Mariusz Iwaniuk

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kernelopts(version)
#`Maple 2022.1, X86 64 WINDOWS, May 26 2022, Build ID 1619613`

(int(sin(x)*exp(-2*I*Pi*f*x)/x, x = -infinity .. infinity) assuming (0 < f));

#signum(0, -2*Pi*f + 1, 0)*Pi/4 + Pi/2 - signum(0, 2*Pi*f - 1, 0)*Pi/4

convert(%, piecewise);

#piecewise(f < 1/(2*Pi), Pi, f = 1/(2*Pi), Pi/2, 1/(2*Pi) < f, 0)

 

Order := 50;
ode := diff(x(u), u, u) + 1/4*exp(-u^2)*x(u) = 0;
sol := convert(dsolve([ode, x(0) = 1, D(x)(0) = 0], x(u), 'series'), polynom);
plot([rhs(sol), diff(rhs(sol), u)], u = 0 .. 2, view = [0 .. 2, -1/2 .. 1])

 

Looks like a bug in integrate.

int(x^n*exp(x), x = 0 .. 2, method = _RETURNVERBOSE); #Can't solve.

I have only a workround.

See attached file:2_ver1.mw

As a infinite sum:

Int(sin(sqrt(-x^2 + 1)), x) = Sum(2^(-1/2 - k)*sqrt(Pi)*x^(1 + 2*k)*BesselY(-1/2 + k, 1)/((1 + 2*k)*GAMMA(1 + k)), k = 0 .. infinity) + C

for: -1<=x<=1

Check:

int(sin(sqrt(-x^2 + 1)), x = 0 .. 1, numeric);
evalf(add(eval(2^(-1/2 - k)*sqrt(Pi)*x^(1 + 2*k)*BesselY(-1/2 + k, 1)/((1 + 2*k)*GAMMA(1 + k)), x = 1), k = 0 .. 2000));

 


 

restart

interface(rtablesize = 100)

eqns := {x(0) = 1, x(t+1) = (1+10^(-6)*(0.4e-2-0.6e-2)/(0.1e-3))*x(t)+0.7e-1*10^(-6)*y(t), y(0) = 857.1428571, y(t+1) = 0.6e-2*10^(-6)*x(t)/(0.1e-3)+(1-0.7e-1*10^(-6))*y(t)}

{x(0) = 1, x(t+1) = .9999800000*x(t)+0.7000000000e-7*y(t), y(0) = 857.1428571, y(t+1) = 0.6000000000e-4*x(t)+.9999999300*y(t)}

(1)

SOL := evalf(rsolve(convert(eqns, rational), {x, y}))

{x(t) = 2.959071538*1.000000138^t-1.959071539*.9999797913^t, y(t) = 851.3060776*1.000000138^t+5.836778775*.9999797913^t}

(2)

seq([rhs(SOL[1]), rhs(SOL[2])], t = 0 .. 20)

[.999999999, 857.1428564], [1.000039997, 857.1428559], [1.000079996, 857.1428555], [1.000119993, 857.1428549], [1.000159988, 857.1428545], [1.000199984, 857.1428540], [1.000239979, 857.1428536], [1.000279972, 857.1428531], [1.000319966, 857.1428526], [1.000359958, 857.1428522], [1.000399950, 857.1428517], [1.000439940, 857.1428513], [1.000479930, 857.1428509], [1.000519919, 857.1428504], [1.000559907, 857.1428499], [1.000599894, 857.1428495], [1.000639882, 857.1428491], [1.000679867, 857.1428487], [1.000719852, 857.1428482], [1.000759837, 857.1428478], [1.000799820, 857.1428474]

