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Magnetic traps in cold-atom physics...

ecterrab 13431 Maple
physics tensor
October 21 2017
3 0

This presentation is about magnetic traps for neutral particles, first achieved for cold neutrons and nowadays widely used in cold-atom physics. The level is that of undergraduate electrodynamics and tensor calculus courses. Tackling this topic within a computer algebra worksheet as shown below illustrates well the kind of advanced computations that can be done today with the Physics package. A new feature minimizetensorcomponents and related functionality is used along the presentation, that requires the updated Physics library distributed at the Maplesoft R&D Physics webpage.
 

 

Magnetic traps in cold-atom physics

 

Pascal Szriftgiser1 and Edgardo S. Cheb-Terrab2 

(1) Laboratoire PhLAM, UMR CNRS 8523, Université Lille 1, F-59655, France

(2) Maplesoft

 

We consider a device constructed with a set of electrical wires fed with constant electrical currents. Those wires can have an arbitrary complex shape. The device is operated in a regime such that, in some region of interest, the moving particles experience a magnetic field that varies slowly compared to the Larmor spin precession frequency. In this region, the effective potential is proportional to the modulus of the field: LinearAlgebra[Norm](`#mover(mi("B"),mo("→"))`(x, y, z)), this potential has a minimum and, close to this minimum, the device behaves as a magnetic trap.

 

 

 

Figure 1: Schematic representation of a Ioffe-Pritchard magnetic trap. It is made of four infinite rods and two coils.

_________________________________________

 

Following [1], we show that:

 

  

a) For a time-independent magnetic field  `#mover(mi("B"),mo("→"))`(x, y, z) in vacuum, up to order two in the relative coordinates X__i = [x, y, z] around some point of interest, the coefficients of orders 1 and 2 in this expansion, `v__i,j` and `c__i,j,k` , respectively the gradient and curvature, contain only 5 and 7 independent components.
  

b) All stationary points of LinearAlgebra[Norm](`#mover(mi("B"),mo("→"))`(x, y, z))^2 (nonzero minima and saddle points) are confined to a curved surface defined by det(`∂`[j](B[i])) = 0.
  

c) The effective potential, proportional to LinearAlgebra[Norm](`#mover(mi("B"),mo("→"))`(x, y, z)), has no maximum, only a minimum.

 

Finally, we draw the stationary condition surface for the case of the widely used Ioffe-Pritchard magnetic trap.

  

 

  

Reference
  

[1] R. Gerritsma and R. J. C. Spreeuw, Topological constraints on magnetostatic traps,  Phys. Rev. A 74, 043405 (2006)
  

 

The independent components of `v__i,j` and `c__i,j,k` entering B[i] = u[i]+v[i, j]*X[j]+(1/2)*c[i, j, k]*X[j]*X[k]

   

The stationary points are within the surface det(`∂`[j](B[i])) = 0

   

U = LinearAlgebra[Norm](`#mover(mi("B",fontweight = "bold"),mo("→",fontweight = "bold"))`)^2 has only minima, no maxima

   

Drawing the Ioffe-Pritchard Magnetic Trap

   


 

MagneticTraps.mw or in pdf format with the sections open: MagneticTraps.pdf

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Examples of pdsolve improvements in Mapl...

ecterrab 13431 Maple
pde
October 05 2017
6 3


 

Maple 2017.1 and 2017.2 introduced several improvements in the solution of PDE & Boundary conditions problems (exact solutions).  Maple 2017.3 includes more improvements in this same area.

 

The following is a set of 25 examples of different PDE & Boundary Conditions problems that are solvable in Maple 2017.3 but not in Maple 2017.2 or previous releases. In the examples that follow, in some cases the PDE is different, in other cases the boundary conditions are of a different kind or solving the problem involves different computational strategies.

As usual, at the end there is a link pointing to the corresponding worksheet.

