# Items tagged with mathmath Tagged Items Feed

### MathematicalFunctions and FunctionAdvisor, one yea...

December 30 2015 Maple
6
7

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

### Approaching Putnam 2015 problems with Maple...

December 20 2015 Maple
2
12

The well known William Lowell Putnam Mathematical Competition (76th edition)  took place this month.
Here is a Maple approach for two of the problems.

1. For each real number x, 0 <= x < 1, let f(x) be the sum of  1/2^n  where n runs through all positive integers for which floor(n*x) is even.
Find the infimum of  f.
(Putnam 2015, A4 problem)

f:=proc(x,N:=100)
local n, s:=0;
for n to N do
if type(floor(n*x),even) then s:=s+2^(-n) fi;
#if floor(n*x) mod 2 = 0  then s:=s+2^(-n) fi;
od;
evalf(s);
#s
end;

plot(f, 0..0.9999);

min([seq(f(t), t=0.. 0.998,0.0001)]);

0.5714285714

identify(%);

So, the infimum is 4/7.
Of course, this is not a rigorous solution, even if the result is correct. But it is a valuable hint.
I am not sure if in the near future, a CAS will be able to provide acceptable solutions for such problems.

2. If the function f  is three times differentiable and  has at least five distinct real zeros,
then f + 6f' + 12f'' + 8f''' has at least two distinct real zeros.
(Putnam 2015, B1 problem)

restart;
F := f + 6*D(f) + 12*(D@@2)(f) + 8*(D@@3)(f);

dsolve(F(x)=u(x),f(x));

