How i can calculate invlaplace?

iman1.mw

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Hi,

How can I find the same result reported in the figure for dsolve the differential equation?

nima.mw

When I input
restart;
a := 2;
b := 3;
p := x^2 - a*x - b*x +a*b ;
I get p := x^2 - 5*x + 6
How can I get x^2 - 2 x - 3x + 6.
General question. I have (a,b) in a list [[2,3],[-3,7],[9,10]]. How to subtitute them to get the result
[[x^2 - 2x - 3x + 6, x^2 + 3x - 7x -21, x^2 - 9x - 10x + 90]? 

I am trying to learn Maple and Fourier transform by calculating the coefficients and the resuting summation as follows:

restart;

assume(n>0);

assume(N>0);

target_f := x -> piecewise(-Pi < x and x < 0, 0, 0 < x and x < Pi, x, Pi < x and x < 2*Pi, 0, 2*Pi < x and x < 3*Pi, x - 2*Pi)

a0 := simplify(int(target_f(x), x = -Pi .. Pi)/(2*Pi))

a_n := int(target_f(x)*cos(n . x), x = -Pi .. Pi)/Pi

b_n := int((target_f(x)) . (sin(n*x)), x = -Pi .. Pi)/Pi

fourier_f := N -> a0 + sum(a_n*cos(n*x) + b_n*sin(n . x), n = 1 .. N)

I am making fourier_f as a function of N becasue I want to see the effect of increasing partial sums. The resulting output of the summation "fourier_f" has a lot of terms. The output looks like this:

               /        /  N      
               |        |-----    
               |        | \       
1         1    |   2    |  )   /  
- Pi + ------- |4 N  Pi | /    |- 
4            2 |        |----- |  
       4 Pi N  \        \n = 1 \  

                                        \                     
                                        |                     
                                        |                     
  sin(n x) (cos(n Pi) n Pi - sin(n Pi))\|   /   2             
  -------------------------------------|| + \I N  Pi LerchPhi(
                   2                   ||                     
                  n  Pi                //                     

                         2                                    
  -exp(-I x), 1, N) - 2 N  LerchPhi(-exp(-I x), 2, N) - I N Pi

      \                  
   + 2/ exp(I N (Pi - x))

   - I N Pi (LerchPhi(-exp(I x), 1, N) N - 1) exp(I N (-Pi + x))

        2                                                 
   - I N  Pi LerchPhi(-exp(-I x), 1, N) exp(-I N (Pi + x))

                                 /   2                            
   + I Pi N exp(-I N (Pi + x)) + \I N  Pi LerchPhi(-exp(I x), 1, N

         2                                       \            
  ) - 2 N  LerchPhi(-exp(I x), 2, N) - I N Pi + 2/ exp(I N (Pi

                          2                          
   + x)) + 2 exp(-I x N) N  LerchPhi(exp(-I x), 2, N)

                   2                         
   + 2 exp(I x N) N  LerchPhi(exp(I x), 2, N)

        2                            2                     
   - 2 N  polylog(2, exp(-I x)) - 2 N  polylog(2, exp(I x))

        2                             2                      
   + 2 N  polylog(2, -exp(-I x)) + 2 N  polylog(2, -exp(I x))

                                 \
                                 |
                                 |
                                 |
   - 2 exp(-I x N) - 2 exp(I x N)|
                                 |
                                 /

The target function is not that complicated. I am using the target function shown in this YouTube video: https://www.youtube.com/watch?v=praNtRezlkw&list=PLPBSZvbAshbxULtiBcygm1qh7BtsDv3DW&index=12

It gave me similar output when the target function was x².  I am not sure what's going on because for a simple square wave, I was able to make this work correctly.

I am deliberately not using existing packages because I want to learn Maple syntax.

Can someone please tell me what's my mistake ?

Wolfram's marketing literature states that a compiled function may generate the dates for the years 1 through 5.7 million in a couple of seconds rather than in minutes (comparing to the "uncompiled implementation").
The given function in this link can be translated into Maple language as follows: 

