Legendre陪関数

P:=(x,n,m)->(-1)^m*(1-x^2)^(m/2)*diff(1/2^n/n!*diff((x^2-1)^n,x$n),x$m);

(x, n, m) -> (-1)^m*(1 - x^2)^(m/2)*diff(((1/2^n)/n!)*diff((x^2 - 1)^n, x $ n), x $ m)

nとmを変化させてみます。

for j from 0 to 5 do
  for i from 0 to j do
    print(hold(P)(x,j,i)=expand(P(x,j,i)));
  end_for
end_for

P(x, 0, 0) = 1
P(x, 1, 0) = x
P(x, 1, 1) = -(1 - x^2)^(1/2)
P(x, 2, 0) = (3*x^2)/2 - 1/2
P(x, 2, 1) = -3*x*(1 - x^2)^(1/2)
P(x, 2, 2) = 3 - 3*x^2
P(x, 3, 0) = (5*x^3)/2 - (3*x)/2
P(x, 3, 1) = (3*(1 - x^2)^(1/2))/2 - (15*x^2*(1 - x^2)^(1/2))/2
P(x, 3, 2) = 15*x - 15*x^3
P(x, 3, 3) = -15*(1 - x^2)^(3/2)
P(x, 4, 0) = (35*x^4)/8 - (15*x^2)/4 + 3/8
P(x, 4, 1) = (15*x*(1 - x^2)^(1/2))/2 - (35*x^3*(1 - x^2)^(1/2))/2
P(x, 4, 2) = - (105*x^4)/2 + 60*x^2 - 15/2
P(x, 4, 3) = -105*x*(1 - x^2)^(3/2)
P(x, 4, 4) = 105*x^4 - 210*x^2 + 105
P(x, 5, 0) = (63*x^5)/8 - (35*x^3)/4 + (15*x)/8
P(x, 5, 1) = (105*x^2*(1 - x^2)^(1/2))/4 - (315*x^4*(1 - x^2)^(1/2))/8 - (15*(1 - x^2)^(1/2))/8
P(x, 5, 2) = - (315*x^5)/2 + 210*x^3 - (105*x)/2
P(x, 5, 3) = (105*(1 - x^2)^(3/2))/2 - (945*x^2*(1 - x^2)^(3/2))/2
P(x, 5, 4) = 945*x^5 - 1890*x^3 + 945*x
P(x, 5, 5) = -945*(1 - x^2)^(5/2)

グラフにしました。

for j from 0 to 5 do
  for i from 0 to j do
    f:=expand(P(x,j,i)):
    print(hold(P)(x,j,i)=f);
    plot(plot::Function2d(f,x=-1..1,Color=RGB::Blue));
  end_for
end_for

P(x, 0, 0) = 1
MuPAD graphics
P(x, 1, 0) = x
MuPAD graphics
P(x, 1, 1) = -(1 - x^2)^(1/2)
MuPAD graphics
P(x, 2, 0) = (3*x^2)/2 - 1/2
MuPAD graphics
P(x, 2, 1) = -3*x*(1 - x^2)^(1/2)
MuPAD graphics
P(x, 2, 2) = 3 - 3*x^2
MuPAD graphics
P(x, 3, 0) = (5*x^3)/2 - (3*x)/2
MuPAD graphics
P(x, 3, 1) = (3*(1 - x^2)^(1/2))/2 - (15*x^2*(1 - x^2)^(1/2))/2
MuPAD graphics
P(x, 3, 2) = 15*x - 15*x^3
MuPAD graphics
P(x, 3, 3) = -15*(1 - x^2)^(3/2)
MuPAD graphics
P(x, 4, 0) = (35*x^4)/8 - (15*x^2)/4 + 3/8
MuPAD graphics
P(x, 4, 1) = (15*x*(1 - x^2)^(1/2))/2 - (35*x^3*(1 - x^2)^(1/2))/2
MuPAD graphics
P(x, 4, 2) = - (105*x^4)/2 + 60*x^2 - 15/2
MuPAD graphics
P(x, 4, 3) = -105*x*(1 - x^2)^(3/2)
MuPAD graphics
P(x, 4, 4) = 105*x^4 - 210*x^2 + 105
MuPAD graphics
P(x, 5, 0) = (63*x^5)/8 - (35*x^3)/4 + (15*x)/8
MuPAD graphics
P(x, 5, 1) = (105*x^2*(1 - x^2)^(1/2))/4 - (315*x^4*(1 - x^2)^(1/2))/8 - (15*(1 - x^2)^(1/2))/8
MuPAD graphics
P(x, 5, 2) = - (315*x^5)/2 + 210*x^3 - (105*x)/2
MuPAD graphics
P(x, 5, 3) = (105*(1 - x^2)^(3/2))/2 - (945*x^2*(1 - x^2)^(3/2))/2
MuPAD graphics
P(x, 5, 4) = 945*x^5 - 1890*x^3 + 945*x
MuPAD graphics
P(x, 5, 5) = -945*(1 - x^2)^(5/2)
MuPAD graphics

