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Mariusz Iwaniuk

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9 years, 164 days

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These are answers submitted by Mariusz Iwaniuk

Another workaround:...

Mariusz Iwaniuk 1571
March 09 2025
3 0
> 

restart
> 

NULL
> 

A := inttrans:-mellin(k*t^k/(1-a*t^k), a, s)

k*t^k*(-1/t^k)^s*Beta(s, 1-s)

 (1)
> 

A1 := `assuming`([simplify(A)], [s > 0, t > 0, k > 0])

(-1)^s*Pi*k*t^(-k*(-1+s))*csc(Pi*s)

 (2)
> 

A2 := sum(A1, k = 1 .. infinity, formal)

t^(1-s)*(-1)^s*Pi*csc(Pi*s)/(t^(1-s)-1)^2

 (3)
> 

A3 := `assuming`([limit((1-t)^2*A2, t = 1, left)], [s > 0])

(-1)^s*Pi/((-1+s)^2*sin(Pi*s))

 (4)
> 

A4 := `assuming`([inttrans:-invmellin(A3, s, a)], [a > 0])

Pi*invmellin((-1)^s/((-1+s)^2*sin(Pi*s)), s, a, -infinity .. infinity, INT)

 (5)
> 

NULL
> 

A5 := `assuming`([residue(A3*a^(-s), s = -m)], [m::posint, m >= 0])

exp(ln(a)*m)/(1+m)^2

 (6)
> 

A6 := sum(eval(A5, a = 1), m = 0 .. infinity)

(1/6)*Pi^2

 (7)
> 

NULL
> 

sum(k*t^k/(1-t^k), k = 1 .. infinity)

sum(k*t^k/(1-t^k), k = 1 .. infinity)

 (8)
> 

NULL
> 

NULL

Download Series_with_limit.mw

Workaround:...

Mariusz Iwaniuk 1571
March 06 2025
2 0


 

> 

restart
> 

kernelopts(version)

`Maple 2024.2, X86 64 WINDOWS, Oct 29 2024, Build ID 1872373`

 (1)
> 

with(InertForm)
> 

NULL
> 

Parse("arctan(2/n^2)") = Parse("int(2/(n^2*(1 + 2*t^2/n^2)), t = 0 .. 1)")

%arctan(`%/`(2, `%^`(n, 2))) = %int(`%/`(2, `%*`(`%^`(n, 2), `%+`(1, `%/`(`%*`(2, `%^`(t, 2)), `%^`(n, 2))))), t = 0 .. 1)

 (2)
> 

Parse("sum(arctan(2/n^2),n=1..infinity)") = `≡`(Parse("sum(int(2/(n^2*(1 + 2*t^2/n^2)), t = 0 .. 1),n=1..infinity)"), Parse("int(sum(2/(n^2*(1 + 2*t^2/n^2)), n=1..infinity), t = 0 .. 1)"))

%sum(%arctan(`%/`(2, `%^`(n, 2))), n = 1 .. infinity) = `≡`(%sum(%int(`%/`(2, `%*`(`%^`(n, 2), `%+`(1, `%/`(`%*`(2, `%^`(t, 2)), `%^`(n, 2))))), t = 0 .. 1), n = 1 .. infinity), %int(%sum(`%/`(2, `%*`(`%^`(n, 2), `%+`(1, `%/`(`%*`(2, `%^`(t, 2)), `%^`(n, 2))))), n = 1 .. infinity), t = 0 .. 1))

 (3)
> 

NULL
> 

S := sum(factor(eval(x/(t^2*x^2+1), x = 2/n^2)), n = 1 .. infinity)

-(1/2)*(Sum(Psi(1-_alpha)/_alpha, _alpha = RootOf(_Z^4+4*t^2)))

 (4)
> 

S1 := `assuming`([simplify(allvalues(S))], [0 <= t and t <= 1])

(-1/4+(1/4)*I)*Pi*(I*cot((1-I)*t^(1/2)*Pi)+cot((1+I)*t^(1/2)*Pi))/t^(1/2)

 (5)
> 

int(S1, t = 0 .. 1)

(3/4)*Pi

 (6)
> 

NULL


 

Download Series.mw

Step by step with Stoltz-Cesaro theorem...

