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int...

vv 12453
November 26 2020
3 0

Why dou you think that the integral can be expressed in terms of known functions?
Note that even int(sqrt(cos(t)^2+cos(2*t)^2),t)  has a complicated form and int(sqrt(cos(t)^2+cos(3*t)^2),t)  is already too much! 

Answer...

vv 12453
November 26 2020
0 0

1. The parantheses are not ignored; they are not displayed, being superfluous in this case, similar to a + (b*c), due to the order of precedence of the Maple operators.

2. Unfortunately Maple does not simplify the set operators; they are used mainly for explicit sets.
If you need to check or simplify expressions in set theory, you can use the Logic package because it works practically in a Boolean algebra.

Answer...

vv 12453
November 25 2020
1 7

If you want the general form of a vector in the kernel, try

B := <<6,3,0>|<4,2,0>|<10,5,0>>:
K := LinearAlgebra:-NullSpace(B):
add(alpha[i]*K[i],i=1..nops(K));

(alpha[i] being arbitrary scalars.)

Shooting...

vv 12453
November 24 2020
1 1

> 

xodephi := {diff(x(t), t) = 16250.25391*(1 - (487*x(t))/168 + 4*Pi*x(t)^(3/2) + (274229*x(t)^2)/72576 - (254*Pi*x(t)^(5/2))/21 + (119.6109573 - (856*ln(16*x(t)))/105)*x(t)^3 + (30295*Pi*x(t)^(7/2))/1728 + (7.617741607 - 23.53000000*ln(x(t)))*x(t)^4 + (535.2001594 - 102.446*ln(x(t)))*x(t)^(9/2) + (413.8828821 + 323.5521650*ln(x(t)))*x(t)^5 + (1533.899179 - 390.2690000*ln(x(t)))*x(t)^(11/2) + (2082.250556 + 423.6762500*ln(x(t)) + 33.2307*ln(x(t)^2))*x(t)^6)*x(t)^5, diff(xphi(t), t) = 5078.204347*x(t)^(3/2), x(0) = 0.03369973351, xphi(0) = a}:  #xphi(10.92469316) = 0}:
> 

sol := dsolve(xodephi, parameters=[a], numeric):
> 

ff:=proc(A)
  sol(parameters=[A]);
  eval(xphi(t), sol(10.92469316))
end:
> 

A:=fsolve(ff);
sol(parameters=[A]);
sol(10.92469316);

-467.1843838

  

[a = -467.1843838]

  

[t = 10.92469316, x(t) = HFloat(0.061040830524973895), xphi(t) = HFloat(1.6909911249030074e-9)]

 (1)
> 

plots:-odeplot(sol, [t,xphi(t)], t=0..11);

 
> 

plots:-odeplot(sol, [t,x(t)], t=0..11);

 
> 

 

 

Generalizations...

vv 12453
November 24 2020
1 0

The term "generalization"  is not always the right one. This happens in maths too. E.g. the Lebesgue integral is a "generalization" of the Riemann integral. However, int(sin(1/x)/x, x=0..1) does not exist  in the Lebesgue sense, but it exists as a Riemann (improper)  integral, and Maple computes it.

Inscribed ellipses of a quadrilateral ...

vv 12453
November 22 2020
3 4

> 

restart;

f maps the unit square onto the quadrilateral; ek = ellipses tangent to the unit square.

> 

Ellipses:=proc(P::listlist(realcons), t::name, k::name)
local a1,b1,c1,a2,b2,c2,b0,c0,ek,
      f:=(x,y)->[a1*x+b1*y+c1,a2*x+b2*y+c2]/~(x+b0*y+c0);
ek:=1/2+k/2*cos(t)-sqrt(1-k^2)/2*sin(t), 1/2+sqrt(1-k^2)/2*sin(t)+k/2*cos(t);
solve(op~([f(0,0)=~P[1], f(1,0)=~P[2], f(1,1)=~P[3], f(0,1)=~P[4]]));
eval([f(ek)[], t=0..2*Pi], %)
end proc:
> 