We are sugested to consider

g:=f(x)*exp(x/2):
g3:=diff(g, x$3); simplify(g3*8*exp(-x/2)); So, F(x) = k(x) * g3 = k(x) * g''' g has 5 distinct zeros implies g''' and hence F have 5-3=2 distinct zeros, q.e.d. ### Alternative to certain Mma commands... December 14 2015 2 3 Hi! In Mathematica 10.0 were introduced regions with functions like TransformedRegion, ReginIntersection, etc. Moreover, it is easy to check if a point is inside a region, etc. I would like to ask if in Maple I could use some API with similar functionality?. For instance, I would like to get integer points which lay inside an intersection of two cubes. How I could do this in Maple? ### PDEs and Boundary Conditions - new developments... September 03 2015 Maple 7 1 The PDE & BC project , a very nice and challenging one, also one where Maple is pioneer in all computer algebra systems, has restarted, including now also the collaboration of Katherina von Bülow. Recapping, the PDE & BC project started 5 years ago implementing some of the basic methods found in textbooks to match arbitrary functions and constants to given PDE boundary conditions of different kinds. At this point we aim to fill gaps, and the first one we tackled is the case of 1st order PDE that can be solved without boundary conditions in terms of an arbitrary function, and where a single boundary condition (BC) is given for the PDE unknown function, and this BC does not depend on the independent variables of the problem. It looks simple ... It can be rather tricky though. The method we implemented is a simple however ingenious use of differential invariants to match the boundary condition. The resulting new code, the portion already tested, is available for download in the Maplesoft R&D webpage for Differential Equations and Mathematical Functions (the development itself is bundled within the library that contains the new developments for the Physics package, in turn within the zip linked in the webpage). The examples that can now be handled, although restricted in generality to "only one 1st order linear or nonlinear PDE and only one boundary condition for the unknown function itself", illustrate well how powerful it can be to use more advanced methods to tackle these tricky situations where we need to match an arbitrary function to a boundary condition. To illustrate the idea, consider first a linear example, among the simplest one could imagine:  >  (1)  >  (2) Input now a boundary condition (bc) for the unknown such that this bc does not depend on the independent variables ; this bc can however depend on arbitrary symbolic parameters, for instance  >  (3) With the recent development, this kind of problem can now be solved in one go:  >  (4) Nice! And how do you verify this result for correctness? With pdetest , which actually also tests the solution against the boundary conditions:  >  (5) And what has been done to obtain the solution (4)? First the PDE was solved regardless of the boundary condition, so in general, obtaining:  >  (6) In a second step, the arbitrary function got determined such that the boundary condition is matched. Concretely, the mapping _F1 is what got determined. You can see this mapping reversing the solving process in two steps. Start taking the difference between the general solution (6) and the solution (4) that matches the boundary condition  >  (7) and isolate here  >  (8) So this is the value that got determined. To see now the actual solving mapping _F1, that takes for arguments and and returns the right-hand side of (8), one can perform a change of variables introducing the two parameters and of the _F1 mapping:  >  (9)  >  (10)  >  (11) So the solving mapping _F1 is  >  (12) Wow! Although this pde & bc problem really look very simple, this solution (12) is highly non-obvious, as is the way to get it just from the boundary condition and the solution (6) too. Let's first verify that this mapping is correct (even when we know, by construction, that it is correct). For that, apply (12) to the arguments of the arbitrary function and we should obtain (8)  >  (13) Indeed this is equal to (8)  >  (14) Skipping the technical details, the key observation to compute a solving mapping is that, given a 1st order PDE where the unknown depends on independent variables, if the boundary condition depends on arbitrary symbolic parameters , one can always seek a "relationship between these parameters and the differential invariants that enter as arguments in the arbitrary function _F1 of the solution", and get the form of the mapping _F1 from this relationship and the bc. The method works in general. Change for instance the bc (3) making its right-hand side be a sum instead of a product  >  (15)  >  (16)  >  (17) An interesting case happens when the boundary condition depends on less than parameters, for instance:  >  (18)  >  (19) As we see in this result, the additional difficulty represented by having few parameters got tackled by introducing an arbitrary constant _C1 (this is likely to evolve into something more general...)  >  (20) Finally, consider a nonlinear example  >  (21)  >  (22) Here we have 2 independent variables, so for illustration purposes use a boundary condition that depends on only one arbitrary parameter  >  (23) All looks OK, but we still have another problem: check the arbitrary function _F1 entering the general solution of pde when tackled without any boundary condition:  >  (24) Remove this RootOf to see the underlying algebraic expression  >  (25) So this is a pde where the general solution is implicit, actually depending on an arbitrary function of the unknown The code handles this problem in the same way, just that in cases like this there may be more than one solution. For this very particular bc (23) there are actually three solutions:  >  (26) Verify these three solutions against the pde and the boundary condition  >  (27) :) Download PDEs_and_Boundary_Conditions.mw Edgardo S. Cheb-Terrab Physics, Differential Equations and Mathematical Functions, Maplesoft ### Nth order derivatives and Faa di Bruno formula... August 28 2015 4 0 In connection with recent developments for symbolic sequences, a number of improvements were implemented regarding symbolic differentiation, that is the computation of order derivatives were n is a symbol, the simplest example being the derivative of the exponential, which of course is the exponential