## 1491 Reputation

8 years, 260 days

## ....

I am no expert in special functions,but adding comand "_EnvLegendreCut"

_EnvLegendreCut := 1 .. infinity;

plot(LegendreQ((1/2)*sqrt(5)-1/2, x), x = -1 .. 1);

?LegendreQ;

## seq,ListTools,select.....

restart;

with(ListTools):

l := seq(n, n = -10 .. 20);

[l];

l1 := Flatten([Transpose([[seq(n, n = -10 .. 20)], [seq(n, n = 1 .. 31)]])]);

l2 := Transpose([[seq(n, n = -10 .. 20)], [seq(n, n = 1 .. 31)]]);

l3 := select(type, [seq(n, n = 115 .. 231)], 'odd');

[l3];

## BUG...

In yours first example Maple gives incorect answer.It's a Bug.!!!

evalf[10](sum(2^n*floor(2^n), n = 1 .. infinity));

#-1.333333333

evalf[10](Sum(2^n*floor(2^n), n = 1 .. infinity));# Big 'S' in Sum

#-1.333333333

assuming([sum(2^n*floor(2^n), n = 1 .. m)], [m > 0]);

#(1/3)*2^(m+1)*floor(2^(m+1))-4/3

limit(%, m = infinity)

#infinity

Second one give a correct answer:

evalf[10](sum(5^(n-1)*floor((1/4)*5^n), n = 1 .. infinity));

#Float(infinity)

evalf[10](Sum(5^(n-1)*floor((1/4)*5^n), n = 1 .. infinity));# Big 'S' in Sum

#If Maple dosen't know the answer then: Returns unevaluated,not infinity in this case.

assuming([sum(5^(n-1)*floor((1/4)*5^n), n = 1 .. m)], [m > 0]);

#Returns unevaluated

Executed in Maple 2018.

## CauchyPrincipalValue....

Using CauchyPrincipalValue =true:

assuming([int(exp(I*x*t)/((x-a)*(x+a)), x = -infinity .. infinity, CauchyPrincipalValue = true)], [a > 0, t > 0])

#(1/2*I)*Pi*(exp(I*a*t)-exp(-I*a*t))/a

int.mw

Exectuted in Maple 2018.

## Another way...

remove[flatten](x-> x = 0, [seq(sin((1/4)*k*Pi), k = 1 .. 8)]);

## ....

The function RootOf is a placeholder for representing all the roots of an equation in one variable.  In particular, it is the standard representation for Maple algebraic numbers, algebraic functions.

?RootOf
sol := solve({x^2+y^2 = 3, x^2+2*y^2 = 3}, {x, y});
allvalues(sol);
evalf(%);

#{x = sqrt(3), y = 0}, {x = -sqrt(3), y = 0}
#{x = 1.732050808, y = 0.}, {x = -1.732050808, y = 0.}


## To rational....

I don't know way it's happens,but if we convert to rational(exact) numbers then works:

eq := [190.0315457*d^6-7.601261828*d^4+344.0*c^2*d^2-2.677685950*c^4, 0.6830134554e-2*c^4+2084.317025*d^8-166.7453620*d^6+3.334907240*d^4-c^2*d^2];
eliminate(convert(eq, rational, exact), c);

#[{c = -(1/14265)*sqrt(13071106675-17435*sqrt(9667390500*d^2+561670891405))*d}, {121*d^4*(582104796124362625620*d^4-46433010457684955790*d^2+2932165094782*sqrt(9667390500*d^2+561670891405)-1272306087125807465)}], [{c = (1/14265)*sqrt(13071106675-17435*sqrt(9667390500*d^2+561670891405))*d}, {121*d^4*(582104796124362625620*d^4-46433010457684955790*d^2+2932165094782*sqrt(9667390500*d^2+561670891405)-1272306087125807465)}], [{c = -(1/14265)*sqrt(13071106675+17435*sqrt(9667390500*d^2+561670891405))*d}, {-121*d^4*(-582104796124362625620*d^4+46433010457684955790*d^2+2932165094782*sqrt(9667390500*d^2+561670891405)+1272306087125807465)}], [{c = (1/14265)*sqrt(13071106675+17435*sqrt(9667390500*d^2+561670891405))*d}, {-121*d^4*(-582104796124362625620*d^4+46433010457684955790*d^2+2932165094782*sqrt(9667390500*d^2+561670891405)+1272306087125807465)}]

