## weak derivation ...

Hello

Can Maple compute the weak derivative of piecewise function like

f:=x->piecewise(0<x and x<0.5,x,0.5<x and x<1,1-x,  elsewhere 0);

Many thanks

## how to convert a equation into functions...

how to convert a^2*b+c to func2(func1(func1(abc[1],abc[1]),abc[2]),abc[3])

when i use custom function func2 to represent plus, func1 to represent multiply

input 3 parameters,

one is a^2*b + c one is [func1, func2] and second is [abc[1],abc[2],abc[3]] corresponding to a, b, c
a^2*b + c = func2(func1(func1(abc[1],abc[1]),abc[2]),abc[3]);

## Digitizing mathematics: ODEs, Special Functions...

by: Maple

The material below was presented in the "Semantic Representation of Mathematical Knowledge Workshop", February 3-5, 2016 at the Fields Institute, University of Toronto. It shows the approach I used for “digitizing mathematical knowledge" regarding Differential Equations, Special Functions and Solutions to Einstein's equations. While for these areas using databases of information helps (for example textbooks frequently contain these sort of databases), these are areas that, at the same time, are very suitable for using algorithmic mathematical approaches, that result in much richer mathematics than what can be hard-coded into a database. The material also focuses on an interesting cherry-picked collection of Maple functionality, that I think is beautiful, not well know, and seldom focused inter-related as here.

 Digitizing of special functions, differential equations, and solutions to Einstein’s equations within a computer algebra system   Edgardo S. Cheb-Terrab Physics, Differential Equations and Mathematical Functions, Maplesoft Editor, Computer Physics Communications

 • Big amounts of knowledge available to everybody in local machines or through the internet
 • Take advantage of basic computer functionality, like searching and editing

 • By digitizing mathematical knowledge inside appropriate computational contexts that understand about the topics, one can use the digitized knowledge to automatically generate more and higher level knowledge
 Challenges 1) how to identify, test and organize the key blocks of information,   2) how to access it: the interface,   3) how to mathematically process it to automatically obtain more information on demand
 Three examples

Mathematical Functions

"Mathematical functions, are defined by algebraic expressions. So consider algebraic expressions in general ..."

"Supporting information on definitions, identities, possible simplifications, integral forms, different types of series expansions, and mathematical properties in general"

 Examples
 General description
 References

Differential equation representation for generic nonlinear algebraic expressions - their use

"Compute differential polynomial forms for arbitrary systems of non-polynomial equations ..."

 The Differential Equations representing arbitrary algebraic expresssions
 Deriving knowledge: ODE solving methods
 Extending the mathematical language to include the inverse functions
 Solving non-polynomial algebraic equations by solving polynomial differential equations
 References

Branch Cuts of algebraic expressions

"Algebraically compute, and visualize, the branch cuts of arbitrary mathematical expressions"

 Examples
 References

Algebraic expresssions in terms of specified functions

"A conversion network for arbitrary mathematical expressions, to rewrite them in terms of different functions in flexible ways"

 Examples
 General description
 References

Symbolic differentiation of algebraic expressions

"Perform symbolic differentiation by combining different algebraic techniques, including functions of symbolic sequences and Faà di Bruno's formula"

 Examples
 References

Ordinary Differential Equations

"Beyond the concept of a database, classify an arbitrary ODE and suggest solution methods for it"

 General description
 Examples
 References

Exact Solutions to Einstein's equations

"The authors of "Exact solutions toEinstein's equations" reviewed more than 4,000 papers containing solutions to Einstein’s equations in the general relativity literature, organized the whole material into chapters according to the physical properties of these solutions. These solutions are key in the area of general relativity, are now all digitized and become alive in a worksheet"

The ability to search the database according to the physical properties of the solutions, their classification, or just by parts of keywords (old paradigm) changes the game.

More important, within a computer algebra system this knowledge becomes alive (new paradigm).

 • The solutions are turned active by a simple call to one commend, called the g_  spacetime metric.
 • Everything else gets automatically derived and set on the fly ( Christoffel symbols  , Ricci  and Riemann  tensors orthonormal and null tetrads , etc.)
 • Almost all of the mathematical operations one can perform on these solutions are implemented as commands in the Physics  and DifferentialGeometry  packages.
 • All the mathematics within the Maple library are instantly ready to work with these solutions and derived mathematical objects.

Finally, in the Maple PDEtools package , we have all the mathematical tools to tackle the equivalence problem around these solutions.

 Examples
 References

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

## Expansion of expression in terms of 'small' functi...

This may be a silly question, but does there exist some simple way of (Taylor) expanding an expression of 'small' functions in terms of these functions.