itself. This post is about these developments, done in collaboration with Katherina von Bülow, and available for download as usual from the Maplesoft R&D web page for Differential Equations and Mathematical functions (the update itself is bundled with the official updates of the Maple Physics package). It is important to note that Maple is pioneer in having an actual implementation of symbolic differentiation, something that works for real, since several releases. The development, however, was somewhat stuck because we were unable to compute the symbolic derivative of a composite function . A formula for this problem is actually known, it is the Faà di Bruno formula, but, in order to implement it, first we were missing the incomplete Bell functions , that got implemented in Maple 15, nice, but then we were still missing differentiating symbolic sequences, and functions whose arguments are symbolic sequences (i.e. the number of arguments of the function is n, a symbol, of unknown value at the time of differentiating). All this got implemented now within the new MathematicalFunctions:-Sequence package, opening the door widely to these improvements in differentiation. The symbolic differentiation code works as mostly all other computer algebra code, by mapping complicated problems into a composition of simpler problems all of which are tractable; what follows is then an illustration of these basic cases. Among the simplest new case that can now be handled there is that of a power where the exponent is linear in the differentiation variable. This is actually an easy problem  >  (1) More complicated, consider the power of a generic function; the corresponding symbolic derivative can be mapped into a sum of symbolic derivatives of powers of with lower degree  >  (2) In some cases where is a known function, the computation can be carried on furthermore. For example, for the result can be expressed using Stirling numbers of the first kind  >  (3) The case of sin and cos are relatively simpler, but then assumptions on the exponent are required in order to proceed further ahead from (2), for example  >  (4) The case of functions of arbitrary number of variables (typical situation where symbolic sequences are required) is now handled properly. This is the pFq hypergeometric function of symbolic order p and q  >  (5) The case of the MeijerG function is more complicated, but in practice, for the computer, once it knows how to handle symbolic sequences, the more involved problem becomes computable  >  (6) Not only the mathematics of this result is correct: the object returned is actually computable to the end (if you provide the values of n, p, m and q), and the typesetting is actually fully readable, as in textbooks, including copy and paste working properly; all this is new. The derivative of a number of mathematical functions that were not implemented before, are now also implemented, covering the gaps, for example:  >  (7)  >  (8)  >  (9)  >  (10) In the same way the fundamental formulas for the derivative of all the 12 elliptic Jacobi functions as well as the four elliptic JacobiTheta functions, the LambertW , LegendreP and some others are now all implemented. Finally there is the "holy grail" of this problem: the derivative of a composite function - this always-unreachable implementation of Faa di Bruno formula. We now have it :)  >  (11) Note the symbolic sequence of symbolic order derivatives of lower degree, both of of f and g, also within the arguments of the IncompleteBellB function. This is a very abstract formula ... And does this really work? Of course it does :). Consider, for instance, a case where the derivatives of and can both be computed by the system:  >  (12) This is the derivative expressed using Faa di Bruno's formula, in turn expressed using symbolic sequences within the IncompleteBellB function  >  (13) These results can all be verified. Take for instance  >  (14) Compute now the inert functions: on the left-hand side this is just the (now explicit) 3rd order derivative, while on the right-hand side we have a sum of IncompleteBellB functions, where the number of arguments, expressed in (13) using symbolic sequences that depend on the summation index k and the differentiation order n, now in (14) depend only on , and get transformed into explicit sequences of arguments when the summation is performed and k assumes integer values  >  (15) Take left-hand side minus right-hand side  >  (16)  > :) Download SymbolicOrderDifferentiation.mw Edgardo S. Cheb-Terrab Physics, Differential Equations and Mathematical Functions, Maplesoft ### How to combine Greek letters and math in labels?... August 13 2015 2 1 Hi, I am trying to use implicit plot. The plot is OK but I want to put labels such as what Latex produces :$\frac{\Omega}{\omega_n}$for x axis and$a_0 \mathrm{(m)}$for y axis How can I apply this in my maple code as below: plot1:=implicitplot(a3, Omega_r=1.5..2.5, a=0.00000001..0.1, labeldirections=[horizontal, horizontal], axes=boxed, labels=["W/w_n",typeset("a_0 (m)")], labelfont=[SYMBOL]): What I have put as bold does not work for me, it is making everything in Greek :) . I want combination of Greek and math. Thasnks, Bahareh ### MathematicalFunctions:-Sequences... July 24 2015 6 2 Symbolic sequences enter in various formulations in mathematics. This post is about a related new subpackage, Sequences, within the MathematicalFunctions package, available for download in Maplesoft's R&D page for Mathematical Functions and Differential Equations (currently bundled with updates to the Physics package). Perhaps the most typical cases of symbolic sequences are: 1) A sequence of numbers - say from n to m - frequently displayed as 2) A sequence of one object, say a, repeated say p times, frequently displayed as 3) A more general sequence, as in 1), but of different objects and not necessarily numbers, frequently displayed as or likewise a sequence of functions In all these cases, of course, none of n, m, or p are known: they are just symbols, or algebraic expressions, representing integer values. These most typical cases of symbolic sequences have been implemented in Maple since day 1 using the $ operator. Cases 1), 2) and 3) above are respectively entered as , , and  or  To have computer algebra representations for all these symbolic sequences is something wonderful, I would say unique in Maple.