## ....

 >
 >
 a_{{p,q}}=\sum _{n=1}^{q}\cos \left( 2\,{\frac {\pi\, \left( p+1  \right)  \left( n-1 \right) }{q}} \right)
 >
 >
 a_{{p,q}}=\sum _{n=1}^{q}\cos \left( 2\,{\frac {\pi\, \left( p+1  \right)  \left( n-1 \right) }{q}} \right)
 >
 >
 a_{{p,q}}=\sum _{n=1}^{q}\cos \left( 2\,{\frac {\pi\, \left( p+1  \right)  \left( n-1 \right) }{q}} \right)

## HINTs...

pdsolve([diff(f2(lam, mu, l, m), m)-(diff(f1(lam, mu, l, m), l))-(diff(f4(lam, mu, l, m), mu))+diff(f3(lam, mu, l, m), lam) = 0, diff(f1(lam, mu, l, m), m)+diff(f2(lam, mu, l, m), l)-(diff(f3(lam, mu, l, m), mu))-(diff(f4(lam, mu, l, m), lam)) = 0], HINT = +);

#A simple solutions.

pdsolve([diff(f2(lam, mu, l, m), m)-(diff(f1(lam, mu, l, m), l))-(diff(f4(lam, mu, l, m), mu))+diff(f3(lam, mu, l, m), lam) = 0, diff(f1(lam, mu, l, m), m)+diff(f2(lam, mu, l, m), l)-(diff(f3(lam, mu, l, m), mu))-(diff(f4(lam, mu, l, m), lam)) = 0], HINT = TWS(sin));

?pdsolve

## ....

Probably you want:

with(Fractals:-LSystem); with(LSystemExamples):

PlotExample(DragonCurve, 15);

Lindenmayer System Plot Generator:

states := "FX";

rules := ["Y" = "FX-Y", "X" = "X+YF"];

cons := ["F" = "draw:1", "+" = "turn:-90", "-" = "turn:90"];

newstate1 := Iterate(states, rules, 10);

LSystemPlot(newstate1, cons);

EDITED:

dragon := proc (k::algebraic, N::integer) local t, i, q1, q2, q3, q4, d; global p; q2 := [k, 0]; q3 := [1-k, 0]; d := evalm(q3-q2); p[0] := plot([[q2[1], q2[2]], [q3[1], q3[2]]]); for i to N do if mod(i, 2) = 0 then t[i] := t[(1/2)*i] else t[i] := (mod(i, 4))-2 end if; q4 := evalm(q3+k*d/(1-2*k)); d := evalm(t[i]*[d[2], -d[1]]); q1 := evalm(q3); q2 := evalm(q4+k*d/(1-2*k)); q3 := evalm(q2+d); p[i] := plot([[q1[1], q1[2]], [q2[1], q2[2]], [q3[1], q3[2]]]) end do; return plots:-display([seq(p[i], i = 0 .. N)]) end proc;

dragon(0.1, 500);

## A numerical approximation for the (parti...

f := x^2*y^2+3*x*y^2;

fdiff(f, [x, y], {x = 1, y = 2});

?fdiff

## Another way:...

restart;

de := diff(y(t), t, t) = -1; ic := y(0) = 1, (D(y))(0) = 0;

Events := [y(t), diff(y(t), t) = -.5*(diff(y(t), t))];

Events2 := [y(t) = -2*(1/1000), halt];

dsol := dsolve({de, ic}, numeric, events = [Events, Events2], range = 0 .. 5);

plots[odeplot](dsol, thickness = 3, color = red)

## Numerically....

You can do it numerically:

c[1]:=1;
c[2]:=1;
int(12.*x^3*c[2]+6.*x^2*c[1]+x^2*exp(x^3*c[2])*exp(x^2*c[1]), x = 0. .. 1.,numeric);

#6.155281446

If you looking a analyticaly solution it's probably not possible.

## Using PDEtools....

PDE := diff(f(x, y), x, x)+diff(f(x, y), y, y) = 0;
sol := PDEtools:-SimilaritySolutions(PDE);
sol[1];

OR:

pdsolve(PDE, HINT = +, build);
pdsolve(PDE, HINT = *, build);

## ....

I doubt there's a closed form for the integral for general variables.

