Advanced Search

Axel Vogt

5906 Reputation

20 Badges

19 years, 329 days
Munich, Bavaria, Germany
Contact Axel Vogt

MaplePrimes Activity


These are answers submitted by Axel Vogt

"normal"...

Axel Vogt 5906
August 29 2024
1 0

How about using normal(e) ? Or simplify(e) ?

suggestion...

Axel Vogt 5906
August 25 2024
1 1

There is a page for downloading as well. For that I use a chromium based browser (instead of Firefox), https://en.wikipedia.org/wiki/Ungoogled-chromium, there exists a portable version for it.

Please ......

Axel Vogt 5906
August 18 2024
0 0

Please write corrections into your original thread instead of opening a new one.

Otherwise it will become quite confusing.

manually...

Axel Vogt 5906
August 17 2024
0 0

Using the valid (!) transformation arcsinh(2*x) = y, i.e. x = sinh(y)/2, 0 <= y gives it after 'simplify'.

no solution (2)...

Axel Vogt 5906
August 08 2024
1 0

If you insert x[2] from g into f then you find x[1] = - 0.668... now feeding g you find x[1] = - 1.552 ...

ctrl + D...

Axel Vogt 5906
June 20 2024
2 1

For me the keyboard combination <ctrl> + <D> works.

= sqrt( cos^2 )...

Axel Vogt 5906
June 09 2024
1 2

I think it can be written as a* sqrt( cos(x+c__1)^2 ) which I consider to be more simple (even if LeafCount is the same as for e2)

MP_238462_some_simplification.mw

trick...

Axel Vogt 5906
May 28 2024
0 8

You may try the following approach:

fsolve usually delivers real solutions only. If you want positive ones you can change your variable, say x, to X^2, fsolve for X and square it.

Find an example bellow. However I will not do it for your problem.

> 

# https://www.mapleprimes.com/questions/238400-Helping-Fsolve
> 

eq:=(x-1)^2-16;   # plot(%);
{fsolve(eq)};     # set of solutions = {-3, +5}

eq := (x-1)^2-16

{-3.00000000000000, 5.00000000000000}

 (1)
> 

EQ:=eval(eq, x=X^2);
# {fsolve(EQ, complex)};     # set of complex solutions for EQ
{fsolve(EQ)};                # set of (real) solutions for EQ
map(q -> q^2, %);            # now back to eq

EQ := (X^2-1)^2-16

{-2.23606797749979, 2.23606797749979}

{5.00000000000000}

 (2)
> 

 


 

Download MP_238400_positive_solutions.mws

try this...

Axel Vogt 5906
May 23 2024
3 0

You can use

(*

  (some code to be de-activated)

*)

CR...

Axel Vogt 5906
May 01 2024
2 2

For example

> 

 
> 

restart; interface(version);

`Classic Worksheet Interface, Maple 2017.3, Windows, Sep 27 2017, Build ID 1265877`

 (1)
> 

 

> 

f:= z -> 1/(z + 2);
f(x+I*y);
u:=unapply(evalc(Re(%)), x,y);
v:=unapply(evalc(Im(%%)), x,y);

f := proc (z) options operator, arrow; 1/(z+2) end proc

1/(x+y*I+2)

u := proc (x, y) options operator, arrow; (x+2)/((x+2)^2+y^2) end proc

v := proc (x, y) options operator, arrow; -y/((x+2)^2+y^2) end proc

 (2)
> 

 

> 

['D[1](u)(x,y) = D[2](v)(x,y)', 'D[2](u)(x,y) = -D[1](v)(x,y)'];
simplify(%): evalc(%);
map(is, %);

[(D[1](u))(x, y) = (D[2](v))(x, y), (D[2](u))(x, y) = -(D[1](v))(x, y)]

[(-x^2+y^2-4*x-4)/(x^2+y^2+4*x+4)^2 = (-x^2+y^2-4*x-4)/(x^2+y^2+4*x+4)^2, (-2*x-4)*y/(x^2+y^2+4*x+4)^2 = (-2*x-4)*y/(x^2+y^2+4*x+4)^2]

[true, true]

 (3)
> 

'[diff(u(x,y),x) = diff(v(x,y),y), diff(u(x,y),y) = -diff(v(x,y),x)]';
evalc(%): simplify(%);
map(is, %);

[diff(u(x, y), x) = diff(v(x, y), y), diff(u(x, y), y) = -(diff(v(x, y), x))]

[(-x^2+y^2-4*x-4)/(x^2+y^2+4*x+4)^2 = (-x^2+y^2-4*x-4)/(x^2+y^2+4*x+4)^2, -2*(x+2)*y/(x^2+y^2+4*x+4)^2 = -2*(x+2)*y/(x^2+y^2+4*x+4)^2]

[true, true]

 (4)
   
 

 


 

Download CR.mws

- 2/7...