 

Legendre陪関数も直交性があります。

hold(int(P(x,5,5)*P(x,4,3),x=-1..1))=int(P(x,5,5)*P(x,4,3),x=-1..1);
hold(int(P(x,7,5)*P(x,8,3),x=-1..1))=int(P(x,7,5)*P(x,8,3),x=-1..1);
hold(int(P(x,9,8)*P(x,9,8),x=-1..1))=int(P(x,9,8)*P(x,9,8),x=-1..1);

int(P(x, 5, 5)*P(x, 4, 3), x = -1..1) = 0
int(P(x, 7, 5)*P(x, 8, 3), x = -1..1) = 0
int(P(x, 9, 8)*P(x, 9, 8), x = -1..1) = 711374856192000/19

xの代わりにcosθとすることが多いようです。

gth:=Symbol::theta;
delete P;
P:=(gth,n,m)->(-1)^m*(1-cos(gth)^2)^(m/2)*diff(1/2^n/n!*diff((cos(gth)^2-1)^n,gth$n),gth$m);
for j from 0 to 5 do
  for i from 0 to j do
    f:=expand(Simplify(P(gth,i,j))):
    f:=subs(f,sqrt(sin(gth)^2)=sin(gth),(sin(gth)^2)^(3/2)=sin(gth)^3,(sin(gth)^2)^(5/2)=sin(gth)^5):
    print(hold(P)(gth,j,i)=f);
  end_for
end_for

`θ`
(gth, n, m) -> (-1)^m*(1 - cos(gth)^2)^(m/2)*diff(((1/2^n)/n!)*diff((cos(gth)^2 - 1)^n, gth $ n), gth $ m)
P(`θ`, 0, 0) = 1
P(`θ`, 1, 0) = 0
P(`θ`, 1, 1) = sin(`θ`) - 2*sin(`θ`)^3
P(`θ`, 2, 0) = 0
P(`θ`, 2, 1) = 4*cos(`θ`)*sin(`θ`)^3
P(`θ`, 2, 2) = 32*sin(`θ`)^6 - 30*sin(`θ`)^4 + 3*sin(`θ`)^2
P(`θ`, 3, 0) = 0
P(`θ`, 3, 1) = 8*sin(`θ`)^5 - 4*sin(`θ`)^3
P(`θ`, 3, 2) = 128*cos(`θ`)^3*sin(`θ`)^4 - 68*cos(`θ`)*sin(`θ`)^4
P(`θ`, 3, 3) = - 972*sin(`θ`)^9 + 1330*sin(`θ`)^7 - 420*sin(`θ`)^5 + 15*sin(`θ`)^3
P(`θ`, 4, 0) = 0
P(`θ`, 4, 1) = -16*cos(`θ`)*sin(`θ`)^5
P(`θ`, 4, 2) = - 512*sin(`θ`)^8 + 504*sin(`θ`)^6 - 60*sin(`θ`)^4
P(`θ`, 4, 3) = 5832*sin(`θ`)*cos(`θ`)^9 - 18008*sin(`θ`)*cos(`θ`)^7 + 19872*sin(`θ`)*cos(`θ`)^5 - 9048*sin(`θ`)*cos(`θ`)^3 + 1352*sin(`θ`)*cos(`θ`)
P(`θ`, 4, 4) = (131072*sin(`θ`)^12)/3 - (235900*sin(`θ`)^10)/3 + 41790*sin(`θ`)^8 - 6300*sin(`θ`)^6 + 105*sin(`θ`)^4
P(`θ`, 5, 0) = 0
P(`θ`, 5, 1) = 16*sin(`θ`)^5 - 32*sin(`θ`)^7
P(`θ`, 5, 2) = 1040*cos(`θ`)*sin(`θ`)^6 - 2048*cos(`θ`)^3*sin(`θ`)^6
P(`θ`, 5, 3) = 34992*sin(`θ`)^11 - 50440*sin(`θ`)^9 + 17640*sin(`θ`)^7 - 840*sin(`θ`)^5
P(`θ`, 5, 4) = - (1048576*cos(`θ`)*sin(`θ`)^12)/3 + 471800*cos(`θ`)*sin(`θ`)^10 - 167160*cos(`θ`)*sin(`θ`)^8 + 12600*cos(`θ`)*sin(`θ`)^6
P(`θ`, 5, 5) = - (7812500*sin(`θ`)^15)/3 + 5811366*sin(`θ`)^13 - 4342800*sin(`θ`)^11 + 1226610*sin(`θ`)^9 - 103950*sin(`θ`)^7 + 945*sin(`θ`)^5

