658217465  -6.68238131631*I
  1.75  -7.02079353295 -10.50925017693*I
  1.80  -1.23729536683 -11.87489146526*I
  1.85   4.51097059951  -9.63931846573*I
  1.90   7.57231949664  -4.46025989870*I
  1.95   6.51285965985   0.83938812900*I
  2.00   2.86295413572   3.22327383639*I
  2.05  -0.08438487542   2.39664112407*I
  2.10  -0.79125260624   0.73812821547*I
  2.15  -0.37658292709  -0.00741162857*I
  2.20  -0.07242825145  -0.06270543954*I
  2.25  -0.00555783783  -0.01596744417*I
  2.30  -0.00024564577  -0.00198110359*I
  2.35  -0.00006363357  -0.00014456247*I
  2.40  -0.00000897682  -0.00000103207*I
  2.45   0.00000017541   0.00000042293*I
  2.50  -0.00000001815  -0.00000002446*I
  2.55   0.00000000365  -0.00000000348*I
  2.60   0.00000000417  -0.00000000273*I
  2.65   0.00000004799  -0.00000005931*I
  2.70   0.00003255138  -0.00000467397*I
  2.75  -0.19083642620  -0.16260934798*I
  2.80      3342.49191        1626.90264
  2.85  -1649460.56280     1879827.35811
  2.90   2741567.93502     -336941.79584
  2.95      6038.48545        2869.75907
  3.00   2.86295413572   3.22327383639*I
  3.05  -0.11194644361  -0.03329948248*I
  3.10   0.15470908504   0.06729995089*I
  3.15   0.54931374228  -0.18745793586*I
  3.20   0.93744764814  -0.14436098094*I
  3.25   1.00498348625  -0.02817250887*I
  3.30   1.00131021153  -0.00310632732*I
  3.35   1.00002907124  -0.00026498357*I
  3.40   0.99998775469  -0.00000908916*I
  3.45   0.99999989914   0.00000076612*I
  3.50   0.99999999413  -0.00000005108*I
  3.55   1.00000000827  -0.00000000201*I
  3.60   1.00000000840  -0.00000000047*I
  3.65   1.00000012021  -0.00000004613*I
  3.70   1.00005151265   0.00002066212*I
  3.75   0.79992815390  -0.33704771994*I
 3.80  -1.91323 E1524 - 1.75181 E1524*I  really huge magnitude ...

[Daniel]

Note I discuss \( ^na=1 \); Tetration research: 1986 - 1991, pg 4.

[tommy1729]

Note I discuss \( ^na=1 \); Tetration research: 1986 - 1991, pg 4.

Yes that is related !

z^^2 = 1 has the solution 2.2136 + 3.1140 i.

I was thinking about solving z^^3 = z today.

z^^3 = z^(z^^2) = z^1 = z.
so 2.2136 + 3.1140 i is a solution.


z^^3 = z = z^(z^z) 
so if z^z = z then z^(z^z) = z^z = z.

therefore 2.86295 + 3.22327 i from the OP is also a solution.

another solution we get from

z^^3 = z
=>
exp( ln(z) * z^z ) = exp(ln(z))
=>
ln(z) * z^z = ln(z) + 2pi i
so z = 2.36678 + 0.617735 i

***

I suspect that z^z^z = z has "2 or 3 rays" of solutions.
whereas z^z = z only has 1 ray.
Look at gottfried's post of the zero's ; the zero's are on a nice curve ... that's what i call a "ray".
informally : Ray's are more or less 1 dimensional set of points as opposed to say the gaussian integers.



In general I think z^^n = z has about n rays of solutions.
I added the informal definition and counterexample to make the statement somewhat falsifiable.

I like that when we pick the smallest element of the rays it is almost like z^^n = z is like solving a polynomial of degree O(n). ( big O notation)

***

Inverting z^^n for fixed n is an interesting function but I have not seen much research for series expansions based upon their fixpoints ?
Maybe Im wrong or maybe that is not interesting ??

In the open problems section there is a conjectured series expansion for it btw !!

***

consider making a list of solutions to z^^k = z for rising index k.

As showed above we meet old solutions (small k) again for larger k.

And we can repeat it.

This is some elementary number theory :

if k-1 factors into k-1 = a*b then the solutions of index a and b are solutions too.

and then a - 1 = c*d etc.

But beware of multiplicities.

Now that is not unique to tetration ofcourse.
Any interesting number theory on this is appreciated !! even experimental !

... it is interesting to understand how this relates to the rays or the rays conjecture ...
if k-1 is primes do the rays shift a bit compared to those for k + a new ray or ??
what if k-1 mod g = small ??

Lots of questions.

Not even mentioned chaos.

***

As sheldon mentioned and as to be expected for chaos and fast speeds ; usually we jump from large to small values and vice versa by iterating.

But some numbers apparantly do not.

such as the fixpoints of order k.

Or limit cycles.

But even limit cycles jump to high and low values.
but not all.

I considered low values and long psuedoperiods with respect to the imput ( size of the base in normed value )

and I came fascinated by the simple base

5 + 9i.

5 + 9i makes nice patterns imho... or absolutely not depending on your view of nice.

That is informal but still.

some gaussian integers make nicer iterations then others and I lack deep understanding of it.


I FELT LIKE SAYING THOSE THINGS BECAUSE THEY MIGHT NOT BE OBVIOUS FROM NUMERICAL METHODS AND PLOTS !!

regards

tommy1729