# Example
> 

P:=[[0,0], [5,0], [4,3], [1,2], [0,0]]: # quadrilateral
> 

Ell:=Ellipses(P, t, k);

[(-4-4*k*cos(t)+(-k^2+1)^(1/2)*sin(t))/(-2-(-k^2+1)^(1/2)*sin(t)), (-3-3*(-k^2+1)^(1/2)*sin(t)-3*k*cos(t))/(-2-(-k^2+1)^(1/2)*sin(t)), t = 0 .. 2*Pi]

 (1)
> 

plots:-display(plot(P,color=blue, thickness=4),
seq(plot(Ell, color=COLOR(HUE,k)), k=0..1,0.05), scaling=constrained, size=[800,800]);

 
> 

plots:-animate(plot, [Ell], k=0..1, background=plot(P,color=blue, thickness=4), scaling=constrained);

 
> 

 

 

Bug in series for JacobiTheta3&4...

vv 12453
November 18 2020
2 3

 

Yes, in the implementation of series for JacobiTheta3 and JacobiTheta4, the term 1 is missing.

 

> 

restart;
> 

FunctionAdvisor(definition, JacobiTheta3);

[JacobiTheta3(z, q) = Sum(2*q^(_k1^2)*cos(2*_k1*z), _k1 = 1 .. infinity)+1, abs(q) < 1]

 (1)
> 

FunctionAdvisor(series, JacobiTheta3);

series(JacobiTheta3(z, q), z, 4) = series(2*(Sum(q^(_k1^2), _k1 = 1 .. infinity))-(4*(Sum(q^(_k1^2)*_k1^2, _k1 = 1 .. infinity)))*z^2+O(z^4),z,4), series(JacobiTheta3(z, q), q, 4) = series(1+(2*cos(2*z))*q+O(q^4),q,4)

 (2)
> 

 

 

Type less...

vv 12453
November 18 2020
1 0

I always prefer to change myself the coordinates. This way I avoid the two distinct conventions used by Maple for spherical.
As a bonus, I type less!

int(eval((x^2 + y^2)*z, [x=sin(u)*cos(v), y=sin(u)*sin(v), z=cos(u)]) * sin(u), u = 0 .. Pi/2, v = 0 .. 2*Pi);

        Pi/2

 

select...

vv 12453
November 17 2020
2 1 
if type(lhs(ode),`*`) and rhs(ode)=0 then 
  ODE:=select(has, lhs(ode), diff) = 0
fi;

If lhs has a factor depending on y, but has not diff, you may keep it, or solve it for y.

examples...

vv 12453
November 16 2020
2 3
> 

restart;
> 

x[n](t) = x0 + int( f(tau, x[n-1](tau)), tau=t0..t);

x[n](t) = x0+int(f(tau, x[n-1](tau)), tau = t0 .. t)

 (1)
> 

# example 1
t0:=0; x0:=1;
f:= (t, x) -> t*x;
x[0]:= t -> x0;

0

  

1

  

proc (t, x) options operator, arrow; x*t end proc

  

proc (t) options operator, arrow; x0 end proc

 (2)
> 

for n to 5 do
x[n] := unapply( x0 + int( f(tau, x[n-1](tau)), tau=t0..t),  t);
print(x[n]= x[n](t));
od:

x[1] = 1+(1/2)*t^2

  

x[2] = 1+(1/8)*t^4+(1/2)*t^2

  

x[3] = 1+(1/48)*t^6+(1/8)*t^4+(1/2)*t^2

  

x[4] = 1+(1/384)*t^8+(1/48)*t^6+(1/8)*t^4+(1/2)*t^2

  

x[5] = 1+(1/3840)*t^10+(1/384)*t^8+(1/48)*t^6+(1/8)*t^4+(1/2)*t^2

 (3)
> 

# exaplle 2 (x[n], f are vector valued)
> 

t0:=0; x0:=<1,2>;
f:= (t, v) -> <t*v[1], v[2]> ;
x[0]:= t -> x0;

t0 := 0

  