グラフにしました。

//θは定義済み
for j from 0 to 5 do
  for i from 0 to j do
    f:=expand(Simplify(P(gth,i,j))):
    f:=subs(f,sqrt(sin(gth)^2)=sin(gth),(sin(gth)^2)^(3/2)=sin(gth)^3,(sin(gth)^2)^(5/2)=sin(gth)^5):
    print(hold(P)(gth,j,i)=f);
    plot(plot::Function2d(f,gth=-PI..PI, Color=RGB::Blue));
  end_for
end_for

P(`θ`, 0, 0) = 1
MuPAD graphics
P(`θ`, 1, 0) = 0
MuPAD graphics
P(`θ`, 1, 1) = sin(`θ`) - 2*sin(`θ`)^3
MuPAD graphics
P(`θ`, 2, 0) = 0
MuPAD graphics
P(`θ`, 2, 1) = 4*cos(`θ`)*sin(`θ`)^3
MuPAD graphics
P(`θ`, 2, 2) = 32*sin(`θ`)^6 - 30*sin(`θ`)^4 + 3*sin(`θ`)^2
MuPAD graphics
P(`θ`, 3, 0) = 0
MuPAD graphics
P(`θ`, 3, 1) = 8*sin(`θ`)^5 - 4*sin(`θ`)^3
MuPAD graphics
P(`θ`, 3, 2) = 128*cos(`θ`)^3*sin(`θ`)^4 - 68*cos(`θ`)*sin(`θ`)^4
MuPAD graphics
P(`θ`, 3, 3) = - 972*sin(`θ`)^9 + 1330*sin(`θ`)^7 - 420*sin(`θ`)^5 + 15*sin(`θ`)^3
MuPAD graphics
P(`θ`, 4, 0) = 0
MuPAD graphics
P(`θ`, 4, 1) = -16*cos(`θ`)*sin(`θ`)^5
MuPAD graphics
P(`θ`, 4, 2) = - 512*sin(`θ`)^8 + 504*sin(`θ`)^6 - 60*sin(`θ`)^4
MuPAD graphics
P(`θ`, 4, 3) = 5832*sin(`θ`)*cos(`θ`)^9 - 18008*sin(`θ`)*cos(`θ`)^7 + 19872*sin(`θ`)*cos(`θ`)^5 - 9048*sin(`θ`)*cos(`θ`)^3 + 1352*sin(`θ`)*cos(`θ`)
MuPAD graphics
P(`θ`, 4, 4) = (131072*sin(`θ`)^12)/3 - (235900*sin(`θ`)^10)/3 + 41790*sin(`θ`)^8 - 6300*sin(`θ`)^6 + 105*sin(`θ`)^4
MuPAD graphics
P(`θ`, 5, 0) = 0
MuPAD graphics
P(`θ`, 5, 1) = 16*sin(`θ`)^5 - 32*sin(`θ`)^7
MuPAD graphics
P(`θ`, 5, 2) = 1040*cos(`θ`)*sin(`θ`)^6 - 2048*cos(`θ`)^3*sin(`θ`)^6
MuPAD graphics
P(`θ`, 5, 3) = 34992*sin(`θ`)^11 - 50440*sin(`θ`)^9 + 17640*sin(`θ`)^7 - 840*sin(`θ`)^5
MuPAD graphics
P(`θ`, 5, 4) = - (1048576*cos(`θ`)*sin(`θ`)^12)/3 + 471800*cos(`θ`)*sin(`θ`)^10 - 167160*cos(`θ`)*sin(`θ`)^8 + 12600*cos(`θ`)*sin(`θ`)^6
MuPAD graphics
P(`θ`, 5, 5) = - (7812500*sin(`θ`)^15)/3 + 5811366*sin(`θ`)^13 - 4342800*sin(`θ`)^11 + 1226610*sin(`θ`)^9 - 103950*sin(`θ`)^7 + 945*sin(`θ`)^5
MuPAD graphics