 

> 

pde[1] := diff(u(x, t), t) = k*(diff(u(x, t), x, x)); bc[1] := u(x, 0) = 6+4*cos(3*Pi*x/L), (D[1](u))(0, t) = 0, (D[1](u))(L, t) = 0

diff(u(x, t), t) = k*(diff(diff(u(x, t), x), x))

  

u(x, 0) = 6+4*cos(3*Pi*x/L), (D[1](u))(0, t) = 0, (D[1](u))(L, t) = 0

 (1)
> 

`assuming`([pdsolve([pde[1], bc[1]], u(x, t))], [L > 0, k > 0])

u(x, t) = 6+4*cos(3*Pi*x/L)*exp(-9*k*Pi^2*t/L^2)

 (2)
> 

pde[2] := diff(g(t, x), t) = diff(g(t, x), x, x)+a*g(t, x); bc[2] := g(0, x) = 1

diff(g(t, x), t) = diff(diff(g(t, x), x), x)+a*g(t, x)

  

g(0, x) = 1

 (3)
> 

pdsolve([pde[2], bc[2]])

g(t, x) = exp(a*t)

 (4)
> 

pde[3] := diff(u(x, t), t) = diff(u(x, t), x, x)
> 

bc[3] := u(x, 0) = f(x), u(-1, t) = 0, u(1, t) = 0

diff(u(x, t), t) = diff(diff(u(x, t), x), x)

  

u(x, 0) = f(x), u(-1, t) = 0, u(1, t) = 0

 (5)
> 

pdsolve([pde[3], bc[3]], u(x, t))

u(x, t) = Sum((Int(f(x)*sin(n*Pi*x), x = -1 .. 1))*sin(n*Pi*x)*exp(-Pi^2*n^2*t), n = 1 .. infinity)

 (6)
> 

pde[4] := diff(u(x, t), t) = k*(diff(u(x, t), x, x)); bc[4] := u(0, t) = 0, u(L, t) = 0, u(x, 0) = piecewise(0 < x and x <= (1/2)*L, 1, (1/2)*L < x and x < L, 2)

pde[4] := diff(u(x, t), t) = k*(diff(u(x, t), x, x))

  

u(0, t) = 0, u(L, t) = 0, u(x, 0) = piecewise(0 < x and x <= (1/2)*L, 1, (1/2)*L < x and x < L, 2)

 (7)
> 

`assuming`([pdsolve([pde[4], bc[4]], u(x, t))], [L > 0])

u(x, t) = Sum((2*cos((1/2)*Pi*n)+2+4*(-1)^(1+n))*sin(n*Pi*x/L)*exp(-k*Pi^2*n^2*t/L^2)/(Pi*n), n = 1 .. infinity)

 (8)
> 

pde[5] := diff(u(x, t), t) = k*(diff(u(x, t), x, x))
> 

bc[5] := (D[1](u))(0, t) = 0, (D[1](u))(L, t) = 0, u(x, 0) = -3*cos(8*Pi*x/L)

diff(u(x, t), t) = k*(diff(diff(u(x, t), x), x))

  

(D[1](u))(0, t) = 0, (D[1](u))(L, t) = 0, u(x, 0) = -3*cos(8*Pi*x/L)

 (9)
> 

`assuming`([pdsolve([pde[5], bc[5]], u(x, t))], [0 < L, 0 < k])

u(x, t) = -3*cos(8*Pi*x/L)*exp(-64*k*Pi^2*t/L^2)

 (10)
> 

pde[6] := diff(u(x, t), t) = k*(diff(u(x, t), x, x))+f(x, t); bc[6] := u(0, t) = 0, u(l, t) = 0, u(x, 0) = g(x)

diff(u(x, t), t) = k*(diff(diff(u(x, t), x), x))+f(x, t)

  

u(0, t) = 0, u(l, t) = 0, u(x, 0) = g(x)