Vector[column](%id = 18446747125427096142)

  

proc (t, v) options operator, arrow; `<,>`(t*v[1], v[2]) end proc

  

proc (t) options operator, arrow; x0 end proc

 (4)
> 

for n to 5 do
x[n] := unapply( x0 + int~( f(tau, x[n-1](tau)), tau=t0..t),  t);
print(x[n]= x[n](t));
od:

x[1] = (Vector(2, {(1) = 1+(1/2)*t^2, (2) = 2+2*t}))

  

x[2] = (Vector(2, {(1) = 1+(1/8)*t^4+(1/2)*t^2, (2) = t^2+2*t+2}))

  

x[3] = (Vector(2, {(1) = 1+(1/48)*t^6+(1/8)*t^4+(1/2)*t^2, (2) = 2+(1/3)*t^3+t^2+2*t}))

  

x[4] = (Vector(2, {(1) = 1+(1/384)*t^8+(1/48)*t^6+(1/8)*t^4+(1/2)*t^2, (2) = 2+2*t+(1/12)*t^4+(1/3)*t^3+t^2}))

  

x[5] = Vector[column](%id = 18446747125427074462)

 (5)
> 

 

NULL
 

Download Picard_iteration-vv.mw

 

Equal without assumptions...

vv 12453
November 08 2020
1 1

> 

restart;
> 

aa := {{y = -2*X1*X2*alpha[1, 8]*alpha[2, 6]/(sqrt((X1^4*alpha[1, 8]^2*alpha[2, 4]^2 + 2*X1*((-2*X1*X2*alpha[3, 6] - 2*X2*alpha[2, 2] + 2*X3)*alpha[2, 6] + X1*X2*alpha[2, 4]*(alpha[2, 8] + alpha[3, 9]))*alpha[1, 8] + X2^2*(alpha[2, 8] + alpha[3, 9])^2)*alpha[1, 8]^2) + alpha[1, 8]^2*alpha[2, 4]*X1^2 + X2*alpha[1, 8]*(alpha[2, 8] + alpha[3, 9])), z = (-sqrt((X1^4*alpha[1, 8]^2*alpha[2, 4]^2 + 2*X1*((-2*X1*X2*alpha[3, 6] - 2*X2*alpha[2, 2] + 2*X3)*alpha[2, 6] + X1*X2*alpha[2, 4]*(alpha[2, 8] + alpha[3, 9]))*alpha[1, 8] + X2^2*(alpha[2, 8] + alpha[3, 9])^2)*alpha[1, 8]^2) - alpha[1, 8]^2*alpha[2, 4]*X1^2 + (-alpha[3, 9] - alpha[2, 8])*X2*alpha[1, 8])/(2*alpha[1, 8]^2*alpha[2, 6]*X1)}, {y = -2*X2*alpha[1, 8]*alpha[2, 6]*X1/(-sqrt((X1^4*alpha[1, 8]^2*alpha[2, 4]^2 + 2*X1*((-2*X1*X2*alpha[3, 6] - 2*X2*alpha[2, 2] + 2*X3)*alpha[2, 6] + X1*X2*alpha[2, 4]*(alpha[2, 8] + alpha[3, 9]))*alpha[1, 8] + X2^2*(alpha[2, 8] + alpha[3, 9])^2)*alpha[1, 8]^2) + alpha[1, 8]^2*alpha[2, 4]*X1^2 + X2*alpha[1, 8]*(alpha[2, 8] + alpha[3, 9])), z = (sqrt((X1^4*alpha[1, 8]^2*alpha[2, 4]^2 + 2*X1*((-2*X1*X2*alpha[3, 6] - 2*X2*alpha[2, 2] + 2*X3)*alpha[2, 6] + X1*X2*alpha[2, 4]*(alpha[2, 8] + alpha[3, 9]))*alpha[1, 8] + X2^2*(alpha[2, 8] + alpha[3, 9])^2)*alpha[1, 8]^2) - alpha[1, 8]^2*alpha[2, 4]*X1^2 + (-alpha[3, 9] - alpha[2, 8])*X2*alpha[1, 8])/(2*alpha[1, 8]^2*alpha[2, 6]*X1)}}:
> 