Mariusz Iwaniuk 1571
March 05 2025
2 1
> 

restart
> 

with(InertForm)
> 

Parse("n/2^(n)*Sum(2^(k)/k,k=1..n)") = Parse("Sum(2^(k)/k,k=1..n)/(2^(n)/n)")

`%*`(`%/`(n, `%^`(2, n)), %Sum(`%/`(`%^`(2, k), k), k = 1 .. n)) = `%/`(%Sum(`%/`(`%^`(2, k), k), k = 1 .. n), `%/`(`%^`(2, n), n))

 (1)
> 

NULL
> 

a := proc (n) options operator, arrow; sum(2^k/k, k = 1 .. n) end proc; b := proc (n) options operator, arrow; 2^n/n end proc

proc (n) options operator, arrow; sum(2^k/k, k = 1 .. n) end proc

  

proc (n) options operator, arrow; 2^n/n end proc

 (2)
> 

S := (a(n+1)-a(n))/(b(n+1)-b(n))

(-(1/2)*2^(n+2)*((1/2)*LerchPhi(2, 1, n)-(1/2)/n-1/(n+1))+(1/2)*2^(n+1)*(LerchPhi(2, 1, n)-1/n))/(2^(n+1)/(n+1)-2^n/n)

 (3)
> 

S1 := simplify(S)

2*n/(n-1)

 (4)
> 

limit(S1, n = infinity)

2

 (5)
 

NULL

Download answer.mw

Try...

Mariusz Iwaniuk 1571
February 26 2025
2 0
> 

restart
> 

int(ln(x)*ln(1-x), x = 0 .. 1)

2-(1/6)*Pi^2

 (1)
> 

`assuming`([int(ln(x)*ln(1-x), x = eps .. 1-delta, AllSolutions = true)], [eps > 0, delta > 0])

piecewise(eps = 1, -2, eps*ln(1-eps)-eps*ln(1-eps)*ln(eps)+ln(eps)*eps-2*eps-dilog(eps)-ln(1-eps))+piecewise(delta = 1, (1/6)*Pi^2, ln(delta)*delta+ln(delta)*ln(1-delta)-ln(delta)*ln(1-delta)*delta-ln(1-delta)+ln(1-delta)*delta+2-2*delta+dilog(1-delta))

 (2)
> 

limit(limit(piecewise(eps = 1, -2, eps*ln(1-eps)-eps*ln(1-eps)*ln(eps)+ln(eps)*eps-2*eps-dilog(eps)-ln(1-eps))+piecewise(delta = 1, (1/6)*Pi^2, ln(delta)*delta+ln(delta)*ln(1-delta)-ln(delta)*ln(1-delta)*delta-ln(1-delta)+ln(1-delta)*delta+2-2*delta+dilog(1-delta)), eps = 0), delta = 0)

2-(1/6)*Pi^2

 (3)
> 

limit(limit(piecewise(eps = 1, -2, eps*ln(1-eps)-eps*ln(1-eps)*ln(eps)+ln(eps)*eps-2*eps-dilog(eps)-ln(1-eps))+piecewise(delta = 1, (1/6)*Pi^2, ln(delta)*delta+ln(delta)*ln(1-delta)-ln(delta)*ln(1-delta)*delta-ln(1-delta)+ln(1-delta)*delta+2-2*delta+dilog(1-delta)), delta = 0), eps = 0)

2-(1/6)*Pi^2

 (4)
 

NULL

Download Int.mw

MmaTranslator or File-Open!...

Mariusz Iwaniuk 1571
October 12 2024
1 0

You may use line by line for example and for simple cases:

with(MmaTranslator);
FromMma("Integrate[x, {x, 0, 1}]")#In Mathematica use InputForm

value(%)

Or: File-Open-> *.nb Mathematica file.

 

Of course, it may not work for very complex expressions.

Another way:...

Mariusz Iwaniuk 1571
September 15 2024
2 1

In Maple 2024:

p := series(x/(-b*x^2 - a*x + 1), x = infinity, oterm = false);
coeff(p, 1/x);

#-1/b

Workaround:...

Mariusz Iwaniuk 1571
August 06 2024
3 1

 

with(MultiSeries):
e := -tanh(sqrt(2)*(a*x + b)):
limit(e, x = 0);
convert(%, tanh);

#-tanh(b*sqrt(2))

Or:

restart:
e := -tanh(sqrt(2)*(a*x + b)); (limit(e, x = 0) assuming (b <> 0));

#(-exp(2*b*sqrt(2)) + 1)/(exp(2*b*sqrt(2)) + 1)

(limit(e, x = 0) assuming (a in real, b in real));

#(-exp(2*b*sqrt(2)) + 1)/(exp(2*b*sqrt(2)) + 1)

 

 

 

 

reply...