Only for a=0,1 and k=-1,0,1,2,3. can be found:

By Maple 2018.

sol := x*a*k*((x-g)/b)^(a-1)*(1+((x-g)/b)^a)^(k+1)/b;

assuming([[seq([int(eval(sol, a = j), x = g .. t)], j = 0 .. 1)]], [a > 0, b > 0, k in integer, k > 0, g > 0, t > 0]);

#[[0], [k*(-g*k+b^(-k)*(b-g+t)^k*k*t-2*b^(-k-1)*(b-g+t)^k*g*k*t+2*b^(-k-1)*(b-g+t)^k*t^2*k+b^(-k-2)*(b-g+t)^k*g^2*k*t-2*b^(-k-2)*(b-g+t)^k*t^2*k*g+b^(-k-2)*(b-g+t)^k*t^3*k+b-3*g-b^(-k+1)*(b-g+t)^k+3*b^(-k)*(b-g+t)^k*g-3*b^(-k-1)*(b-g+t)^k*g^2+3*b^(-k-1)*(b-g+t)^k*t^2+b^(-k-2)*(b-g+t)^k*g^3-3*b^(-k-2)*(b-g+t)^k*t^2*g+2*b^(-k-2)*(b-g+t)^k*t^3)/(k^2+5*k+6)]]

assuming([[seq([int(eval(sol, k = j), x = g .. t)], j = -1 .. 3)]], [a > 0, b > 0, g > 0, t > 0, a > 0]);

