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Mariusz Iwaniuk

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These are answers submitted by Mariusz Iwaniuk

Use initialvalue....

Mariusz Iwaniuk 1476
September 26 2017
1 5


 

> 

X := [seq(i, i = 0 .. 24)]; Y := [1154, 1156, 1156, 1155, 1152, 1143, 1105, 1069, 1051, 1077, 1117, 1154, 1154, 1156, 1158, 1157, 1155, 1152, 1128, 1089, 1058, 1059, 1092, 1130, 1163]

[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]

  

[1154, 1156, 1156, 1155, 1152, 1143, 1105, 1069, 1051, 1077, 1117, 1154, 1154, 1156, 1158, 1157, 1155, 1152, 1128, 1089, 1058, 1059, 1092, 1130, 1163]

 (1)
> 

with(Statistics)
> 

f := Fit(a*sin(b*x+c)+d, X, Y, x, initialvalues = [a = 50, b = -1/2, c = 2, d = 1100])

-HFloat(48.70292499204853)*sin(HFloat(0.5173979580818461)*x-HFloat(2.5826651644523233))+HFloat(1124.950148688867)

 (2)
> 

plot({f(x), [seq([X[i], Y[i]], i = 1 .. 25)]}, x = 0 .. 24, style = [line, point])

 
> 

``


 

Download fit.mw

Using another initalvalues:

f := Fit(a*sin(b*x+c)+d, X, Y, x, initialvalues = [a = 50, b = 1/2, c = 1/2, d = 1120]);

#f := 48.7029442510649*sin(0.517394221899901*x+0.558920097492524)+1124.95011131587

Maple gives almost the same as Geogebra

Works fine,but where is the problem?...

Mariusz Iwaniuk 1476
September 25 2017
0 0


Maple 2017.2 output:

 

dsolve({B(t)*(diff(B(t), t, t))*A(t)-A(t)*(diff(B(t), t))^2-(diff(A(t), t, t))*B(t)^2+(diff(A(t), t))*B(t)*(diff(B(t), t))-A(t) = 0, diff(A(t), t) = 0});

#[{B(t) = B(t)}, {A(t) = 0}], [{A(t) = _C3}, {B(t) = (1/2)*_C1*(1/((exp(_C2/_C1))^2*(exp(t/_C1))^2)+1)*exp(_C2/_C1)*exp(t/_C1), B(t) = (1/2)*_C1*((exp(_C2/_C1))^2*(exp(t/_C1))^2+1)/(exp(_C2/_C1)*exp(t/_C1))}]

 

......

Mariusz Iwaniuk 1476
September 24 2017
1 0 
Digits := 20;
sol := 126*0.9 = int(14*t*exp(-(1/3)*t), t = 0 .. x);
solve({sol, x > 0}, {x})

#{x = 11.669160509602287174}

As procedure:

UpperLimit := proc (percent) rhs(solve({(126/100)*percent = int(14*t*exp(-(1/3)*t), t = 0 .. x), 0 < x}, x)[1]) end proc;
UpperLimit(90)

#-3-3*LambertW(-1, -(1/10)*exp(-1))

evalf(UpperLimit(90))# 90%

# 11.669160509602287174

evalf(UpperLimit(50))# 50%

#5.0350409700499819602

evalf(UpperLimit(10))# 10%

#1.5954348251688360603

 

Inner and outer Sum...