bb := {{y = -2*X2*alpha[2, 6]*X1/(-sqrt(X1^4*alpha[1, 8]^2*alpha[2, 4]^2 + 2*X2*alpha[1, 8]*((alpha[2, 8] + alpha[3, 9])*alpha[2, 4] - 2*alpha[2, 6]*alpha[3, 6])*X1^2 - 4*alpha[1, 8]*alpha[2, 6]*(X2*alpha[2, 2] - X3)*X1 + X2^2*(alpha[2, 8] + alpha[3, 9])^2) + (alpha[2, 8] + alpha[3, 9])*X2 + alpha[1, 8]*alpha[2, 4]*X1^2), z = (sqrt(X1^4*alpha[1, 8]^2*alpha[2, 4]^2 + 2*X2*alpha[1, 8]*((alpha[2, 8] + alpha[3, 9])*alpha[2, 4] - 2*alpha[2, 6]*alpha[3, 6])*X1^2 - 4*alpha[1, 8]*alpha[2, 6]*(X2*alpha[2, 2] - X3)*X1 + X2^2*(alpha[2, 8] + alpha[3, 9])^2) - alpha[1, 8]*alpha[2, 4]*X1^2 + (-alpha[3, 9] - alpha[2, 8])*X2)/(2*alpha[1, 8]*alpha[2, 6]*X1)}, {y = -2*X2*alpha[2, 6]*X1/(sqrt(X1^4*alpha[1, 8]^2*alpha[2, 4]^2 + 2*X2*alpha[1, 8]*((alpha[2, 8] + alpha[3, 9])*alpha[2, 4] - 2*alpha[2, 6]*alpha[3, 6])*X1^2 - 4*alpha[1, 8]*alpha[2, 6]*(X2*alpha[2, 2] - X3)*X1 + X2^2*(alpha[2, 8] + alpha[3, 9])^2) + alpha[1, 8]*alpha[2, 4]*X1^2 + (alpha[2, 8] + alpha[3, 9])*X2), z = (-sqrt(X1^4*alpha[1, 8]^2*alpha[2, 4]^2 + 2*X2*alpha[1, 8]*((alpha[2, 8] + alpha[3, 9])*alpha[2, 4] - 2*alpha[2, 6]*alpha[3, 6])*X1^2 - 4*alpha[1, 8]*alpha[2, 6]*(X2*alpha[2, 2] - X3)*X1 + X2^2*(alpha[2, 8] + alpha[3, 9])^2) - alpha[1, 8]*alpha[2, 4]*X1^2 + (-alpha[3, 9] - alpha[2, 8])*X2)/(2*alpha[1, 8]*alpha[2, 6]*X1)}}:
> 

aa:=simplify(rationalize(aa)):
bb:=simplify(rationalize(bb)):
> 

indets(aa);

{X1, X2, X3, y, z, alpha[1, 8], alpha[2, 2], alpha[2, 4], alpha[2, 6], alpha[2, 8], alpha[3, 6], alpha[3, 9], ((X1^4*alpha[1, 8]^2*alpha[2, 4]^2+2*X1*((-2*X1*X2*alpha[3, 6]-2*X2*alpha[2, 2]+2*X3)*alpha[2, 6]+X1*X2*alpha[2, 4]*(alpha[2, 8]+alpha[3, 9]))*alpha[1, 8]+X2^2*(alpha[2, 8]+alpha[3, 9])^2)*alpha[1, 8]^2)^(1/2)}

 (1)
> 

indets(bb);