Until recently, however, the typesetting of these symbolic sequences was frankly poor, input like  or  just being echoed in the display. More relevant: too little could be done with these objects; the rest of Maple didn't know how to add, multiply, differentiate or map an operation over the elements of the sequence, nor for instance count the sequence's number of elements.

All this has now been implemented.  What follows is a brief illustration.

 >

First of all, now these three types of sequences have textbook-like typesetting:

 >
 (1)
 >
 (2)

For the above, a$p works the same way  >  (3) Moreover, this now permits textbook display of mathematical functions that depend on sequences of paramateters, for example:  >  (4)  >  (5) More interestingly, these new developments now permit differentiating these functions even when their arguments are symbolic sequences, and displaying the result as in textbooks, with copy and paste working properly, for instance  >  (6) It is very interesting how much this enhances the representation capabilities; to mention but one, this makes 100% possible the implementation of the Faa-di-Bruno formula for the nth symbolic derivative of composite functions (more on this in a post to follow this one). But the bread-and-butter first: the new package for handling sequences is  >  (7) The five commands that got loaded do what their name tells. Consider for instance the first kind of sequences mentione above, i.e  >  (8) Check what is behind this nice typesetting  >  $(n .. m)

All OK. How many operands (an abstract version of Maple's nops  command):

 >
 (9)

That was easy, ok. Add the sequence

 >
 (10)

Multiply the sequence

 >
 (11)

Map an operation over the elements of the sequence

 >
 (12)
 >
 $(f(j), j = n .. m) Map works as map, i.e. you can map extra arguments as well  >  (13) All this works the same way with symbolic sequences of forms , and . For example:  >  (14)  >  $(a, p)
 >
 (15)
 >
 (16)
 >
 (17)

Differentation also works

 >
 (18)
 >
 (19)
 >
 (20)

For a symbolic sequence of type 3)

 >
 (21)
 >
 (22)
 >
 (23)
 >
 (24)

The following is nontrivial: differentiating the sequence , with respect to  should return 1 when  (i.e the running index has the value k), and 0 otherwise, and the same regarding m and k. That is how it works now:

 >
 (25)
 >
 $(piecewise(k = i, 1, 0), i = n .. m)  >  (26)  >  (27)  >  $`((diff(f(a[i]), a[i]))*piecewise(k = i, 1, 0), i = n .. m)

And that is it. Summarizing: in addition to the former implementation of symbolic sequences, we now have textbook-like typesetting for them, and more important: Add, Multiply, Differentiate, Map and Nops. :)

The first large application we have been working on taking advantage of this is symbolic differentiation, with very nice results; I will see to summarize them in a post to follow in a couple of days.

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

### Plot - which curve is which!?...

May 15 2015
0 12

Hi is there a way to identify curves in your plot? Especially when you dont know it yourself?

For example, I have 10 polynomials all order 5 and up, whilst I can trial and error identify which one is which it is very inefficient.

I used to use Mathematica, and I could manually label the curves with color which allows to see which curve is which. I assume there is something of the sort for Maple?

### How can I fix my math expressions in T.A.?...

May 07 2015
1 6

I am writing here because of a problem with writing mathematical expressions in Maple T.A.. I have been using it since two years, and I have a lot if questions created in version 9.0 and 9.5. A few months ago my university installed T.A. 10. At the beginning there were problems with the connections between T.A. and the Maple server. After the administrators got over these, there were another problems.