Mariusz Iwaniuk 1476
September 19 2017
1 0


 

> 

Sum(Sum((t+1)*Q[h]^2*(1-Q[h])^(t+T)/(t+1+(R[h]/S[h])^sigma*(T+1)), t = 0 .. infinity), T = 0 .. infinity)

Sum(Sum((t+1)*Q[h]^2*(1-Q[h])^(t+T)/(t+1+(R[h]/S[h])^sigma*(T+1)), t = 0 .. infinity), T = 0 .. infinity)

 (1)
> 

NULL
> 

func := (t+1)*Q[h]^2*(1-Q[h])^(t+T)/(t+1+(R[h]/S[h])^sigma*(T+1))

(t+1)*Q[h]^2*(1-Q[h])^(t+T)/(t+1+(R[h]/S[h])^sigma*(T+1))

 (2)
> 

s1 := `assuming`([sum(func, t = 0 .. infinity, formal)], [Q[h] > 0, R[h] > 0, S[h] > 0, sigma > 0])

Q[h]^2*(1-Q[h])^T*((R[h]^sigma*T+S[h]^sigma+R[h]^sigma)*S[h]^(-sigma)/Q[h]-R[h]^sigma*(R[h]^sigma*T^2+S[h]^sigma*T+2*R[h]^sigma*T+S[h]^sigma+R[h]^sigma)*(S[h]^(-sigma))^2*LerchPhi(1-Q[h], 1, (R[h]^sigma*T+S[h]^sigma+R[h]^sigma)*S[h]^(-sigma)))/((R[h]/S[h])^sigma*T+(R[h]/S[h])^sigma+1)

 (3)
> 

evalf(Sum(s1, T = 0 .. infinity))

Sum(Q[h]^2*(1-Q[h])^T*((R[h]^sigma*T+S[h]^sigma+R[h]^sigma)*S[h]^(-sigma)/Q[h]-R[h]^sigma*(R[h]^sigma*T^2+S[h]^sigma*T+2*R[h]^sigma*T+S[h]^sigma+R[h]^sigma)*(S[h]^(-sigma))^2*LerchPhi(1-Q[h], 1, (R[h]^sigma*T+S[h]^sigma+R[h]^sigma)*S[h]^(-sigma)))/((R[h]/S[h])^sigma*T+(R[h]/S[h])^sigma+1), T = 0 .. infinity)

 (4)
> 

s2 := `assuming`([sum(func, T = 0 .. infinity, formal)], [Q[h] > 0, R[h] > 0, S[h] > 0, sigma > 0])

(t+1)*Q[h]^2*(1-Q[h])^t*(t*S[h]^sigma+S[h]^sigma+R[h]^sigma)*R[h]^(-sigma)*LerchPhi(1-Q[h], 1, (t*S[h]^sigma+S[h]^sigma+R[h]^sigma)*R[h]^(-sigma))/(t+1+(R[h]/S[h])^sigma)

 (5)
> 

evalf(Sum(s2, t = 0 .. infinity))

Sum((t+1)*Q[h]^2*(1-Q[h])^t*(t*S[h]^sigma+S[h]^sigma+R[h]^sigma)*R[h]^(-sigma)*LerchPhi(1-Q[h], 1, (t*S[h]^sigma+S[h]^sigma+R[h]^sigma)*R[h]^(-sigma))/(t+1+(R[h]/S[h])^sigma), t = 0 .. infinity)

 (6)
> 

NULL


 

Download Sums.mw

Mathematica can solve....

Mariusz Iwaniuk 1476
September 04 2017
0 0

Using SeriesCoefficient function:

Weakness in convert...

Mariusz Iwaniuk 1476
September 03 2017
0 1

Well, Maples convert(func,FPS) or convert(func,Sum) function is not strong enough.

Using Maxima http://maxima.sourceforge.net/  powerseries function:

tanh(x) = Sum((4^n-1)*4^n*bernoulli(2*n)*x^(2*n-1)/factorial(2*n), n = 0 .. infinity)

tanh(x+1) = Sum((4^n-1)*4^n*bernoulli(2*n)*(x+1)^(2*n-1)/factorial(2*n), n = 0 .. infinity)

 

Regards Mariusz

Or symbolic solution....