(3)

````

rtable([%], subtype = Vector[column])

Vector[column](%id = 36893490192963129204)

(4)

NULL


 

Download SystemRecursive_new2.mw

.

f := (x, y) -> piecewise(x <> 0 and y <> 0, (y^2 + x^2)^x, x = 0 and y = 0, 1);
DX := diff(f(x, y), x);
DY := diff(f(x, y), y);
[eval(DX, [x = 1, y = 1]), eval(DY, [x = 1, y = 1])]

#at x=1 and y=1;
#[2*ln(2) + 2, 2]

 


 

NULL

Vector(3, {(1) = (l+s(t))*cos(beta(t))+xo+s(t)-x(t), (2) = sin(beta(t))*(l+s(t))*sin(alpha(t))-y(t), (3) = -sin(beta(t))*(l+s(t))*cos(alpha(t))-z(t)})

Vector[column](%id = 36893489901319436636)

(1)

"solve(?[2],[beta(t)])[][]"

beta(t) = arcsin(y(t)/(sin(alpha(t))*(l+s(t))))

(2)

"subs(beta(t) = arcsin(y(t)/sin(alpha(t))/(l+s(t))),?[3])"

-sin(arcsin(y(t)/(sin(alpha(t))*(l+s(t)))))*(l+s(t))*cos(alpha(t))-z(t)

(3)

solve(-sin(arcsin(y(t)/(sin(alpha(t))*(l+s(t)))))*(l+s(t))*cos(alpha(t))-z(t), [alpha(t)])

[[alpha(t) = -arctan(y(t)/z(t))]]

(4)

" solve(simplify(subs(beta(t) = arcsin(y(t)/sin(alpha(t))/(l+s(t))),?[3])),[alpha(t)])"

[[alpha(t) = -arctan(y(t)/z(t))]]

(5)

NULL

NULL


 

Download Arctan_ver2.mw

 

restart;
with(LinearAlgebra);
w := (2*Pi)/14;
v := Vector([1, sin(w*t), cos(w*t)]);
simplify(eval(sum(v . (Transpose(v)), t = k .. k + n), n = 13));

 

restart;
ODE := diff(W(x), x $ 2) + (A*ohm^2*ro - kd)*W(x)/T = p;
sol1 := dsolve(ODE, W(x));
assign(sol1);

subs(_C1 = 0, W(x));

#or:

eval(W(x), _C1 = 0);


#sin(sqrt(A*ohm^2*ro - kd)*x/sqrt(T))*_C2 + p*T/(A*ohm^2*ro - kd)

 

 

int(exp(-2*r)*cos(theta)^3*r^2*sin(theta), phi = 0 .. 2*Pi, r = 0 .. infinity, theta = 0 .. Pi);
#O

 

 

We can find solution with series representation:

restart;
odeSystem := {diff(y1(x), x) = -x*y2(x) - (1 + x)*y3(x), diff(y2(x), x) = -x*y1(x) - (1 + x)*y4(x), diff(y3(x), x) = -x*y1(x) - (1 + x)*y4(x) - 5*x*cos(1/2*x^2), diff(y4(x), x) = -x*y2(x) - (1 + x)*y3(x) + 5*x*sin(1/2*x^2), y1(0) = 5, y2(0) = 1, y3(0) = -1, y4(0) = 0};
Order := 10;
systemSol := dsolve(odeSystem, [y1(x), y2(x), y3(x), y4(x)], series);
F := convert(systemSol, polynom);
plot([rhs(F[1]), rhs(F[2]), rhs(F[3]), rhs(F[4])], x = 0 .. 1, color = [red, blue, green, gold], legend = [y1(x), y2(x), y3(x), y4(x)]);

A workaround with Mathematica:

f[k_] := Sum[(-1)^i (k - i + 1)^(2 k + 4)/((i!)*((2 k - i + 2)!)), {i,0, k}];
L = Table[f[k], {k, 0, 1000}];
FindSequenceFunction[L, k]

(*1/12 (k + 3 k^2 + 2 k^3)*)

With Maple:

restart;
with(gfun):
f := k -> sum((-1)^i*(k - i + 1)^(2*k + 4)/(i!*(2*k - i + 2)!), i = 0 .. k);
l := [seq(f(k), k = 0 .. 1000)];
rec := listtorec(l, u(k), [ogf]);
rsolve(op(1, rec), u(k));

#1/6*k^3 + 3/4*k^2 + 13/12*k + 1/2

 

See attached file.

integral.mw

Sum((-1)^k*sin(k*x)/k, k = -infinity .. infinity) = evalc(sum((-1)^k*sin(k*x)/k, k = -infinity .. infinity, formal))

 

f := x -> convert(sin(theta), x);
map(f, [cos, tan, cot, csc, sec]);

#[-cos(theta + Pi/2), 2*tan(theta/2)/(1 + tan(theta/2)^2), 2*cot(theta/2)/(cot(theta/2)^2 + 1), 1/csc(theta), -1/sec(theta + Pi/2)]

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