A simple example: Assume that diff(f(x),x) and g(x) are two functions both with range, say, in [-a,+a], where a << 1, and consider the following expression:

`sqrt(1 + diff(f(x),x)) * (2 + g(x));`

Its expansion to first order in terms of diff(f(x),x) and g(x) should be 2 + diff(f(x),x) + g(x). My problem is that mtaylor does not accept functions as variables to expand on, and I would prefer not to have to substitute back and forth with some 'placeholders'.

## how do I plot a solution with series representatio...

Hi! I'm trying to find the way to plot the solution with series representation. I need some help to find the easiest way.

Note: I realized some typing errors, which do not change the question a lot ,and I corrected them.

plot.mw

## Set Theory, how to create 'set functions' in Maple...

Typically sets are created like:

A:={a,b,c};

B:={c,q,w,e};

and then you can carryout A union B or B\A

what if  you wanted to create the set as

A:={values in some three dimensional space};

B:={volume, based on values taken from A};

Can these relationships be set up in Maple? If so, how? If there are commands that specifically handle these types of sets, what does maple call them?  I've seen the term 'set function' but what might Maple call them?

Note: I am not even sure i 'tagged' this correctly because it I am not sure the proper terms for these functions/sets.

Thanks in advance for any help.

## Argument, absolute results display...

I am working in Document mode, 2D input.  I have doing some complex number evalutions

Lets say I have a complex number Z= 2+3i

when I ask maple for the "abs(Z)" I get the display of "abs(2+3i)" rather than the numberic answer. This occurs with all the complex operation arg, abs, polar conversion,

polar(abs(2+3*i), argument(2+3*i))   this is from Maple.

How do I display the numeric value for these functions in my document

Thanks Bill

## Composition of functions...

I have entered three functions into Maple and I would like to create a set of all possible two-function and three-function compositions involving the three functions. For instance, for my three functions f(x), g(x) and h(x), the set would contain f(g(x)), g(f(x)), f(g(h(x))), f(f(f(x))), etc.

I'm also looking for a method that will be generalisable to larger numbers of initial functions than just three.

## Creating a sequence of functions ...

i'm using maple in a research but i want to add a recursive function h_m(t) in 2 case : if m is integer positive and not,
la formule est donnée comme suit :
if (mod(m,1) = 0  and m>0) then  h:=proc(m,t)  local  t ;  h[0,t]:=t ;   for  i from -4 to  m  by  2 do  h [m,t]:= h[0, t]-(GAMMA(i/(2)))/(2*GAMMA((i+1)/(2)))*cos(Pi*t)*sin(Pi*t)  od:  fi:  end;
and i wanna to know how to programmate a Gaus Hypegeometric function. Thank You

## Conditions on general functions ...

I am fairly new to Maple and cannot figure this out.

I'm having some trouble in computing formally the derivatives of
some composite functions

h:= x -> f(g(x)),

imposing conditions on the unknown functions f and g.

For example, I cannot find a way to impose g'(0)=0,
such that the computation of

h'(0)

simply gives 0.

## `assuming` not working for functions...

This is a stripped down example of something I've been doing. Basically I'm building matrices which I then, using unapply, convert into functions of some variables of t.
.... but found that simplify seems to often not work as i'd wish.

restart:
mm:=Matrix([[cos(sqrt(g__1^2)*t), (-I*g__1*sin(sqrt(g__1^2)*t))*(1/sqrt(g__1^2))], [(-I*g__1*sin(sqrt(g__1^2)*t))*(1/sqrt(g__1^2)) ,cos(sqrt(g__1^2)*t)]]);

#great - simplifies as i'd expect:
simplify(mm) assuming g__1::positive;

Do the same thing but when matrix is a function of t
mmFun:=unapply(mm, t);

#the function works - gives what i'd expect
mmFun(3); mmFun(t);

#but now the simplification does not work - why the g__1 in the argument of cos does not get properly simplified?
simplify(mmFun(t)) assuming g__1::positive;

Any ideas if this is a bug? I'm using maple 2015.2 on linux 64-bit.

here is the worksheet: simplify_issue.mw

thanks

EDIT:

as a side note once can sometimes overcome this with mapping simplify  as in :

map(simplify, resultMatrix ) assuming g__1::positive;

but this is not optimal, and sometimes does not work when i first multiply the matrix by say a vector.

## How to form equation of specific kind?...

Hello,

I would like to create a function from an equation.
I have define 4 functions u[1](t), ..., u[4](t).
From these 4 functions, I would like to define 4 equations. I would like to obtain this result:
Equ[1]:=u[1](t)=L/2*cos(w*t+phi[1])
Equ[2]:=u[2](t)=L/2*cos(w*t+phi[2])
Equ[3]:=u[3](t)=L/2*cos(w*t+phi[3])
Equ[4]:=u[4](t)=L/2*cos(w*t+phi[4])

My code is the following:

for i to 4
do
u[i]:=unapply(L/2*cos(w*t+phi[i]),t);
Equ[i]:='u[i](t)'=u[i](t);
end do;

These lines are not working because the left member is not incremented (u[i](t) stays at each iteration u[i](t))

Generally, how can I transform a function to an equation ?
And, in this specific case, how can I obtain the four equations mentioned above ?