(*
  Note that this is only a mathematical program that outputs some data,
   hence 'Easter(-2, 1)' will never return real Gregorian Easter dates!
*)
Easter:=proc(BEGIN::integer[4],END::integer[4],$)::Array(BEGIN..END,[integer[1..12],integer[1..31]]);# the parent function
	description "https://www.wolfram.com/language/12/code-compilation/compute-the-date-of-easter.html";
	local computus::procedure[[integer[1 .. 12], integer[1 .. 31]]](integer):=proc(Year::integer,` $`)::[integer[1..12],integer[1..31]];# the child function
		options threadsafe;
		local a::nonnegint,b::integer,c::nonnegint,d::integer,e::nonnegint,f::integer,g::nonnegint,h::nonnegint,i::nonnegint,j::nonnegint,k::nonnegint,Month::integer[1..12],Day::integer[1..31];
		(* For compatibility, when `Year` is nonpositive, the command `iquo` must be replaced with slower `floor`. *)
		if Year<=0 then
			a,b,c:=Year mod 19,floor(Year/100),Year mod 100;
			d,e,f:=floor(b/4),b mod 4,floor((8*b+13)/25);
			g,h,i:=19*a+b-d-f+15 mod 30,floor(c/4),c mod 4;
			j:=floor((a+11*g)/319);k:=2*e+2*h-i-g+j+32 mod 7;
			Month:=floor((g-j+k+90)/25);Day:=g-j+k+Month+19 mod 32
		else
			a,b,c:=irem(Year,19),iquo(Year,100),irem(Year,100):
			d,e,f:=iquo(b,4),irem(b,4),iquo(8*b+13,25);
			g,h,i:=irem(19*a+b-d-f+15,30),iquo(c,4),irem(c,4);
			j:=iquo(a+11*g,319);k:=irem(2*e+2*h-i-g+j+32,7);
			Month:=iquo(g-j+k+90,25);Day:=irem(g-j+k+Month+19,32)
		fi;
		[Month,Day]
	end;
	Array(BEGIN..END,computus)
end:

However, as "no nested procedures can be translated" to optimized native machine code (cf. ), executing Easter(1, 5700000) has to take at least two minutes: 

Is there some workaround that can provide improved performance for such a numerical procedure that contains a nested procedure? In other words, is it possible to produce the `result` (without modifying the algorithm) in two seconds in modern Maple as that Wolfram marketing literature claims?

I think this is a simple thing but I cannot find the solution. I have lots of complicated computations in Maple 2015 (with different functions of different packages), but I need the final result to be real numbers/functions. Is there a way to prevent maple following along with computations if it detects a potential complex number? For example, if in some internal step is taking sqrt(x) it should stop and tell something like: "you must assume x is positive".

Dear Maple experts

I am using the following two implicitplot commands successfully for variables diffr1 and diffr2:

implicitplot(diffr1, 0. .. 1, 0. .. 1.0, filledregions = true, coloring = [cyan, yellow]);

implicitplot(diffr2, 0. .. 1, 0. .. 1.0, filledregions = true, coloring = [cyan, yellow]);

Now I want to have a plot that mixes them. For example, if diffr1>0 and diffr2>0 I get a color. if diffr1>0 and diffr2<0 I get another color, etc. Is there any way to do that? 

Could someone provide me with the references used when implementing the Statistics:-PredictiveLeastSquares function?

TIA

The pendulum and the cantilever share simple-looking ordinary differential equations (ODEs), but they are challenging to solve:

This post derives solutions from Lawden and Bisshopp by Maple commands, which (to the best of my knowledge) have not been published providing not only results but also the accompanying computer algebra techniques. A tabulated format has been chosen to better highlight similarities and differences.

Both solutions have in common that in a first step, the unknown function is integrated and then in a second step the inverse of the unknown function (i.e., the independent variable) is integrated. Only in combination with a well-chosen set of initial/boundary conditions solutions are possible. This makes these two cases difficult to handle by generic integration methods.

Originally, I was not looking for this insight. I was more interested in an exact solution for nonlinear deformations to benchmark numerical simulation results.  Relatively quickly, I was able to achieve this with the help of this forum, but after that I was left with some nagging questions:

Why does Maple not provide a solution for the pendulum although one exists?

Why isn’t there an explicit solution for the cantilever when there is one for the pendulum?

Why is it so difficult to proof that elliptic expressions are equal?

Repeatedly, whenever there was time, I came back to these questions and got more and more a better understanding of the two problems and the overall context. It also required me to learn more of Maple, and I had to revisit fundamentals of functions, differential equations, and integration, which was entirely possible within Maples help system. Today, I am satisfied to the point that I think it is too much to expect Maple to provide a high-level general integration method for such problems.

I am also satisfied that I was able to combine all my findings scattered across many documents and Mapleprime questions into a single executable textbook-style document with hidden Maple code that:

Exclusively uses and manipulates equation expressions (no assignment operators := were required),

Avoids differentials that are often used in textbooks but (for good reasons) are not supported by Maple,

Exclusively applies high-level commands (i.e. no extraction of subexpression, manipulation
           and subsequent re-assembly of expressions was needed).