#[[-b^(-a)*(t-g)^a*(a*t+g)/(a+1)], [0], [(6*b^(-3*a)*(t-g)^(3*a)*a^3*t+2*b^(-3*a)*(t-g)^(3*a)*a^2*g+9*b^(-3*a)*(t-g)^(3*a)*a^2*t+18*b^(-2*a)*(t-g)^(2*a)*a^3*t+3*b^(-3*a)*(t-g)^(3*a)*a*g+3*b^(-3*a)*(t-g)^(3*a)*a*t+9*b^(-2*a)*(t-g)^(2*a)*a^2*g+24*b^(-2*a)*(t-g)^(2*a)*a^2*t+18*b^(-a)*(t-g)^a*a^3*t+b^(-3*a)*(t-g)^(3*a)*g+12*b^(-2*a)*(t-g)^(2*a)*a*g+6*b^(-2*a)*(t-g)^(2*a)*a*t+18*b^(-a)*(t-g)^a*a^2*g+15*b^(-a)*(t-g)^a*a^2*t+3*g*b^(-2*a)*(t-g)^(2*a)+15*b^(-a)*(t-g)^a*a*g+3*b^(-a)*(t-g)^a*a*t+3*g*b^(-a)*(t-g)^a)/(3*(6*a^3+11*a^2+6*a+1))], [(168*b^(-3*a)*(t-g)^(3*a)*a^3*t+56*b^(-3*a)*(t-g)^(3*a)*a^2*g+84*b^(-3*a)*(t-g)^(3*a)*a^2*t+228*b^(-2*a)*(t-g)^(2*a)*a^3*t+28*b^(-3*a)*(t-g)^(3*a)*a*g+12*b^(-3*a)*(t-g)^(3*a)*a*t+114*b^(-2*a)*(t-g)^(2*a)*a^2*g+96*b^(-2*a)*(t-g)^(2*a)*a^2*t+104*b^(-a)*(t-g)^a*a^3*t+48*b^(-2*a)*(t-g)^(2*a)*a*g+12*b^(-2*a)*(t-g)^(2*a)*a*t+104*b^(-a)*(t-g)^a*a^2*g+36*b^(-a)*(t-g)^a*a^2*t+36*b^(-a)*(t-g)^a*a*g+4*b^(-a)*(t-g)^a*a*t+4*b^(-3*a)*(t-g)^(3*a)*g+6*g*b^(-2*a)*(t-g)^(2*a)+4*g*b^(-a)*(t-g)^a+11*b^(-4*a)*(t-g)^(4*a)*a^2*g+24*b^(-4*a)*(t-g)^(4*a)*a^2*t+6*b^(-4*a)*(t-g)^(4*a)*a*g+4*b^(-4*a)*(t-g)^(4*a)*a*t+96*b^(-a)*(t-g)^a*a^3*g+24*b^(-4*a)*(t-g)^(4*a)*a^4*t+6*b^(-4*a)*(t-g)^(4*a)*a^3*g+44*b^(-4*a)*(t-g)^(4*a)*a^3*t+96*b^(-3*a)*(t-g)^(3*a)*a^4*t+32*b^(-3*a)*(t-g)^(3*a)*a^3*g+144*b^(-2*a)*(t-g)^(2*a)*a^4*t+72*b^(-2*a)*(t-g)^(2*a)*a^3*g+96*b^(-a)*(t-g)^a*a^4*t+b^(-4*a)*(t-g)^(4*a)*g)/(2*(24*a^4+50*a^3+35*a^2+10*a+1))], [(3*(b^(-5*a)*(t-g)^(5*a)*g+600*b^(-a)*(t-g)^a*a^4*g+600*b^(-a)*(t-g)^a*a^5*t+120*b^(-5*a)*(t-g)^(5*a)*a^5*t+1470*b^(-3*a)*(t-g)^(3*a)*a^3*t+490*b^(-3*a)*(t-g)^(3*a)*a^2*g+360*b^(-3*a)*(t-g)^(3*a)*a^2*t+1180*b^(-2*a)*(t-g)^(2*a)*a^3*t+120*b^(-3*a)*(t-g)^(3*a)*a*g+30*b^(-3*a)*(t-g)^(3*a)*a*t+590*b^(-2*a)*(t-g)^(2*a)*a^2*g+260*b^(-2*a)*(t-g)^(2*a)*a^2*t+355*b^(-a)*(t-g)^a*a^3*t+130*b^(-2*a)*(t-g)^(2*a)*a*g+20*b^(-2*a)*(t-g)^(2*a)*a*t+355*b^(-a)*(t-g)^a*a^2*g+70*b^(-a)*(t-g)^a*a^2*t+70*b^(-a)*(t-g)^a*a*g+5*b^(-a)*(t-g)^a*a*t+10*b^(-3*a)*(t-g)^(3*a)*g+10*g*b^(-2*a)*(t-g)^(2*a)+5*g*b^(-a)*(t-g)^a+205*b^(-4*a)*(t-g)^(4*a)*a^2*g+220*b^(-4*a)*(t-g)^(4*a)*a^2*t+55*b^(-4*a)*(t-g)^(4*a)*a*g+20*b^(-4*a)*(t-g)^(4*a)*a*t+770*b^(-a)*(t-g)^a*a^3*g+1220*b^(-4*a)*(t-g)^(4*a)*a^4*t+305*b^(-4*a)*(t-g)^(4*a)*a^3*g+820*b^(-4*a)*(t-g)^(4*a)*a^3*t+2340*b^(-3*a)*(t-g)^(3*a)*a^4*t+780*b^(-3*a)*(t-g)^(3*a)*a^3*g+2140*b^(-2*a)*(t-g)^(2*a)*a^4*t+1070*b^(-2*a)*(t-g)^(2*a)*a^3*g+770*b^(-a)*(t-g)^a*a^4*t+5*b^(-4*a)*(t-g)^(4*a)*g+24*b^(-5*a)*(t-g)^(5*a)*a^4*g+250*b^(-5*a)*(t-g)^(5*a)*a^4*t+50*b^(-5*a)*(t-g)^(5*a)*a^3*g+175*b^(-5*a)*(t-g)^(5*a)*a^3*t+35*b^(-5*a)*(t-g)^(5*a)*a^2*g+50*b^(-5*a)*(t-g)^(5*a)*a^2*t+10*b^(-5*a)*(t-g)^(5*a)*a*g+5*b^(-5*a)*(t-g)^(5*a)*a*t+600*b^(-4*a)*(t-g)^(4*a)*a^5*t+150*b^(-4*a)*(t-g)^(4*a)*a^4*g+1200*b^(-3*a)*(t-g)^(3*a)*a^5*t+400*b^(-3*a)*(t-g)^(3*a)*a^4*g+1200*b^(-2*a)*(t-g)^(2*a)*a^5*t+600*b^(-2*a)*(t-g)^(2*a)*a^4*g))/(5*(120*a^5+274*a^4+225*a^3+85*a^2+15*a+1))]]

Mathematica 11.3 can find solution for: - infinity > a >= 2 and -1<= k < infinity

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