Thanks a lot for your help.

by: Maple

The year 2015 has been one with interesting and relevant developments in the MathematicalFunctions  and FunctionAdvisor projects.

 • Gaps were filled regarding mathematical formulas, with more identities for all of BesselI, BesselK, BesselY, ChebyshevT, ChebyshevU, Chi, Ci, FresnelC, FresnelS, GAMMA(z), HankelH1, HankelH2, InverseJacobiAM, the twelve InverseJacobiPQ for P, Q in [C,D,N,S], KelvinBei, KelvinBer, KelvinKei, KelvinKer, LerchPhi, arcsin, arcsinh, arctan, ln;
 • Developments happened in the Mathematical function package, to both compute with symbolic sequences and symbolic nth order derivatives of algebraic expressions and functions;
 • The input  now returns both the first derivative (old behavior) and the nth symbolic derivative (new behavior) of a mathematical function;
 • A new topic, plot, used as , now returns 2D and 3D plots for each mathematical function, following the NIST Digital Library of Mathematical Functions;
 • The previously existing  got redesigned, so that it now displays more information about any mathematical function, and organized into a Section with subsections for each of the different topics, making it simpler to find the information one needs without getting distracted by a myriad of formulas that are not related to what one is looking for.

More mathematics

More mathematical knowledge is in place, more identities, differentiation rules of special functions with respect to their parameters, differentiation of functions whose arguments involve symbolic sequences with an indeterminate number of operands, and sum representations for special functions under different conditions on the functions' parameters.

 Examples

More powerful symbolic differentiation (nth order derivative)

Significative developments happened in the computation of the nth order derivative of mathematical functions and algebraic expressions involving them.

 Examples

Mathematical handling of symbolic sequences

Symbolic sequences enter various formulations in mathematics. Their computerized mathematical handling, however, was never implemented - only a representation for them existed in the Maple system. In connection with this, a new subpackage, Sequences , within the MathematicalFunctions package, has been developed.

 Examples

Visualization of mathematical functions

When working with mathematical functions, it is frequently desired to have a rapid glimpse of the shape of the function for some sampled values of their parameters. Following the NIST Digital Library of Mathematical Functions, a new option, plot, has now been implemented.

 Examples

Section and subsections displaying properties of mathematical functions

Until recently, the display of a whole set of mathematical information regarding a function was somehow cumbersome, appearing all together on the screen. That display was and is still available via entering, for instance for the sin function,  . That returns a table of information that can be used programmatically.

With time however, the FunctionAdvisor evolved into a consultation tool, where a better organization of the information being displayed is required, making it simpler to find the information we need without being distracted by a screen full of complicated formulas.

To address this requirement, the FunctionAdvisor now returns the information organized into a Section with subsections, built using the DocumentTools package. This enhances the presentation significantly.

 Examples

These developments can be installed in Maple 2015 as usual, by downloading the updates (bundled with the Physics and Differential Equations updates) from the Maplesoft R&D webpage for Mathematical Functions and Differential Equations

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

## How to diff(g(y, t), y(t))?...

I have functions like

g := (y, t)->sin(y(t))

diff(g(y, t), t) # ok

How to determine the derivative below with maple :

diff(g(y, t), y(t))

Thanks

## How to plot two functions of two variables?...

How plot two functions of two variables to (2-dimension)?

restart;
approximate := exp(-2*t)+x*exp(-2*t)+(1/2)*exp(-2*t)*x^2+(1/6)*exp(-2*t)*x^3+(1/24)*exp(-2*t)*x^4+(1/120)*exp(-2*t)*x^5+(1/720)*exp(-2*t)*x^6+(1/5040)*exp(-2*t)*x^7+(1/40320)*exp(-2*t)*x^8+(1/362880)*exp(-2*t)*x^9+(1/3628800)*exp(-2*t)*x^10+(1/39916800)*exp(-2*t)*x^11+(1/479001600)*exp(-2*t)*x^12+(1/6227020800)*exp(-2*t)*x^13+(1/87178291200)*exp(-2*t)*x^14+(1/1307674368000)*exp(-2*t)*x^15+(1/20922789888000)*exp(-2*t)*x^16+(1/355687428096000)*exp(-2*t)*x^17+(1/6402373705728000)*exp(-2*t)*x^18+(1/121645100408832000)*exp(-2*t)*x^19+(1/2432902008176640000)*exp(-2*t)*x^20+(1/51090942171709440000)*exp(-2*t)*x^21;
apr := unapply(approximate, x, t);
Exact := (x, t)-> exp(x-2*t) ;
plot3d(exp(x-2*t)-approximate, x = 0 .. 5, t = 0 .. 5);

plot.mw

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