Mariusz Iwaniuk 1476
June 19 2017
2 0

restart;

k := 5;

EQ := diff(u(x, t), t) = k*(diff(u(x, t), x$2));

ibc := u(0, t) = 0, u(1, t) = 0, u(x, 0) = x;

sol := pdsolve({EQ, ibc});

J := eval(subs(_Z1 = n, sol), infinity = 100);

int(value(eval(rhs(J), x = .5)), t = 0 .. 10)

 

(*0.01249999355*)

Use 'explicit' :...

Mariusz Iwaniuk 1476
June 17 2017
0 1

If You type:

sol := dsolve(diff(ln(y(x)), x) = y(x)^(1/(1-y(x))), y(x), 'implicit')

solution is implicit form.You must type:

sol := dsolve(diff(ln(y(x)), x) = y(x)^(1/(1-y(x))), y(x), 'explicit')

sol := y(x) = RootOf(x-Intat(_a^(-(-2+_a)/(-1+_a)), _a = _Z)+_C1)

form more details see:

https://www.maplesoft.com/support/help/maple/view.aspx?path=dsolve%2fdetails

but You can't solve this integral int(x^x,x) ,symbolic(analitically) solution does not exist yet,maybe in the future.

 

See this :

https://en.wikipedia.org/wiki/Nonelementary_integral

 

Use assume......

Mariusz Iwaniuk 1476
June 16 2017
1 0


 

NULL

> 

restart
> 

`assuming`([int(1/(x^2*sqrt(1-x^4/b^4)), x = a .. b)], [b > a])

(-a^4+b^4)^(1/2)/(b^2*a)+2^(1/2)*signum(b)*EllipticK((1/2)*2^(1/2))/b-2*2^(1/2)*signum(b)*EllipticE((1/2)*2^(1/2))/b-((1/2)*I)*2^(1/2)*signum(b)*EllipticPi((-a^2+b^2)^(1/2)*signum(b)/b, 1, (1/2)*2^(1/2))/b

 (1)
> 

`assuming`([int(1/(x^2*sqrt(1-x^4/b^4)), x = a .. b)], [b > a, b > 0, a > 0])

(-a^4+b^4)^(1/2)/(b^2*a)+(1/2)*2^(1/2)*EllipticF((-a^2+b^2)^(1/2)/b, (1/2)*2^(1/2))/b-2^(1/2)*EllipticE((-a^2+b^2)^(1/2)/b, (1/2)*2^(1/2))/b

 (2)
> 

int(1/(x^2*sqrt(1-x^4/b^4)), x = a .. b, AllSolutions)

piecewise(a < b, piecewise(b = 0, infinity, (EllipticK(I)-EllipticE(I))/b)+piecewise(And(0 < b, a < 0), infinity, 0)+piecewise(And(0 < b, a < -b), -(EllipticF(signum(b), I)-EllipticE(signum(b), I))*(b^(1/2)*(-1/b)^(1/2)-(1/b)^(1/2)*(-b)^(1/2))*signum(b)^2/(b^(1/2)*(-b)^(1/2)), 0)-piecewise(a = 0, -infinity, -b = a, -(1/b)^(1/2)*(EllipticF(signum(b), I)-EllipticE(signum(b), I))*signum(b)^2/b^(1/2), -((-a^2+b^2)^(1/2)*(a^2+b^2)^(1/2)*EllipticE(a/abs(b), I)*signum(b)*a*b-(-a^2+b^2)^(1/2)*(a^2+b^2)^(1/2)*EllipticF(a/abs(b), I)*signum(b)*a*b-a^4+b^4)/((-a^4+b^4)^(1/2)*a*abs(b)^2)), b = a, 0, b < a, -piecewise(a = 0, infinity, -b = a, -(-1/b)^(1/2)*(EllipticF(signum(b), I)-EllipticE(signum(b), I))/(-b)^(1/2), -((-a^2+b^2)^(1/2)*(a^2+b^2)^(1/2)*EllipticE(a/abs(b), I)*signum(b)*a*b-(-a^2+b^2)^(1/2)*(a^2+b^2)^(1/2)*EllipticF(a/abs(b), I)*signum(b)*a*b-a^4+b^4)/((-a^4+b^4)^(1/2)*a*abs(b)^2))-piecewise(And(0 < a, b < 0), infinity, 0)-piecewise(And(b < 0, -b < a), -(EllipticF(signum(b), I)-EllipticE(signum(b), I))*(b^(1/2)*(-1/b)^(1/2)-(1/b)^(1/2)*(-b)^(1/2))*signum(b)^2/(b^(1/2)*(-b)^(1/2)), 0)+piecewise(b = 0, -infinity, (EllipticK(I)-EllipticE(I))/b))

