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janhardo

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How to plot this complex function?...

janhardo 695 Maple
complex 3d plotting
December 27 2021
2 12
  • How to get the ln(z) 3d plot example in the FunctionAdvisor /plot ?
  • a table plot, to show different plots together 

How to perform this complex substitution...

janhardo 695 Maple
complex-analysis arctan
December 23 2021
1 30
> 

 

Some more difficult calculation with complex numbers

Starting from the equation     
A = arctan(z/b)/b and arctan(z/b)/b = int(1/(b^2+z^2), z = 0 .. z)
made the substitution                               
z = i*b(t-1)/(t+1)
Thus obtaining NULL
A = -(int(1/t, t = 1 .. t))/(2*bi) and -(int(1/t, t = 1 .. t))/(2*bi) = log[10](t)/(2*bi) and log[10](t)/(2*bi) = log[10]((b*i+z)/(b*i-z))/(2*bi)
One of the important relations between logarithms and inverse trigonometrische functions

Download logaritmisc_functie_betrekking_goniometrische_functie.mw

Complex number commands in Maple ?...

janhardo 695 Maple
complex plotting duplicate-question
December 20 2021
3 20

I am studying something about complex numbers. 

What commands are specific to find in Maple for complex numbers ?

  • complexplot()
  • conformal()
  • conformal3d()

It seems that there are a lot of standard calculus statements can be used by adding the word : complex 
Iam using here a package downloaded from Maple website : complex analysis for mathematics and engineering

Got the impression that some modern plot commands for complex numbers are not yet in this book present ..and how about other commands?

an_introduction_to_complex_numbers.mws

Subprocedures in a main procedure ...

janhardo 695 Maple
array procedure programming
December 17 2021
2 15

How to add any/many subprocedure in a mainprocedure?
Somehow the mainprocedure must deliver the input for the mainprocedure

Here is a example 

Note: probably can this old procedure be rewritten in modern Maple programming language ?  

> 

restart;

 

Code to Find the Max and Min Values in a List A

 

> 

maxmin:=proc(A,maxv::evaln,minv::evaln)
> 

   local i;
> 

   maxv:=A[1]; minv:=A[1];
> 

   for i from 2 to nops(A) do
> 

     if eval(maxv) < A[i] then maxv:=A[i] end if;
> 

     if eval(minv) > A[i] then minv:=A[i] end if;
> 

   end do;
> 

   RETURN()
> 

end proc:
> 

 

Now we can call the maxmin procedure from a procedure named optimize, which is designed to create the polygonal approximation, use maxmin to find the largest and smallest y-values from among the vertices, and produce some graphic output.

See the documentation in the book.

 

Code to Approximate the Max and Min Values of a Function  f :

 

> 

optimize:=proc(f,a,b,N,pic::evaln)
> 

  local X,Y,L,i,A,xmax,xmin;
> 

  X:=array(0..N);Y:=array(0..N);L:=array(0..N);
> 

  with(plots,display);
> 

  for i from 0 to N do
> 

    X[i]:=evalf(a+i*(b-a)/N);

> 

    Y[i]:=f(X[i]);
> 

    L[i]:=plot([[X[i],0],[X[i],Y[i]]],color=black):
> 

  end do;
> 

  A:=[seq(Y[i-1],i=1..N+1)];
> 

  maxmin(A,maxv,minv); # USE OF SECOND PROCEDURE --------------
> 

  xmax:={};xmin:={};
> 

  for i from 0 to N do
> 

    if Y[i]=maxv then xmax:=xmax union {X[i]} end if;
> 

    if Y[i]=minv then xmin:=xmin union {X[i]} end if;
> 

  end do; #----------------------------------------------------
> 

  pic:=display({seq(L[i],i=1..N)}):
> 

  print(`maximum y value is`,maxv,`and occurs at these x values`,xmax);
> 

  print(`minimum y value is`,minv,`and occurs at these x values`,xmin);
> 

  end proc:
> 

 

Note that you must execute the code that defines maxmin before the procedure optimize will work. This only make sense.

 

We test the optimize procedure with the following function

 

> 

f:=x->3+10*(-x^2+x^4)*exp(-x^2);

proc (x) options operator, arrow; 3+10*(-x^2+x^4)*exp(-x^2) end proc

 (1)

on the interval [-1, 4].

 

> 

optimize(f,-1,4,150,pic);

`maximum y value is`, 6.088066652, `and occurs at these x values`, {1.633333333}

  

`minimum y value is`, 1.391538737, `and occurs at these x values`, {-.6333333333, .6333333333}

 (2)
> 

pic;

 

 

Download procedure_en_subprocedures.mw

Making a procedure for saddlepoints ...

janhardo 695 Maple
extrema plotting multivariate-calculus
December 12 2021
4 23
> 

restart;
> 

f:=(x,y)->x^4-3*x^2-2*y^3+3*y+0.5*x*y;

proc (x, y) options operator, arrow; x^4-3*x^2-2*y^3+3*y+.5*y*x end proc

 (1)
> 

 

Critical points: tangentline in x direction and y direction are  0  for some points on the surface  f(x,y)  

 

> 

CriticalPoints:= [ solve( {diff(f(x,y),x)=0,diff(f(x,y),y)=0}, {x,y}) ];

[{x = 0.5935572277e-1, y = .7105957432}, {x = 1.191117672, y = .7741187286}, {x = 1.255942703, y = -.7776000848}, {x = -0.5877159444e-1, y = -.7036351094}, {x = -1.197482099, y = -.6326213916}, {x = -1.250162405, y = .6291421140}]

 (2)
> 

         

Seems that maple calculate 6 points on the surface with a default domain?

let me make a  3D plot of f  ( not specifying the domain values)

> 

plot3d(f);

 
> 

 

I calculated 6 critical points :how to show them in the plot? : i calculate the functionvalues for f
So now i do have x,y,z values for a 3D point
plot3D ({ f, points}) ?

> 

?student
> 

 

Some information to find in

• 

 Student[MultivariateCalculus]  ..link?
• 

Multivariate Calculus Study Guide

In order to make a procedure ( to make it general) out of the wanted calculated steps , i must first calculate these steps
List of wanted steps:

1. 

calculate (6 )critical points : i did

2. 

investigate of the found critical points :are there saddle points among them?
Show this in a table

3. 

show a critical point with 2 section lines in this critical point intersected in a plot
4. 

show these critical points in a plot

===========================================

How to make table out of this list.CriticalPoints. for easy reading.?

==========================================

> 

CriticalPoints:= [ solve( {diff(f(x,y),x)=0,diff(f(x,y),y)=0}, {x,y}) ];

[{x = 0.5935572277e-1, y = .7105957432}, {x = 1.191117672, y = .7741187286}, {x = 1.255942703, y = -.7776000848}, {x = -0.5877159444e-1, y = -.7036351094}, {x = -1.197482099, y = -.6326213916}, {x = -1.250162405, y = .6291421140}]

 (3)
> 

 

Download zadelpunten_procedure.mw

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