There are a lot of questions where I use greek letters, for example Sigma. Earlier it was easy, I wrote

in the 'Algorithm' section, and I could write

in the Text of the question. The letter had a perfect italic and bold style, like as it would be created with Equation Editor.

Now I have to write

sig2=maple("MathML:-ExportPresentation(\$sig1)");

### How to solve the following equation in Maple?...

March 07 2015
1 2

I have the following mathematical expression

where a=0.2, b=0.09, c=0.57, p=0.3 and q=1-p=0.7, Now i want to find the value of n for which the value of the expression will be 1, i.e., for what value of n , A_n will be 1?

### Maximizing Planck's Intensity Function...

February 08 2015
1 1

I am attempting to maximize Planck's intesnity function for any given temperature T with respect to lambda. The function is as follows:

My initial attempt was to simply take the derivative of this function with respect to labda, set it equal to zero, and then solve for lamda. Theoretically this should give me the value of lambda that maximizes the intesity with respect to the constants / values h, c, k, and T (where T is temperature).

However doing this in maple just leads to the result "warning, solutions may have been lost".

Does anyone know how I can go about maximizing this?

Thanks!

### how to seq and sum over a range of subscript...

January 27 2015
1 11

we always have subscript variable in the math book, but how could this be natral done in maple

I want to get a seq aaa3

seq(a[i],i=1..3)

but how could I get a  aij

seq(a[i_j],i=1..3);

and

seq(a[ij],i=1..3);  both was not right

### How do you insert a left bracket containing multip...

December 21 2014
2 1

Is there a way with Maple 18 to place a bracket as shown below to contain two DE's? I am not trying to solve them. Only for note-taking purposes. If there is can you please share how to do it?

### Automatic simplification and a new Assume (as in "...

December 09 2014
6
6

Hi
Two new things recently added to the latest version of Physics available on Maplesoft's R&D Physics webpage are worth mentioning outside the framework of Physics.

• automaticsimplification. This means that after "Physics:-Setup(automaticsimplification=true)", the output corresponding to every single input (literally) gets automatically simplified in size before being returned to the screen. This is fantastically convenient for interactive work in most situations.

• Add Physics:-Library:-Assume, to perform the same operations one typically performs with the  assume command, but without the side effect that the variables get redefined. So the variables do not get redefined, they only receive assumptions.

This new Assume implements the concept of an "extended assuming". It permits re-using expressions involving the variables being assumed, expressions that were entered before the assumptions were placed, as well as reusing all the expressions computed while the variables had assumptions, even after removing the variable's assumptions. None of this is possible when placing assumptions using the standard assume. The new routine also permits placing assumptions on global variables that have special meaning, that cannot be redefined, e.g. the cartesian, cylindrical or spherical coordinates sets, or the coordinates of a coordinate spacetime system within the Physics package, etc.

Examples:

 >

This is Physics from today:

 >
 (1.1)
 • Automatic simplification is here. At this point automaticsimplification is OFF by default.
 >
 (1.2)

Hence, for instance, if you input the following expression, the computer just echoes your input:

 >
 (1.3)

There is however some structure behind (1.3) and, in most situations, it is convenient to have these structures
apparent, in part because they frequently provide hints on how to proceed ahead, but also because a more
compact expression is, roughly speaking, simpler to understand. To see this
automaticsimplification in action,
turn it ON:

 >
 (1.4)

Recall this same expression (you could input it with the equation label (1.3) as well)

 >
 (1.5)

What happened: this output, as everything else after you set  and with no
exceptions, is now further processed with simplify/size before being returned. And enjoy computing with frankly
shorter expressions all around! And no need anymore for "simplify(%, size)" every three or four input lines.