Mariusz Iwaniuk 1571
August 02 2024
1 1


 

> 

restart
> 

ODE := diff(u(x), x, x)-(v-2)*u(x)-(v+v1/mu)*(-u(x)^3+u(x)^5) = 0

diff(diff(u(x), x), x)-(v-2)*u(x)-(v+v1/mu)*(-u(x)^3+u(x)^5) = 0

 (1)
> 

with(PDEtools)
> 

A1 := `assuming`([simplify(dchange({x = c*t+xi/sqrt(k), u(x) = W(xi)^(1/2)}, ODE, [W(xi), xi], params = {c, k, t}))], [k > 0])

(1/4)*(-4*W(xi)^4*mu*v-4*W(xi)^4*v1+4*W(xi)^3*mu*v+4*W(xi)^3*v1-4*W(xi)^2*mu*v+2*W(xi)*k*(diff(diff(W(xi), xi), xi))*mu+8*W(xi)^2*mu-(diff(W(xi), xi))^2*k*mu)/(W(xi)^(3/2)*mu) = 0

 (2)
> 

collect(expand(A1*sqrt(W(xi))), W(xi))

(-v-v1/mu)*W(xi)^3+(v+v1/mu)*W(xi)^2+(-v+2)*W(xi)+(1/2)*k*(diff(diff(W(xi), xi), xi))-(1/4)*(diff(W(xi), xi))^2*k/W(xi) = 0

 (3)
> 

A2 := `assuming`([simplify(dchange({x = c*t+xi/sqrt(k), u(x) = -W(xi)^(1/2)}, ODE, [W(xi), xi], params = {c, k, t}))], [k > 0])

(-(1/2)*W(xi)*k*(diff(diff(W(xi), xi), xi))*mu+(1/4)*(diff(W(xi), xi))^2*k*mu+W(xi)^2*((mu*v+v1)*W(xi)^2+(-mu*v-v1)*W(xi)+mu*(v-2)))/(W(xi)^(3/2)*mu) = 0

 (4)
> 

collect(expand(A2*sqrt(W(xi))), W(xi))

(v+v1/mu)*W(xi)^3+(-v-v1/mu)*W(xi)^2+(v-2)*W(xi)-(1/2)*k*(diff(diff(W(xi), xi), xi))+(1/4)*(diff(W(xi), xi))^2*k/W(xi) = 0

 (5)
> 

NULL


 

Download ODE.mw

Integral solution....

Mariusz Iwaniuk 1571
July 04 2024
0 0 
Int((1 - sigma*sin(2*Pi*x))^k, x = 0 .. n) = n*hypergeom([1/2 - k/2, -k/2], [1], sigma^2);

f := (k, n, sigma) -> int((1 - sigma*sin(2*Pi*x))^k, x = 0 .. n):
g := (k, n, sigma) -> n*hypergeom([1/2 - 1/2*k, -1/2*k], [1], sigma^2):

f(11/2, 6, -2/13):
evalf(%);

g(11/2, 6, -2/13):
evalf(%);

Solution only for:  n for Integers !!! 

Regards M.I.

 

Download Integral.mw

Without use rsolve:...

Mariusz Iwaniuk 1571
June 25 2024
0 0

Solution without use rsolve:

N := 30;
B := array(0 .. N):
B[0] := 1:
B[1] := -1/2:
for n from 2 to N do
    B[n] := -add(binomial(n + 1, k)*B[k], k = 0 .. n - 1)/(n + 1):
end do:
[B[i] $ (i = 0 .. N)];

[1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, 0, -691/2730, 0, 7/6, 0, -3617/510, 0, 43867/798, 0, -174611/330, 0, 854513/138, 0, -236364091/2730, 0, 8553103/6, 0, -23749461029/870, 0, 8615841276005/14322]

Analytical solution: for:n >=1 is: -n*Zeta(1 - n).

[seq(-n*Zeta(1 - n), n = 2 .. N)];

[1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, 0, -691/2730, 0, 7/6, 0, -3617/510, 0, 43867/798, 0, -174611/330, 0, 854513/138, 0, -236364091/2730, 0, 8553103/6, 0, -23749461029/870, 0, 8615841276005/14322, 0, -7709321041217/510, 0, 2577687858367/6, 0, -26315271553053477373/1919190, 0, 2929993913841559/6, 0, -261082718496449122051/13530, 0, 1520097643918070802691/1806, 0, -27833269579301024235023/690, 0, 596451111593912163277961/282, 0, -5609403368997817686249127547/46410, 0, 495057205241079648212477525/66]

 Why doesn’t this give me any solution ?, because Maple is not a magic box that'll spit out a solution to any problem..All computer algebra systems, including Maple, are limited in their capabilities.