 (3)

``


 

Download Integral_ver2.mw

 

Try this...

Mariusz Iwaniuk 1476
June 14 2017
2 2 
sol4 := fsolve([sol1, sol2], {T = 0...0.5, W = 0...30});

 {T = 0.3216117634, W = 29.46435118}

Mathematica says the same what a Maple.

 

maple_solution.mw

Do it this way:...

Mariusz Iwaniuk 1476
June 10 2017
0 2

 

 eval(sum(sum(x^(q-p), p = 0 .. q), q = 0 .. 10), x = 0)

 

or:

 

eval(value(Sum(Sum(x^(q-p), p = 0 .. q), q = 0 .. 10)), x = 0)

In Maple 2017 maybe like this: J := `a...

Mariusz Iwaniuk 1476
June 05 2017
0 0

In Maple 2017 maybe like this:

J := `assuming`([int(ln(t)^n, t = 0 .. x, AllSolutions = true)], [n::posint])

for n=1:

value(eval(J, n = 1))

is:

x*ln(x)-x

 

Only works for alpha 1 and 2- symbolic s...

Mariusz Iwaniuk 1476
June 02 2017
1 6


 

> 

alpha := 1;

1

 (1)
> 

f := proc (x, y) options operator, arrow; x*y-V^alpha*W/(-a*x-b*y+V)^alpha end proc

proc (x, y) options operator, arrow; y*x-V^alpha*W/(-a*x-b*y+V)^alpha end proc

 (2)
> 

sol := solve([diff(f(x, y), x), diff(f(x, y), y)], {x, y})

{x = b*RootOf(4*_Z^3*b^2-4*V*_Z^2*b+V^2*_Z-V*W*a)/a, y = RootOf(4*_Z^3*b^2-4*V*_Z^2*b+V^2*_Z-V*W*a)}

 (3)
> 

allvalues(sol)