Another  example, typical in computer algebra where expressions become uncomfortably large and difficult to
read: convert the following input to 2D math input mode first, in order to compare what is being entered with the
automatically simplified output on the screen

 >
 (1.6)

You can turn automaticsimplification OFF the same way

 >
 (1.7)
 • New  facility; welcome to the world of "extended assuming" :)

Consider a generic variable, x. Nothing is known about it

 >

Each variable has associated a number that depends on the session, and the computer (internally) uses this
number to refer to the variable.

 >
 (1.8)

When using the assume  command to place assumptions on a variable, this number, associated to it, changes,
for example:

 >
 >
 (1.9)

Indeed, the variable x got redefined and renamed, it is not anymore the variable x referenced in (1.8).

 >
 Originally x, renamed x~:   is assumed to be: RealRange(Open(0),Open(1/2*Pi))

The semantics may seem confusing but that is what happened, you enter x and the computer thinks x~, not x
anymore.This means two things:

1) all the equations/expressions, entered before placing the assumptions on x using assume, involve a variable x
that is different than the one that exists after placing the assumptions, and so these previous expressions
cannot
be reused
. They involve a different variable.

2) Also, because, after placing the assumptions using assume, x refers to a different object, programs that depend
on the
x that existed before placing the assumptions will not recognize the new x redefined by assume .

For example, if x was part of a coordinate system and the spacetime metric depends on it, the new variable x
redefined within assume, being a different symbol, will not be recognized as part of the dependency of  This
posed constant obstacles to working with curved spacetimes that depend on parameters or on coordinates that
have a restricted range. These problems are resolved entirely with this new
Library:-Assume, because it does not
redefine the variables. It only places assumptions on them, and in this sense it works like
assuming , not assume .
As another example, all the
Physics:-Vectors commands look for the cartesian, cylindrical or spherical coordinates
sets
in order to determine how to proceed, but these variables disappear if you use
assume to place assumptions on them. For that reason, only assuming  was fully compatible with Physics, not assume.

To undo assumptions placed using the assume command one reassigns the variable x to itself:

 >
 (1.10)

Check the numerical address: it is again equal to (1.8)

 >
 (1.11)

·All these issues get resolved with the new Library:-Assume, that uses all the implementation of the existing
assume command but with a different approach: the variables being assumed do not get redefined, and hence:
a) you can reuse expressions/equations entered before placing the assumptions, you can also undo the
assumptions and reuse results obtained with assumptions. This is the concept of an
extended assuming. Also,
commands that depend on these assumed variables will all continue to work normally, before, during or after
placing the assumption, because
the variables do not get redefined.

Example:

 >

So this simplification attempt accomplishes nothing

 >
 (1.12)

Let's assume now that

 >
 (1.13)

The new command echoes the internal format representing the assumption placed.

a) The address is still the same as (1.8)

 >
 (1.14)

So the variable did not get redefined. The system however knows about the assumption - all the machinery of the
assume command is being used

 >
 Originally x, renamed x:   is assumed to be: RealRange(Open(0),Open(1/2*Pi))

Note that the renaming is to the variable itself - i.e. no renaming.

Hence, expressions entered before placing assumptions can be reused. For example, for (1.12), we now have

 >
 (1.15)

To clear the assumptions on x, you can use either of Library:-Assume(x=x) or Library:-Assume(clear = {x, ...}) in
the case of many variables being cleared in one go, or in the case of a single variable being cleared:

 >
 >

The implementation includes the additionally functionality, for that purpose add the keyword
anywhere in the calling sequence. For example:

 >
 (1.16)
 >
 Originally x, renamed x:   is assumed to be: RealRange(Open(0),infinity)
 >
 (1.17)
 >

In summary, the new Library:-Assume command implements the concept of an extended assuming, that can be
turned ON and OFF at will at any moment without changing the variables involved.

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

### substitution a sequnce of numbers in a function...

November 18 2014
0 6

Hi all

Assume that we have a function, say f(t) and we want to substitute t in it where t is:

t=[0,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1]

by subs or other better command, how can we do it?

best wishes