 (11)
> 

pdsolve([pde[6], bc[6]], u(x, t))

u(x, t) = Sum(2*(Int(g(tau1)*sin(Pi*n1*tau1/l), tau1 = 0 .. l))*sin(Pi*n1*x/l)*exp(-k*Pi^2*n1^2*t/l^2)/l, n1 = 1 .. infinity)+Int(Sum(2*(Int(f(x, tau1)*sin(Pi*n*x/l), x = 0 .. l))*sin(Pi*n*x/l)*exp(-k*Pi^2*n^2*(t-tau1)/l^2)/l, n = 1 .. infinity), tau1 = 0 .. t)

 (12)
> 

pde[7] := diff(u(x, t), t) = diff(u(x, t), x, x); bc[7] := u(x, 0) = f(x), u(-1, t) = 0, u(1, t) = 0

diff(u(x, t), t) = diff(diff(u(x, t), x), x)

  

u(x, 0) = f(x), u(-1, t) = 0, u(1, t) = 0

 (13)
> 

pdsolve([pde[7], bc[7]], u(x, t))

u(x, t) = Sum((Int(f(x)*sin(n*Pi*x), x = -1 .. 1))*sin(n*Pi*x)*exp(-Pi^2*n^2*t), n = 1 .. infinity)

 (14)
> 

pde[8] := diff(u(x, t), t) = k*(diff(u(x, t), x, x))-h*u(x, t); bc[8] := u(x, 0) = sin(x), u(-Pi, t) = u(Pi, t), (D[1](u))(-Pi, t) = (D[1](u))(Pi, t)

diff(u(x, t), t) = k*(diff(diff(u(x, t), x), x))-h*u(x, t)

  

u(x, 0) = sin(x), u(-Pi, t) = u(Pi, t), (D[1](u))(-Pi, t) = (D[1](u))(Pi, t)

 (15)
> 

pdsolve([pde[8], bc[8]], u(x, t))

u(x, t) = sin(x)*exp(-t*(k+h))

 (16)
> 

pde[9] := diff(u(x, t), t) = diff(u(x, t), x, x)
> 

bc[9] := u(0, t) = 20, u(1, t) = 50, u(x, 0) = 0

diff(u(x, t), t) = diff(diff(u(x, t), x), x)

  

u(0, t) = 20, u(1, t) = 50, u(x, 0) = 0

 (17)
> 

pdsolve([pde[9], bc[9]], u(x, t))

u(x, t) = 20+Sum((-40+100*(-1)^n)*sin(n*Pi*x)*exp(-Pi^2*n^2*t)/(Pi*n), n = 1 .. infinity)+30*x

 (18)
> 

pde[10] := diff(u(x, y), x, x)+diff(u(x, y), y, y) = 0; bc[10] := u(x, 0) = 0, u(x, 1) = 0, u(0, y) = y^2-y, u(1, y) = 0

diff(diff(u(x, y), x), x)+diff(diff(u(x, y), y), y) = 0

  

u(x, 0) = 0, u(x, 1) = 0, u(0, y) = y^2-y, u(1, y) = 0

 (19)
> 

pdsolve([pde[10], bc[10]], u(x, y))

u(x, y) = Sum(-4*((-1)^n-1)*sin(Pi*y*n)*(exp(Pi*n*(2*x-1))-exp(Pi*n))*exp(-Pi*n*(x-1))/((exp(2*Pi*n)-1)*Pi^3*n^3), n = 1 .. infinity)

 (20)
> 

pde[11] := diff(u(x, y), x, x)+diff(u(x, y), y, y) = 0
> 

bc[11] := (D[1](u))(0, y) = 0, (D[1](u))(L, y) = 0, u(x, H) = f(x), u(x, 0) = 0

diff(diff(u(x, y), x), x)+diff(diff(u(x, y), y), y) = 0

  