Axel Vogt 5906
March 25 2024
1 4

It should simplify to - 2/7
 

> 

# https://www.mapleprimes.com/questions/236590-Error-in-Trignormalsincosargs-Too
restart; kernelopts(version);

`Maple 2024.0, X86 64 WINDOWS, Feb 05 2024, Build ID 1785419`

 (1)
> 

expr := -1/7 - (-1/7*7^(5/7)*exp(2/7*Pi*I)*sin(1/7*Pi)*I - 1/7*cos(1/7*Pi)*7^(5/7)*exp(2/7*Pi*I))^(7/2):
> 

# evalf[20](expr): fnormal(%): identify(%); # = -2/7
> 

# evalc(expr): simplify(%);                 # = -2/7
> 

simplify(expr):
evalc(%): simplify(%);                      # = -2/7

-2/7

 (2)
> 

# convert(expr, trig): simplify(%);         # = -2/7


 

Download MP_236590.mw

Real and Imaginary...

Axel Vogt 5906
March 16 2024
1 0

It often helps to care for spurious numerical imaginary results:

[Re(res), Im(res)]:
plot(%,t=-5..1, color=[red,blue]);

formally...

Axel Vogt 5906
February 19 2024
2 3

For example like this (upload does not work ...):

NumericEventHandler(invalid_operation = `Heaviside/EventHandler`(value_at_zero = 0)):
assume(u::real, v::real);

[Int(Dirac(u-v), [u, v]) , min(u,v)]; value(%);
convert(%, piecewise, u);
%[1] - %[2]; # = 0

[Int(Dirac(u+v-1), [u, v]), max(u+v-1, 0)]; value(%);
convert(%, piecewise, u);
%[1] - %[2]; # = 0

 

 

hm...

Axel Vogt 5906
February 18 2024
1 5

I think that limit(expr, +oo) = 24 is false

Proof...

Axel Vogt 5906
January 29 2024
1 0


 

> 

# https://www.mapleprimes.com/questions/237741-Integral-Of-Dirac
restart; kernelopts(version);

`Maple 2023.2, X86 64 WINDOWS, Nov 24 2023, Build ID 1762575`

 (1)
> 

g:= x -> x^2+y^2-1;
#g:= x -> x^2-a^2; 'y^2-1 = eval(-a^2, a=sqrt(1-y^2))'; is(%);

proc (x) options operator, arrow; x^2+y^2-1 end proc

 (2)
> 

Int(Dirac('g'(x)), x=-infinity ... infinity): '%'= value(%);

Int(Dirac(g(x)), x = -infinity .. infinity) = 1/abs(y^2-1)^(1/2)

 (3)

 

This is correct using a quite common definition for composing Dirac and appropriate functions, see Maple's help

or https://en.wikipedia.org/wiki/Dirac_delta_function#Composition_with_a_function referencing to I M Gelfand

 

> 

0='g'(x);
Zeros:={solve(%,x)};
x1,x2:=op(Zeros);

0 = g(x)

  

{(-y^2+1)^(1/2), -(-y^2+1)^(1/2)}

  

(-y^2+1)^(1/2), -(-y^2+1)^(1/2)

 (4)

 

The zeros of g are simple iff y <> -1 . Therefore the composition can be defined as

 

> 

delta('g'(x)) = Sum('delta(x-xi)/abs(D(g)(xi))', xi in 'Zeros');
subs(Sum=add, %): %;
Dirac_of_g(x):=eval(rhs(%), delta=Dirac);

delta(g(x)) = Sum(delta(x-xi)/abs((D(g))(xi)), `in`(xi, Zeros))

  

delta(x^2+y^2-1) = (1/2)*delta(x-(-y^2+1)^(1/2))/abs(y^2-1)^(1/2)+(1/2)*delta(x+(-y^2+1)^(1/2))/abs(y^2-1)^(1/2)

  

(1/2)*Dirac(x-(-y^2+1)^(1/2))/abs(y^2-1)^(1/2)+(1/2)*Dirac(x+(-y^2+1)^(1/2))/abs(y^2-1)^(1/2)

 (5)

 

Then the integral works out as asserted (since Int(Dirac(x+constant), x = -infinity .. infinity) = 1 ):

 

> 

Int(Dirac('g'(x)), x=-infinity ... infinity);
``=Int(Dirac_of_g(x), x=-infinity .. infinity);
value(%);

Int(Dirac(g(x)), x = -infinity .. infinity)

  

`` = Int((1/2)*Dirac(x-(-y^2+1)^(1/2))/abs(y^2-1)^(1/2)+(1/2)*Dirac(x+(-y^2+1)^(1/2))/abs(y^2-1)^(1/2), x = -infinity .. infinity)

  

`` = 1/abs(y^2-1)^(1/2)

 (6)
 

 


 

Download MP_237741-Integral-Of-Dirac.mw

1 2 3 4 5 6 7 Last Page 1 of 93