{x = b*((1/6)*(V*(27*W*a*b+3*(-3*W*a*(-27*W*a*b+2*V^2)/b)^(1/2)*b-V^2))^(1/3)/b+(1/6)*V^2/(b*(V*(27*W*a*b+3*(-3*W*a*(-27*W*a*b+2*V^2)/b)^(1/2)*b-V^2))^(1/3))+(1/3)*V/b)/a, y = (1/6)*(V*(27*W*a*b+3*(-3*W*a*(-27*W*a*b+2*V^2)/b)^(1/2)*b-V^2))^(1/3)/b+(1/6)*V^2/(b*(V*(27*W*a*b+3*(-3*W*a*(-27*W*a*b+2*V^2)/b)^(1/2)*b-V^2))^(1/3))+(1/3)*V/b}, {x = b*(-(1/12)*(V*(27*W*a*b+3*(-3*W*a*(-27*W*a*b+2*V^2)/b)^(1/2)*b-V^2))^(1/3)/b-(1/12)*V^2/(b*(V*(27*W*a*b+3*(-3*W*a*(-27*W*a*b+2*V^2)/b)^(1/2)*b-V^2))^(1/3))+(1/3)*V/b+((1/2)*I)*3^(1/2)*((1/6)*(V*(27*W*a*b+3*(-3*W*a*(-27*W*a*b+2*V^2)/b)^(1/2)*b-V^2))^(1/3)/b-(1/6)*V^2/(b*(V*(27*W*a*b+3*(-3*W*a*(-27*W*a*b+2*V^2)/b)^(1/2)*b-V^2))^(1/3))))/a, y = -(1/12)*(V*(27*W*a*b+3*(-3*W*a*(-27*W*a*b+2*V^2)/b)^(1/2)*b-V^2))^(1/3)/b-(1/12)*V^2/(b*(V*(27*W*a*b+3*(-3*W*a*(-27*W*a*b+2*V^2)/b)^(1/2)*b-V^2))^(1/3))+(1/3)*V/b+((1/2)*I)*3^(1/2)*((1/6)*(V*(27*W*a*b+3*(-3*W*a*(-27*W*a*b+2*V^2)/b)^(1/2)*b-V^2))^(1/3)/b-(1/6)*V^2/(b*(V*(27*W*a*b+3*(-3*W*a*(-27*W*a*b+2*V^2)/b)^(1/2)*b-V^2))^(1/3)))}, {x = b*(-(1/12)*(V*(27*W*a*b+3*(-3*W*a*(-27*W*a*b+2*V^2)/b)^(1/2)*b-V^2))^(1/3)/b-(1/12)*V^2/(b*(V*(27*W*a*b+3*(-3*W*a*(-27*W*a*b+2*V^2)/b)^(1/2)*b-V^2))^(1/3))+(1/3)*V/b-((1/2)*I)*3^(1/2)*((1/6)*(V*(27*W*a*b+3*(-3*W*a*(-27*W*a*b+2*V^2)/b)^(1/2)*b-V^2))^(1/3)/b-(1/6)*V^2/(b*(V*(27*W*a*b+3*(-3*W*a*(-27*W*a*b+2*V^2)/b)^(1/2)*b-V^2))^(1/3))))/a, y = -(1/12)*(V*(27*W*a*b+3*(-3*W*a*(-27*W*a*b+2*V^2)/b)^(1/2)*b-V^2))^(1/3)/b-(1/12)*V^2/(b*(V*(27*W*a*b+3*(-3*W*a*(-27*W*a*b+2*V^2)/b)^(1/2)*b-V^2))^(1/3))+(1/3)*V/b-((1/2)*I)*3^(1/2)*((1/6)*(V*(27*W*a*b+3*(-3*W*a*(-27*W*a*b+2*V^2)/b)^(1/2)*b-V^2))^(1/3)/b-(1/6)*V^2/(b*(V*(27*W*a*b+3*(-3*W*a*(-27*W*a*b+2*V^2)/b)^(1/2)*b-V^2))^(1/3)))}

 (4)
> 

``


 

Download Worksheet.mw

for alpha=3,4,5,6,7......

and for alpha>0 ,alpha in real

numeric calculation see worksheet:

Numeric_solution.mw

 

 

 

QDifferenceEquations packed is only do ...

Mariusz Iwaniuk 1476
May 07 2017
0 0

QDifferenceEquations packed is only do algebraic calculation.

From wikipedia: qgamma = (1-q)^(1-z)*(product((1-q^(n+1))/(1-q^(n+z)), n = 0 .. infinity))

 

qgamma := proc (z, q) options operator, arrow; (1-q)^(1-z)*(product((1-q^(n+1))/(1-q^(n+z)), n = 0 .. 1000)) end proc

z := .8; q := .9

evalf(qgamma(z, q)) =1.156991553

qgamma.mw

eq1 := 1 = abs(1/(I*Pi*z*(1+I*Pi*a))); e...

Mariusz Iwaniuk 1476
May 03 2017
0 1

eq1 := 1 = abs(1/(I*Pi*z*(1+I*Pi*a)));
eq2 := -(5/9)*Pi = argument(1/(I*Pi*z*(1+I*Pi*a)));

fsolve([eq1, eq2], {a = 0 .. 1, z = 0 .. 1})

{a = 0.5612662116e-1, z = 0.3134740437}

PS: Mathematica gives symbolic answer:

 

 

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