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Why does Maple not simplify to infinity?...

C_R 3412 Maple
integration simplification
May 29 2022
2 2

Evaluating this integral

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

Int(1/((1-x)^(1/2)*(-x^2+1)^(1/2)), x = 0 .. 1) = infinity*(Pi+1)

 (1)

simplify(Int(1/((1-x)^(1/2)*(-x^2+1)^(1/2)), x = 0 .. 1) = infinity*(Pi+1))

Int(1/((1-x)^(1/2)*(-x^2+1)^(1/2)), x = 0 .. 1) = infinity

 (2)

NULL

produces an infinite product as output. Why does Maple not automatically simplify to infinity. Can the extra information (1+pi) be of any use?


Download Infinity_times_something.mw

 

Why is simplify ineffective with these c...

C_R 3412 Maple
simplify assumptions arctan
May 15 2022
1 2

The expression

f := arctan(y, x)+arctan(-y, x)

arctan(y, x)+arctan(-y, x)

 (1)

simplifies to zero in the real range if y=0 is excluded.

x < 0, y::real, y <> 0; `assuming`([simplify(f), is(f = 0)], [%])

0, true

 (2)

x >= 0, y::real, y <> 0; `assuming`([simplify(f), is(f = 0)], [%])

0, true

 (3)

Combining the above assumptions as attempted bellow does not simplify to zero

x::real, y::real, y <> 0; `assuming`([simplify(f), is(f = 0)], [%])

arctan(y, x)+arctan(-y, x), FAIL

 (4)

`or`(x < 0, x >= 0), y::real, y <> 0; `assuming`([simplify(f), is(f = 0)], [%])

arctan(y, x)+arctan(-y, x), FAIL

 (5)

Or(x < 0, x >= 0), y::real, y <> 0; `assuming`([simplify(f), is(f = 0)], [%])

arctan(y, x)+arctan(-y, x), FAIL

 (6)

`and`(-infinity <= x, x <= infinity), y::real, y <> 0; `assuming`([simplify(f), is(f = 0)], [%])

arctan(y, x)+arctan(-y, x), FAIL

 (7)

interface(version)

`Standard Worksheet Interface, Maple 2022.0, Windows 10, March 8 2022 Build ID 1599809`

 (8)

 

  

Download assuming_and_differently_combined_inequalities.mw

is(-arctan(-x, y) = arctan(x, y))?...

C_R 3412 Maple
arctan
May 07 2022
1 3

In the positive range Maple confirms that this is true.
In the real range Maple fails to provide an answer (see attachments).

Is this identity correct?

restart

kernelopts(version)

`Maple 2022.0, X86 64 WINDOWS, Mar 8 2022, Build ID 1599809`

 (1)

NULL

is(arctan(-x) = -arctan(x))

true

 (2)

`assuming`([is(arctan(x, y) = -arctan(-x, y))], [x::real, y::real])

FAIL

 (3)

`assuming`([is(arctan(x, y) = -arctan(-x, y))], [x::positive, y::positive])

true

 (4)

`assuming`([simplify(arctan(-x, y)+arctan(x, y), trig)], [x::real, y::real])

arctan(-x, y)+arctan(x, y)

 (5)

But

plot3d([arctan(-x, y)+arctan(x, y)], x = -1000000 .. 1000000, y = -1000000 .. 1000000, title = arctan(-x, y)+arctan(x, y))

  

On a unit circle

x = cos(alpha), y = sin(alpha)

x = cos(alpha), y = sin(alpha)

 (6)

subs(x = cos(alpha), y = sin(alpha), arctan(-x, y)+arctan(x, y))

arctan(-cos(alpha), sin(alpha))+arctan(cos(alpha), sin(alpha))

 (7)

`assuming`([simplify(%)], [alpha::real])

arctan(-cos(alpha), sin(alpha))+arctan(cos(alpha), sin(alpha))

 (8)

`assuming`([simplify(%)], [alpha::positive])

arctan(-cos(alpha), sin(alpha))+arctan(cos(alpha), sin(alpha))

 (9)

`assuming`([simplify(%)], [-Pi < alpha and alpha < Pi])

arctan(-cos(alpha), sin(alpha))+arctan(cos(alpha), sin(alpha))

 (10)

plot(arctan(-cos(alpha), sin(alpha))+arctan(cos(alpha), sin(alpha)), alpha = -2*Pi .. 2*Pi, axes = boxed, color = red)

  

NULL


Download arctan_xy_simplify.mw

and another maybe related case where simplification does not work

arctan_xy_simplify_2.mw

Solve & Trig: Why is the output differen...

C_R 3412 Maple
solve arctan inverse-kinematics
April 25 2022
2 6

Solve produces different output in the attachment depending on how it is used. Why is that and how can simplification to arctan(y/z) be avoided? Arctan(y/z) only gives correct angles for positive y and z.  I prefer arctan(y,z) output that I can subsequently simplify to the y and z ranges of interest (if possible). Imagine “wrong” simplification of complex algebraic output (e.g., from inverse kinematics).

Arctan.mw

How to get `&D;` into the favorites pale...

C_R 3412 Maple
April 15 2022
0 0

I got a solution to use D as a symbol that prints well in italic. For convienience I like to have it in my favorite palette.
When I drag `&D;` from a Maple Input line to the favorites palette, the ampersand and the statement operator are removed. Copy and paste have the same effect on the pasted selection. If  these characters are removed, `&D;` becomes the differential operator D. That's not what I want.
Is it possible at all to get `&D;` or simliar expressions using special characters within left single quotes into the favorite palette?

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