(D[1](u))(0, y) = 0, (D[1](u))(L, y) = 0, u(x, H) = f(x), u(x, 0) = 0

 (21)
> 

`assuming`([pdsolve([pde[11], bc[11]], u(x, y))], [0 < L, 0 < H])

u(x, y) = Sum(2*(Int(cos(Pi*x*n/L)*f(x), x = 0 .. L))*cos(Pi*x*n/L)*exp(Pi*n*(H-y)/L)*(exp(2*Pi*y*n/L)-1)/(L*(exp(2*Pi*H*n/L)-1)), n = 1 .. infinity)

 (22)
> 

pde[12] := diff(u(x, y), x, x)+diff(u(x, y), y, y) = 0
> 

bc[12] := (D[1](u))(L, y) = 0, u(x, H) = 0, u(x, 0) = 0, (D[1](u))(0, y) = g(y)

diff(diff(u(x, y), x), x)+diff(diff(u(x, y), y), y) = 0

  

(D[1](u))(L, y) = 0, u(x, H) = 0, u(x, 0) = 0, (D[1](u))(0, y) = g(y)

 (23)
> 

`assuming`([pdsolve([pde[12], bc[12]], u(x, y))], [0 < x, x <= L, 0 < y, y <= H])

u(x, y) = Sum(-2*(Int(sin(Pi*y*n/H)*g(y), y = 0 .. H))*sin(Pi*y*n/H)*(exp(-Pi*n*(L-2*x)/H)+exp(Pi*L*n/H))*exp(Pi*n*(L-x)/H)/(Pi*n*(exp(2*Pi*L*n/H)-1)), n = 1 .. infinity)

 (24)
> 

pde[13] := diff(u(x, y), x, x)+diff(u(x, y), y, y) = 0
> 

bc[13] := (D[1](u))(0, y) = 0, u(x, 0) = 0, u(x, H) = 0, u(L, y) = g(y)

diff(diff(u(x, y), x), x)+diff(diff(u(x, y), y), y) = 0

  

(D[1](u))(0, y) = 0, u(x, 0) = 0, u(x, H) = 0, u(L, y) = g(y)

 (25)
> 

`assuming`([pdsolve([pde[13], bc[13]], u(x, y))], [0 < L, 0 < H])

u(x, y) = Sum(2*(Int(sin(Pi*y*n/H)*g(y), y = 0 .. H))*sin(Pi*y*n/H)*exp(Pi*n*(L-x)/H)*(exp(2*Pi*x*n/H)+1)/(H*(exp(2*Pi*L*n/H)+1)), n = 1 .. infinity)

 (26)
> 

pde[14] := diff(u(x, y), x, x)+diff(u(x, y), y, y) = 0
> 

bc[14] := u(0, y) = g(y), u(L, y) = 0, (D[2](u))(x, 0) = 0, u(x, H) = 0

diff(diff(u(x, y), x), x)+diff(diff(u(x, y), y), y) = 0

  

u(0, y) = g(y), u(L, y) = 0, (D[2](u))(x, 0) = 0, u(x, H) = 0

 (27)
> 

`assuming`([pdsolve([pde[14], bc[14]], u(x, y))], [0 < x, x <= L, 0 < y, y <= H])

u(x, y) = Sum(-2*(exp(-(L-2*x)*(1/2+n)*Pi/H)-exp((1/2)*Pi*(1+2*n)*L/H))*cos((1/2)*Pi*(1+2*n)*y/H)*(Int(cos((1/2)*Pi*(1+2*n)*y/H)*g(y), y = 0 .. H))*exp((1/2)*Pi*(1+2*n)*(L-x)/H)/(H*(exp(Pi*(1+2*n)*L/H)-1)), n = 0 .. infinity)

 (28)
> 

pde[15] := diff(u(x, y), x, x)+diff(u(x, y), y, y) = 0
> 

bc[15] := u(0, y) = 0, u(L, y) = 0, u(x, 0) = (D[2](u))(x, 0), u(x, H) = f(x)

diff(diff(u(x, y), x), x)+diff(diff(u(x, y), y), y) = 0

  

u(0, y) = 0, u(L, y) = 0, u(x, 0) = (D[2](u))(x, 0), u(x, H) = f(x)

