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In fsolve: Would it make sense to implic...

C_R 3042 Maple
assumptions fsolve range
March 19 2023
3 9

In the following example, the information in which range fsolve should search for a solution and the range in which a function is defined is somehow redundant. (This example has been adopted from here where fsolve without assumptions does not throw an error but evaluates forever.)

Why can’t fsolve not always assume that the range equals the domain of interest? This would make life easier and provide more solutions to inexperienced users who have not yet learned the need for assumptions.

If there are good reasons (and there probably are) that such an implicit assumption (i.e. range equals domain) would be too restrictive: can’t fsolve give a hint or provide an new option to use ranges as assumptions?

Example with incomplete elliptic integral of the first kind

f := proc (x__0) options operator, arrow; int(1/(sqrt(x__0-x)*sqrt(-x^2+1)), x = 0 .. x__0) end proc

proc (x__0) options operator, arrow; int(1/(sqrt(x__0-x)*sqrt(-x^2+1)), x = 0 .. x__0) end proc

 (1)

Range := 0 .. 1; plot(f(x__0), x__0 = Range, labels = [x__0, 'f(x__0)'])

  

f(.5) = 1.524886838NULL

Defining the inverse of f

g := proc (y) options operator, arrow; fsolve(`assuming`([f(x__0) = y, x__0 = Range], [lhs(Range) <= x__0 and x__0 <= rhs(Range)])) end proc

proc (y) options operator, arrow; fsolve(`assuming`([f(x__0) = y, x__0 = Range], [lhs(Range) <= x__0 and x__0 <= rhs(Range)])) end proc

 (2)

x__0 = g(f(.5)) → x__0 = .5000000000NULL

Now without assumptions

h := proc (y) options operator, arrow; fsolve(f(x__0) = y, x__0 = Range) end proc

proc (y) options operator, arrow; fsolve(f(x__0) = y, x__0 = Range) end proc

 (3)

h(f(.5))

Error, (in fsolve) cannot determine if this expression is true or false: abs(Re(x))+abs(Im(x)) <= 0.

  

``

Download ranges_as_assumptions.mw

Why can't the commands in these document...

C_R 3042 Maple 2023
document-block
March 13 2023
0 0

 

ODE for electrical circuit (right click on Documentblock, unselect show command does not work. Command still visble)u(t) = T*(diff(`&varphi;`(t), t))+L*(diff(i(t), t))+i(t)*R

u(t) = T*(diff(varphi(t), t))+L*(diff(i(t), t))+i(t)*R

 (1)

 

ODE for motor (toggle Documentblock, unselect show command is only effective on equations  3 and 4)i(t)*T = J*(diff(`&varphi;`(t), t, t))

i(t)*T = J*(diff(diff(varphi(t), t), t))

 (2)

Isolate i(t) and taking the derivative

i(t) = J*(diff(diff(varphi(t), t), t))/T

 (3)

``

diff(i(t), t) = J*(diff(diff(diff(varphi(t), t), t), t))/T

 (4)

``

Download Document_Block_hide_command.mw

Some unit questions about angles...

C_R 3042 Maple
units radians
March 11 2023
0 3

Q1: In the above, why can I only convert radians to radians. Or: Why does the menu "Chose unit" not offer arcdeg?

Q2: In the above, why is nothing happening when I enter arcdeg in the field "Enter Unit"?

Q3: How to change the displayed symbol for arcdeg to ° (in the attachment are failed attempts)?

arcdeg.mw

How to shift the index in a sum...

C_R 3042 Maple
sum index
March 06 2023
3 5

For example from

sum(a[k]*(k+r)*(k+r-1)*x^(k+r-1), k = 0 .. infinity)

sum(a[k]*(k+r)*(k+r-1)*x^(k+r-1), k = 0 .. infinity)

 (1)

by substituting k by k+1 to

sum(a[k+1]*(k+1+r)*(k+r)*x^(k+r), k = -1 .. infinity)

sum(a[k+1]*(k+1+r)*(k+r)*x^(k+r), k = -1 .. infinity)

 (2)

Doing

subs(k = k+1, sum(a[k]*(k+r)*(k+r-1)*x^(k+r-1), k = 0 .. infinity))

sum(a[k+1]*(k+1+r)*(k+r)*x^(k+r), k+1 = 0 .. infinity)

 (3)

leads to output that can't be processed further  

subsop(2 = 'k = -1 .. infinity', sum(a[k+1]*(k+1+r)*(k+r)*x^(k+r), k+1 = 0 .. infinity))

Error, (in sum) second argument must be a name, name=a..b, name=RootOf, or name=algebraic

  

NULL

Doing it programmatically this way

shift := k = k+1; rsd := (lhs-rhs)(shift); subs(shift, [op(sum(a[k]*(k+r)*(k+r-1)*x^(k+r-1), k = 0 .. infinity))]); lhs(%[2])+rsd = map(`+`, rhs(%[2]), rsd); sum(subsop(2 = %, `%%`)[])

sum(a[k+1]*(k+1+r)*(k+r)*x^(k+r), k = -1 .. infinity)

 (4)

is not realy elegant and easy to understand.

Download shifting_index_in_sums.mw

Are there better ways?

Why no warning is returned with 2D input...

C_R 3042 Maple 2023
integration int warning
February 19 2023
0 5

(Deleted because not reproducible on a different PC)

With 1D

> 

int(1/(sqrt(x__0 - x)*sqrt(-x^2 + 1)), x = 0 .. x__0)

Warning, unable to determine if -1 is between 0 and x__0; try to use assumptions or use the AllSolutions option

  

Warning, unable to determine if 1 is between 0 and x__0; try to use assumptions or use the AllSolutions option

  

int(1/((x__0-x)^(1/2)*(-x^2+1)^(1/2)), x = 0 .. x__0)

 (1)
> 

NULL

With 2D

int(1/(sqrt(x__0-x)*sqrt(-x^2+1)), x = 0 .. x__0)

int(1/((x__0-x)^(1/2)*(-x^2+1)^(1/2)), x = 0 .. x__0)

 (1)

NULL

The root cause might be the same as for this open question.

int_warning_2D.mw

int_warning_1D.mw

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