I doubt there's a closed form for the in...

Mariusz Iwaniuk 1571
May 06 2024
1 0

See attached file.

Download Int.mw

Maybe this little helps...

Mariusz Iwaniuk 1571
May 03 2024
1 0

Download SOL.mw

I doubt there are simpler form....

Mariusz Iwaniuk 1571
April 11 2024
1 0

Another formula:

-(-1 + 2*k)*sqrt(1 - csc(omega*T)^2)*(csc(omega*T)^2)^(-1/2 + k)*int(t^(k - 1)/sqrt(1 - t), t = 0 .. 1/csc(omega*T)^2)/sqrt(-1 + csc(omega*T)^2) + 2*sqrt(Pi)*sqrt(1 - csc(omega*T)^2)*(csc(omega*T)^2)^(-1/2 + k)*GAMMA(k)/(sqrt(-1 + csc(omega*T)^2)*GAMMA(-1/2 + k))

Integral is incomplete beta function ,but Maple dosen't know, solve as hypergeometric function.

I have only:...

Mariusz Iwaniuk 1571
March 15 2024
1 0


 

> 

restart
> 

kernelopts(version)

`Maple 2024.0, X86 64 WINDOWS, Mar 01 2024, Build ID 1794891`

 (1)
> 

F := int(1/(u*sqrt(1+p1*u^2/(2*p2))), u)

-arctanh(2^(1/2)/(2+p1*u^2/p2)^(1/2))

 (2)
> 

G := `assuming`([simplify(F)], [p1 > 0, p2 < 0])

arctanh(2^(1/2)*p2/((p1*u^2+2*p2)*p2)^(1/2))

 (3)
> 

S := solve(G = x-x[0], u)

(-2*p1*p2*(tanh(x-x[0])^2-1))^(1/2)/(p1*tanh(x-x[0])), -(-2*p1*p2*(tanh(x-x[0])^2-1))^(1/2)/(p1*tanh(x-x[0]))

 (4)
> 

`assuming`([simplify({evalc(Im(S[1])), evalc(Im(S[2]))})], [p1 > 0, p2 < 0, x-x[0] > 0])

{2^(1/2)*(-p2)^(1/2)*csch(x-x[0])/p1^(1/2), -2^(1/2)*(-p2)^(1/2)*csch(x-x[0])/p1^(1/2)}

 (5)
> 

NULL


 

Download start_v1.mw

Maple is not figure it out to give solut...

Mariusz Iwaniuk 1571
February 28 2024
1 5
> 

restart;
> 

eq1 := 2^(-m/2-n/2)*exp(lambda^2*sigma^2/4)/sqrt(n!)/sqrt(m!*Pi)*int(HermiteH(m,s+lambda*sigma/2)*HermiteH(n,s+lambda*sigma/2)*exp(-s^2), s=-infinity..infinity);;

2^(-(1/2)*m-(1/2)*n)*exp((1/4)*lambda^2*sigma^2)*(int(HermiteH(m, s+(1/2)*lambda*sigma)*HermiteH(n, s+(1/2)*lambda*sigma)*exp(-s^2), s = -infinity .. infinity))/(factorial(n)^(1/2)*(factorial(m)*Pi)^(1/2))

 (1)
> 

eq2 := 2^(-(1/2)*n+(1/2)*m)*exp((1/4)*lambda^2*sigma^2)*sqrt(factorial(m))*(lambda*sigma)^(-m+n)*LaguerreL(m, -m+n, -(1/2)*lambda^2*sigma^2)/sqrt(factorial(n))
> 

eq1 = eq2

2^(-(1/2)*m-(1/2)*n)*exp((1/4)*lambda^2*sigma^2)*(int(HermiteH(m, s+(1/2)*lambda*sigma)*HermiteH(n, s+(1/2)*lambda*sigma)*exp(-s^2), s = -infinity .. infinity))/(factorial(n)^(1/2)*(factorial(m)*Pi)^(1/2)) = 2^(-(1/2)*n+(1/2)*m)*exp((1/4)*lambda^2*sigma^2)*factorial(m)^(1/2)*(lambda*sigma)^(-m+n)*LaguerreL(m, -m+n, -(1/2)*lambda^2*sigma^2)/factorial(n)^(1/2)

 (2)
> 

NULL
> 

NULL
> 

m := 2; n := 5; lambda := 1/6; sigma := 2/3
> 

evalf[20](eq1)

0.62998679107780885597e-3

 (3)
> 

evalf[20](eq2)

0.62998679107780885597e-3

 (4)
> 

``

Download Q2.mw

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