 (29)
> 

`assuming`([pdsolve([pde[15], bc[15]], u(x, y))], [0 < x, x <= L, 0 < y, y <= H])

u(x, y) = Sum(2*exp(Pi*n*(H-y)/L)*sin(Pi*x*n/L)*((Pi*n+L)*exp(2*Pi*y*n/L)+Pi*n-L)*(Int(sin(Pi*x*n/L)*f(x), x = 0 .. L))/(L*((Pi*n+L)*exp(2*Pi*H*n/L)+Pi*n-L)), n = 1 .. infinity)

 (30)
> 

pde[16] := diff(u(x, t), t, t) = c^2*(diff(u(x, t), x, x))
> 

bc[16] := u(0, t) = 0, (D[1](u))(L, t) = 0, (D[2](u))(x, 0) = 0, u(x, 0) = f(x)

diff(diff(u(x, t), t), t) = c^2*(diff(diff(u(x, t), x), x))

  

u(0, t) = 0, (D[1](u))(L, t) = 0, (D[2](u))(x, 0) = 0, u(x, 0) = f(x)

 (31)
> 

`assuming`([pdsolve([pde[16], bc[16]], u(x, t))], [0 < x, x <= L])

u(x, t) = Sum(2*cos((1/2)*c*Pi*(1+2*n)*t/L)*(Int(sin((1/2)*Pi*(1+2*n)*x/L)*f(x), x = 0 .. L))*sin((1/2)*Pi*(1+2*n)*x/L)/L, n = 0 .. infinity)

 (32)
> 

pde[17] := diff(w(x1, x2, x3, t), t) = diff(w(x1, x2, x3, t), x1, x1)+diff(w(x1, x2, x3, t), x2, x2)+diff(w(x1, x2, x3, t), x3, x3)
> 

bc[17] := w(x1, x2, x3, 0) = exp(x1)+x2*x3^5

diff(w(x1, x2, x3, t), t) = diff(diff(w(x1, x2, x3, t), x1), x1)+diff(diff(w(x1, x2, x3, t), x2), x2)+diff(diff(w(x1, x2, x3, t), x3), x3)

  

w(x1, x2, x3, 0) = exp(x1)+x2*x3^5

 (33)
> 

pdsolve([pde[17], bc[17]])

w(x1, x2, x3, t) = Sum(t^n*((proc (U) options operator, arrow; diff(diff(U, x1), x1)+diff(diff(U, x2), x2)+diff(diff(U, x3), x3) end proc)@@n)(exp(x1)+x2*x3^5)/factorial(n), n = 0 .. infinity)

 (34)
> 

pde[18] := diff(u(x, t), t)+u(x, t)*(diff(u(x, t), x)) = -x
> 

bc[18] := u(x, 0) = x

diff(u(x, t), t)+u(x, t)*(diff(u(x, t), x)) = -x

  

u(x, 0) = x

 (35)
> 

pdsolve([pde[18], bc[18]], u(x, t))

u(x, t) = x/tan(t+(1/4)*Pi)

 (36)
> 

pde[19] := diff(u(x, t), t)-u(x, t)^2*(diff(u(x, t), x)) = 3*u(x, t)
> 

bc[19] := u(x, 0) = x

diff(u(x, t), t)-u(x, t)^2*(diff(u(x, t), x)) = 3*u(x, t)

  

u(x, 0) = x

 (37)
> 

pdsolve([pde[19], bc[19]], u(x, t))

u(x, t) = -exp(3*t)*((-6*exp(6*t)*x+6*x+9)^(1/2)-3)/(exp(6*t)-1)