{X1, X2, X3, y, z, alpha[1, 8], alpha[2, 2], alpha[2, 4], alpha[2, 6], alpha[2, 8], alpha[3, 6], alpha[3, 9], (X1^4*alpha[1, 8]^2*alpha[2, 4]^2+2*X2*alpha[1, 8]*(-2*alpha[3, 6]*alpha[2, 6]+alpha[2, 4]*(alpha[2, 8]+alpha[3, 9]))*X1^2-4*alpha[1, 8]*alpha[2, 6]*(X2*alpha[2, 2]-X3)*X1+X2^2*(alpha[2, 8]+alpha[3, 9])^2)^(1/2)}

 (2)
> 

ra:=indets(aa,radical)[]:
rb:=indets(bb,radical)[]:
> 

simplify(ra^2/rb^2);

alpha[1, 8]^2

 (3)
> 

AA:=eval(aa, [ra=R*s*t, alpha[1,8]=t]):  # So, s = +1 or -1
> 

BB:=eval(bb, [rb=R, alpha[1,8]=t]):
> 

simplify( eval(y, AA[1])-eval(y,BB[1]) ), simplify( eval(z, AA[1])-eval(z,BB[1]) );
simplify( eval(y, AA[2])-eval(y,BB[2]) ), simplify( eval(z, AA[2])-eval(z,BB[2]) );

(1/2)*X2*R*(s-1)/(t*((X1*alpha[3, 6]+alpha[2, 2])*X2-X3)), -(1/2)*R*(s-1)/(t*alpha[2, 6]*X1)

  

-(1/2)*X2*R*(s-1)/(t*((X1*alpha[3, 6]+alpha[2, 2])*X2-X3)), (1/2)*R*(s-1)/(t*alpha[2, 6]*X1)

 (4)
> 

simplify( eval(y, AA[1])-eval(y,BB[2]) ), simplify( eval(z, AA[1])-eval(z,BB[2]) );
simplify( eval(y, AA[2])-eval(y,BB[1]) ), simplify( eval(z, AA[2])-eval(z,BB[1]) );

(1/2)*X2*R*(s+1)/(t*((X1*alpha[3, 6]+alpha[2, 2])*X2-X3)), -(1/2)*R*(s+1)/(t*alpha[2, 6]*X1)

  

-(1/2)*X2*R*(s+1)/(t*((X1*alpha[3, 6]+alpha[2, 2])*X2-X3)), (1/2)*R*(s+1)/(t*alpha[2, 6]*X1)

 (5)
> 

# ==> aa = bb even for complex parameters
> 

 

 

StringTools...

vv 12453
November 03 2020
1 1 
restart;
ic:=[y(0)=0,D(y)(0)=3];
s:=Physics:-Latex(ic, output=string);
s1:=StringTools:-SubstituteAll(s, "D\\left(y \\right)", "y'");   # or,
s2:=StringTools:-SubstituteAll(s, "D\\left(y \\right)", "y^{\\prime}");

 

Polya...

vv 12453
October 21 2020
2 0

Try to prove (without Maple) the nice formulae due to G. Polya.

> 

Int(x^(-x), x=0..1) = Sum(n^(-n), n=1..infinity);
Int(x^(x), x=0..1) = Sum((-1)^(n-1)*(n)^(-n), n=1..infinity);

Int(x^(-x), x = 0 .. 1) = Sum(n^(-n), n = 1 .. infinity)

  

Int(x^x, x = 0 .. 1) = Sum((-1)^(n-1)*n^(-n), n = 1 .. infinity)

 (1)
> 

evalf([%%,%])

[1.291285997 = 1.291285997, .7834305107 = .7834305107]

 (2)

 

Unstable...

vv 12453
October 20 2020
1 2

JordanForm does non accept numeric matrices containing floats, because the Jordan form is very unstable numericallly (and SingularValues is recommended for numeric tasks).

(Rank is also unstable, but it accepts floats; probably it is considered "less unstable").

Of course it's possible to cheat e.g. JordanForm(A+'sin(0)')  or convert to rationals.

 

syntax...

vv 12453
October 18 2020
1 0

@Stretto You want to use your own syntax! Note that Maple already has the operators `--` and `++`.

a:=10: b:=100:
a--+b;  [a,b];
                             a := 9
                              110
                            [9, 100]
 
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