The solutions for the pendulum and the cantilever are substantially different although the ODEs and essential derivation steps are similar. I think that an explicit solution for the cantilever, as it exists for the pendulum, is impossible (using elliptic integrals and functions). I leave it open to comments: whether this is correct and whether it is attributable to the set of initial and boundary conditions, the different symmetry of the sine and cosine functions in the ODEs, or both. I hope that the tabular presentation will provide an easy overview, allowing to form an own opinion.

If you are patient enough to work through the table, you will find a link between the cantilever and the pendulum that you are probably not expecting. 

Finally, I have to give credit to Bisshopp, who was probably the first to provide a solution for the cantilever. The clarity and compactness on only 3 pages and the way how the inverse of functions was determined before the age of computers makes this paper worth studying. Also, Lawden has to be mentioned, who did the same on 3 pages for the pendulum in a marvelous book on elliptic functions and applications. It happens that he is overlooked in more recent publications and it’s unclear to me if he was the first who published an explicit solution. His book might be one of the last of its kind in this age of computers, and for that reason alone, it is worth enjoying as he enjoyed writing it.

Download Cantilever_and_pendulum_side_by_side_-_input_hidden.mw

Hi,
Apparently I have a problem but I can't find it. Please advise what is the source of the error?
Please see the attached worksheet.
1.mw

I have a great deal of output from a procedure that I wrote clumsily.  How can I delete that output without highlighting and pressing the delete button.  Maple help is often of no help whatsoever.

What is the simplest way to direct all Maple output, and only Maple output, into a PDF?  

My preference would be to include a command at the beginning of a worksheet so that causes all output returned by Maple, print and graphics, to be directed to a named PDF.  Does such a command or set of commands exist, and if so, what is the process to get it to work?

An alternative is to have a command at the end of a worksheet that causes Maple to print the worksheet to a PDF.  Does that exist, and if so, what is it?

For example: n=78 (is squarefree because 2 x 3 x 13)

Possible periodlenghts in different bases for 1/78  are  {2,3,4,6,12} this is calculated with Multiplicative-Order (Maple).

Now we pick a period from above, for example 6.

The question is:

Which possible bases exist for 1/78 with  periodlenght of 6 ?

The result is: possible bases calculated with Multiplicative-Order
are {17,23,29,35,43,49}.

Question:
Is there a formular to calculate the bases directly or with less calculations than the Maple-Command ?

Thanks for helping :)

hi i am trying to create a procedure for newton rhapson method, which finds roots of a function. the first procedure i made work but now i want it to show an error message for when the function diverges at x0 and when the derivative at x0 = 0. can someone pls advise me how to fix it? thank you:)

NM1 := proc(f, xold)
    local x1, x0, precision;
    x0 := xold;
    x1 := evalf(x0 - f(x0)/D(f)(x0));
    precision := 10^(-3);
        if limit(f,(x0)) = 0 then
            error("Newton's Method cannot do the calculation");
        end if;
        if D(f)(x0) = 0 then
            error("Newton's Method cannot do the calculation");
        else
        while abs(x1-x0) > precision do
            x0 := x1;
            print(x0);        
            x1 := evalf(x0 - f(x0)/D(f)(x0));            
        end do;
        end if;

   return x1;
end proc:

Dear users! 

I hope everyone is fine here. I have the following expression:

r*y[0, 1]+y[0, 0]+(1/6)*r*(r-1)*(1+r)*y[-1, 3]+(1/2)*r*(r-1)*y[-1, 2]+(1/120)*r*(r-1)*(r-2)*(1+r)*(2+r)*y[-2, 5]+(1/24)*r*(r-1)*(r-2)*(1+r)*y[-2, 4]+(1/720)*r*(r-1)*(r-2)*(r-3)*(r-4)*(1+r)*y[-3, 8]+(1/360)*r*(r-1)^2*(r-2)*(r-3)*(1+r)*y[-3, 7]+(1/720)*r*(r-1)*(r-2)*(r-3)*(1+r)*(2+r)*y[-3, 6]:

What is the procedure to select some terms in the above expression for example for N=2 I just want the following terms:

y[0, 0]+r*y[0, 1]+(1/2)*r*(r-1)*y[-1, 2];

and for N=3 I just want the following terms:

y[0, 0]+r*y[0, 1]+(1/2)*r*(r-1)*y[-1, 2]+(1/6)*r*(r-1)*(1+r)*y[-1, 3];

and for N=4 I want:

(1/24)*r*(r-1)*(r-2)*(1+r)*y[-2, 4]+y[0, 0]+r*y[0, 1]+(1/2)*r*(r-1)*y[-1, 2]+(1/6)*r*(r-1)*(1+r)*y[-1, 3];

and so on,

in the descending order in first suffices and ascending order in second suffices (like term having y[0,0],  y[0,1], y[-1,2], y[-1,3], y[-2,4]). I am waiting for your response. Thanks.