 (38)
> 

pde[20] := diff(u(x, t), t)-x*u(x, t)*(diff(u(x, t), x)) = 0
> 

bc[20] := u(x, 0) = x

diff(u(x, t), t)-x*u(x, t)*(diff(u(x, t), x)) = 0

  

u(x, 0) = x

 (39)
> 

pdsolve([pde[20], bc[20]], u(x, t))

u(x, t) = -LambertW(-t*x)/t

 (40)
> 

pde[21] := diff(u(x, t), t)+u(x, t)*(diff(u(x, t), x)) = 0
> 

bc[21] := u(x, 0) = x

diff(u(x, t), t)+u(x, t)*(diff(u(x, t), x)) = 0

  

u(x, 0) = x

 (41)
> 

pdsolve([pde[21], bc[21]], u(x, t))

u(x, t) = x/(t+1)

 (42)
> 

pde[22] := (t+u(x, t))*(diff(u(x, t), x))+t*(diff(u(x, t), t)) = 0; bc[22] := u(x, 1) = x

(t+u(x, t))*(diff(u(x, t), x))+t*(diff(u(x, t), t)) = 0

  

u(x, 1) = x

 (43)
> 

pdsolve([pde[22], bc[22]], u(x, t))

u(x, t) = (t-x-1)/(ln(1/t)-1)

 (44)
> 

pde[23] := (diff(r*(diff(u(r, theta), r)), r))/r+(diff(u(r, theta), theta, theta))/r^2 = 0; bc[23] := u(a, theta) = f(theta), u(r, -Pi) = u(r, Pi), (D[2](u))(r, -Pi) = (D[2](u))(r, Pi)

(diff(u(r, theta), r)+r*(diff(diff(u(r, theta), r), r)))/r+(diff(diff(u(r, theta), theta), theta))/r^2 = 0

  

u(a, theta) = f(theta), u(r, -Pi) = u(r, Pi), (D[2](u))(r, -Pi) = (D[2](u))(r, Pi)

 (45)
> 

`assuming`([pdsolve([pde[23], bc[23]], u(r, theta), HINT = boundedseries)], [a > 0])

u(r, theta) = (1/2)*(2*(Sum(r^n*(sin(n*theta)*(Int(f(theta)*sin(n*theta), theta = -Pi .. Pi))+cos(n*theta)*(Int(f(theta)*cos(n*theta), theta = -Pi .. Pi)))*a^(-n)/Pi, n = 1 .. infinity))*Pi+Int(f(theta), theta = -Pi .. Pi))/Pi

 (46)
> 

pde[24] := diff(g(t, x), t) = diff(g(t, x), x, x)+a*g(t, x); bc[24] := g(0, x) = f(x)

diff(g(t, x), t) = diff(diff(g(t, x), x), x)+a*g(t, x)

  

g(0, x) = f(x)

 (47)
> 

pdsolve([pde[24], bc[24]])

g(t, x) = exp(a*t)*invfourier(fourier(f(x), x, s1)*exp(-t*s1^2), s1, x)

 (48)
> 

pde[25] := diff(u(x, y), x, x)+diff(u(x, y), y, y) = 0; bc[25] := u(0, y) = y*(-y+1), u(1, y) = 0, (D[2](u))(x, 0) = 0, (D[2](u))(x, 1) = 0

diff(diff(u(x, y), x), x)+diff(diff(u(x, y), y), y) = 0

  

u(0, y) = y*(-y+1), u(1, y) = 0, (D[2](u))(x, 0) = 0, (D[2](u))(x, 1) = 0

 (49)
> 

pdsolve([pde[25], bc[25]], u(x, y))

u(x, y) = Sum(2*((-1)^n+1)*cos(Pi*y*n)*(exp(Pi*n*(2*x-1))-exp(Pi*n))*exp(-Pi*n*(x-1))/((exp(2*Pi*n)-1)*Pi^2*n^2), n = 1 .. infinity)

 (50)
> 

``


 

Download ImprovementsInPdsolve.mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Nobel prize in physics 2017 awarded for ...

ecterrab 13431 Maple
education
October 03 2017
2 4

And the Nobel prize in physics 2017 went for work in General Relativity ! Actually, experimental work involving sophisticated detectors and Numerical Relativity, one of the branches of GR. The prize was awarded to Rainer Weiss (85 years old, 1/2 of the prize), Barry Barish (81 years old, 1/4 of the prize) and Kip Thorne (77 years old, 1/4 of the prize) who have "shaken the world again" with their work on Ligo experiment, which was able to detect ripples in the fabric of spacetime.

