Mariusz Iwaniuk

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These are replies submitted by Mariusz Iwaniuk

@briceM 

 

The integral is divergent for a>0,N>0,sigma>0.


Do you have any reason to think there is a closed form? Most integrals don't have one. Maybe the best you can do is numerical methods.

@Mohamed19 

I think that that does not have finite closed-form expression in terms of very large class of special functions.

@Mohamed19 

For first question function j does not apears,but give correct values.

For second question see below:


 

restart

J := proc (alpha, u) options operator, arrow; simplify(sum(GAMMA(alpha+1)*(-1)^m*((1/2)*u)^(2*m)/(factorial(m)*GAMMA(alpha+m+1)), m = 0 .. infinity)) end proc

proc (alpha, u) options operator, arrow; simplify(sum(GAMMA(alpha+1)*(-1)^m*((1/2)*u)^(2*m)/(factorial(m)*GAMMA(alpha+m+1)), m = 0 .. infinity)) end proc

(1)

eval(J(alpha, u), u = sqrt(u^2+v^2-2*uv*cos(phi)))

BesselJ(alpha, (u^2+v^2-2*uv*cos(phi))^(1/2))*GAMMA(alpha+1)*2^alpha*((u^2+v^2-2*uv*cos(phi))^(1/2))^(-alpha)

(2)

simplify(diff(BesselJ(alpha, (u^2+v^2-2*uv*cos(phi))^(1/2))*GAMMA(alpha+1)*2^alpha*((u^2+v^2-2*uv*cos(phi))^(1/2))^(-alpha), u))

-u*GAMMA(alpha+1)*BesselJ(alpha+1, (u^2+v^2-2*uv*cos(phi))^(1/2))*((1/4)*u^2+(1/4)*v^2-(1/2)*uv*cos(phi))^(-(1/2)*alpha)/(u^2+v^2-2*uv*cos(phi))^(1/2)

(3)

``


 

Download BesselI_n-th_derivative_3.mw

@Mohamed19 


 

restart

J := proc (alpha, u) options operator, arrow; simplify(sum(GAMMA(alpha+1)*(-1)^m*((1/2)*u)^(2*m)/(factorial(m)*GAMMA(alpha+m+1)), m = 0 .. infinity)) end proc

proc (alpha, u) options operator, arrow; simplify(sum(GAMMA(alpha+1)*(-1)^m*((1/2)*u)^(2*m)/(factorial(m)*GAMMA(alpha+m+1)), m = 0 .. infinity)) end proc

(1)

J(alpha, u)

BesselJ(alpha, u)*GAMMA(alpha+1)*2^alpha*u^(-alpha)

(2)

NULL

evalf(eval(diff(J(alpha, u), u), [u = 1, alpha = 1]))

-.2298069698

(3)

evalf(eval(-u*J(alpha+1, u)/(2*(alpha+1)), [u = 1, alpha = 1]))

-.2298069698

(4)

Diff('J(alpha, u)', `$`(u, n)) = sum(simplify(diff(GAMMA(alpha+1)*(-1)^m*((1/2)*u)^(2*m)/(factorial(m)*GAMMA(alpha+m+1)), `$`(u, n))), m = 0 .. infinity)

Diff(J(alpha, u), `$`(u, n)) = hypergeom([1/2, 1], [alpha+1, 1-(1/2)*n, -(1/2)*n+1/2], -(1/4)*u^2)/(u^n*GAMMA(-n+1))

(5)

eval(eval(rhs(Diff(J(alpha, u), `$`(u, n)) = hypergeom([1/2, 1], [alpha+1, 1-(1/2)*n, -(1/2)*n+1/2], -(1/4)*u^2)/(u^n*GAMMA(-n+1))), [alpha = 1, u = 1]), n = .999999999)

-.2298069689

(6)

eval(eval(rhs(Diff(J(alpha, u), `$`(u, n)) = hypergeom([1/2, 1], [alpha+1, 1-(1/2)*n, -(1/2)*n+1/2], -(1/4)*u^2)/(u^n*GAMMA(-n+1))), [alpha = 1, u = 1]), n = 1.000000001)

-.2298069709

(7)

limit(eval(rhs(Diff(J(alpha, u), `$`(u, n)) = hypergeom([1/2, 1], [alpha+1, 1-(1/2)*n, -(1/2)*n+1/2], -(1/4)*u^2)/(u^n*GAMMA(-n+1))), [alpha = 1, u = 1]), n = 1)

limit(hypergeom([1/2, 1], [2, 1-(1/2)*n, -(1/2)*n+1/2], -1/4)/GAMMA(-n+1), n = 1)

(8)

plot(hypergeom([1/2, 1], [2, 1-(1/2)*n, -(1/2)*n+1/2], -1/4)/GAMMA(-n+1), n = 1/2 .. 3/2)

 

``


 

Download BesselI_n-th_derivative_2.mw

@NickH 

Try this.

@Oliveira 

Try add evalf command:

evalf(eval(-Gradient(uval(r, t), 'cylindrical'[r, theta, z]), [r = 2, t = 4]));

 

I don't know why there is a problem here,but you use workaround:

eval(solve({-1 < b, a < -1, b < 0, -z < a}), z = infinity);

#{-1 < b, a < -1, b < 0, -a < infinity}

It works for me (see the attached file).

plot2019.1.mw

WHY2019.1.mw

Maybe this helps

@Carl Love 

 

I honestly don't know why it worked.

First I tried Maple help: ?dsolve,system and I use option: useInt , but it dosen't work(bug or what?).If I use useint it worked.That's all.

@Bernard Afful 

Maybe this helps.

@minhthien2016 

 

restart;

Digits := 20;

sol := solve(simplify(expand(convert(sin(9*x - 1/3*Pi) = sin(5*x - 1/6*Pi), exp))), [x], explicit, allsolutions);

seq([x = simplify(allvalues(identify(evalf(simplify(Re(rhs(sol[j][1])))), all)))], j = 1 .. nops(sol));

#[x = Pi*_Z7 + 1/4*Pi], [x = Pi*_Z6 + 1/24*Pi], [x = Pi*_Z6 - 11/24*Pi], [x = Pi*_Z5 - 1/28*Pi], [x = Pi*_Z5 + 3/28*Pi], [x = #Pi*_Z5 + 11/28*Pi], [x = Pi*_Z5 - 13/28*Pi], [x = Pi*_Z5 - 9/28*Pi], [x = Pi*_Z5 - 5/28*Pi]

@Carl Love 

I copy and paste code to Maple and dosen't work?

@Carl Love 

It make sense to me.I don't  mean a discrete derivative.

@vv 

Yes I want differentiate with respect to a natural number.

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