General Relativity continues to be at the center of work in theoretical and experimental physics. I take this opportunity to note that, in Maple 2017, among the several improvements in the Physics package regarding General Relativity, there is a new package, Physics:-ThreePlusOne, all dedicated to the symbolic manipulations necessary to formulate problems in Numerical Relativity.

The GR functionality implemented in Physics, Physics:-Tetrads and Physics:-ThreePlusOne is unique in computer algebra systems and reflects the Maplesoft intention, for several years now, to provide the very best possible computer algebra environment for Physics, regarding current research activity as well as related education in advanced mathematical-physics methods.

For what is going on in theoretical physics nowadays and its connection with General Relativity, in very short: the unification of gravity with the other forces, check for instance this map from Aug/2015 (by the way a very nice summary for whoever is interested):

It is also interesting the article behind this map of topics as well as this brilliant and accessible presentation by Nima Arkani-Hamed (Princeton):  Quantum Mechanics and Spacetime in the 21st Century, given in the Perimeter Institute for Theoretical Physics (Waterloo), November 2014. 

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

ACA 2017 - The Appell doubly hypegeometr...

ecterrab 13431 Maple
numeric
August 02 2017
3 0

I'm back from presenting work in the "23rd Conference on Applications of Computer Algebra -2017" . It was a very interesting event. This fifth presentation, about "The Appell doubly hypergeometric functions", describes a very recent project I've been working at Maple, i.e. the very first complete computational implementation of the Appell doubly hypergeometric functions. This work appeared in Maple 2017. These functions have a tremendous potential in that, at the same time, they have a myriad of properties, and include as particular cases most of the existing mathematical language, and so they have obvious applications in integration, differential equations, and applied mathematics all around. I think these will be the functions of this XXI century, analogously to what happened with hypergeometric functions in the previous century.

At the end, there is a link to the presentation worksheet, with which one could open the sections and reproduce the presentation examples.
 

The four double-hypergeometric Appell functions,

a complete implementation in a computer algebra system

 

Edgardo S. Cheb-Terrab

Physics, Differential Equations and Mathematical Functions, Maplesoft

 

Abstract:
The four multi-parameter Appell functions, AppellF1 , AppellF2 , AppellF3  and AppellF4  are doubly hypergeometric functions that include as particular cases the 2F1 hypergeometric  and some cases of the MeijerG  function, and with them most of the known functions of mathematical physics. Appell functions have been popping up with increasing frequency in applications in quantum mechanics, molecular physics, and general relativity. In this talk, a full implementation of these functions in the Maple computer algebra system, including, for the first time, their numerical evaluation over the whole complex plane, is presented, with details about the symbolic and numerical strategies used.

Appell Functions (symbolic)

 

 

The main references:

• 

P. Appel, J.Kamke de Feriet, "Fonctions hypergeometriques et Hyperspheriques", 1926

• 

H. Srivastava, P.W. Karlsson, "Multiple Gaussian Hypergeometric Series", 1985

• 

24 papers in the literature, ranging from 1882 to 2015

 

Definition and Symmetries

   

Polynomial and Singular Cases

   

Single Power Series with Hypergeometric Coefficients

   

Analytic Extension from the Appell Series to the Appell Functions

   

Euler-Type and Contiguity Identities

   

Appell Differential Equations

   

Putting all together

   

Problem: some formulas in the literature are wrong or miss the conditions indicating when are they valid (exchange with the Mathematics director of the DLMF - NIST)

   

Appell Functions (numeric)

 

 

Goals

 

• 

Compute these Appell functions over the whole complex plane
• 

Considering that this is a research problem, implement different methods and flexible optional arguments to allow for:

a) comparison between methods (both performance and correctness),

b) investigation of a single method in different circumstances.

• 

Develop a computational structure that can be reused with other special functions (abstract code and provide the main options), and that could also be translated to C (so: only one numerical implementation, not 100 special function numerical implementations)

Limitation: the Maple original evalf command does not accept optional arguments

 

The cost of numerically evaluating an Appell function

 

• 

If it is a special hypergeometric case, then between 1 to 2 hypergeometric functions

• 

Next simplest case (series/recurrence below) 3 to 4 hypergeometric functions plus adding somewhat large formulas that involve only arithmetic operations up to 20,000 times (frequently less than 100 times)

• 

Next simplest case: the formulas themselves are power series with hypergeometric function coefficients; these cases frequently converge rapidly but may involve the numerical evaluation of up to hundreds of hypergeometric functions to get the value of a single Appell function.

 

Strategy for the numerical evaluation of Appell functions (or other functions ...)

 

 

The numerical evaluation flows orderly according to:

1) check whether it is a singular case

2) check whether it is a special value

3) compute the value using a series derived from a recurrence related to the underlying ODE

4) perform an sum using an infinite sum formula, checking for convergence

5) perform the numerical integration of the ODE underlying the given Appell function

6) perform a sequence of concatenated Taylor series expansions

Examples

   

Series/recurrence

   

Numerical integration of an underlying differential equation (ODEs and dsolve/numeric)

   

Concatenated Taylor series expansions covering the whole complex plane

   

Subproducts

 

Improvements in the numerical evaluation of hypergeometric functions

   

Evalf: an organized structure to implement the numerical evaluation of special functions in general

   

To be done

   


 

Download Appell_Functions.mw   
Download Appell_Functions.pdf

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

ACA 2017 - The FunctionAdvisor, beyond a...

ecterrab 13431 Maple
August 02 2017
2 1

I'm back from presenting work in the "23rd Conference on Applications of Computer Algebra -2017" . It was a very interesting event. This fourth presentation, about "The FunctionAdvisor: extending information on mathematical functions with computer algebra algorithms", describes the FunctionAdvisor project at Maple, a project I started working during 1998, where the key idea I am trying to explore is that we do not need to collect a gazillion of formulas but just core blocks of mathematical information surrounded by clouds of algorithms able to derive extended information from them. In this sense this is also unique piece of software: it can derive properties for rather general algebraic expressions, not just well known tabulated functions. The examples illustrate the idea.

At the end, there is a link to the presentation worksheet, with which one could open the sections and reproduce the presentation examples.
 

The FunctionAdvisor: extending information on mathematical functions

with computer algebra algorithms

 

Edgardo S. Cheb-Terrab

Physics, Differential Equations and Mathematical Functions, Maplesoft

 

Abstract:

A shift in paradigm is happening, from: encoding information into a database, to: encoding essential blocks of information together with algorithms within a computer algebra system. Then, the information is not only searchable but can also be recreated in many different ways and actually used to compute. This talk focuses on this shift in paradigm over a real case example: the digitizing of information regarding mathematical functions as the FunctionAdvisor project of the Maple computer algebra system.

The FunctionAdvisor (basic)

   

Beyond the concept of a database

 
  

" Mathematical functions, are defined by algebraic expressions. So consider algebraic expressions in general ..."

Formal power series for algebraic expressions

   

Differential polynomial forms for algebraic expressions

   

Branch cuts for algebraic expressions

   

The nth derivative problem for algebraic expressions

   

Conversion network for mathematical and algebraic expressions

   

References

   


 

Download FunctionAdvisor.mw

Download FunctionAdvisor.pdf

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

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