#Legendre polynomials as functions of x
#
P0:=1:
P01:=x:
P02:=simplify((3*x*P01-1*P0)/2):
P03:=simplify((5*x*P02-2*P01)/3):
P04:=simplify((7*x*P03-3*P02)/4):
P05:=simplify((9*x*P04-4*P03)/5):
P06:=simplify((11*x*P05-5*P04)/6):
P07:=simplify((13*x*P06-6*P05)/7):
P08:=simplify((15*x*P07-7*P06)/8):
P09:=simplify((17*x*P08-8*P07)/9):
#
#Simplified forms of Legendre polynomials multiplied by constants to
#remove factors of 2,8,16 etc.
#
P1:=unapply(P01,x):
P2:=unapply(simplify(P02),x):
P3:=unapply(simplify(P03),x):
P4:=unapply(simplify(P04),x):
P5:=unapply(simplify(P05),x):
P6:=unapply(simplify(P06),x):
P7:=unapply(simplify(P07),x):
P8:=unapply(simplify(P08),x):
P9:=unapply(simplify(P09),x):
#
R1:=sqrt(r^2+z^2):
#
#Harmonic axisymmetric bounded potentials in terms of r,z
#
F1:=expand(simplify(R1*P1(z/R1)));
F2:=expand(simplify(R1^2*P2(z/R1)));
F3:=expand(simplify(R1^3*P3(z/R1)));
F4:=expand(simplify(R1^4*P4(z/R1))):
F5:=expand(simplify(R1^5*P5(z/R1))):
F6:=expand(simplify(R1^6*P6(z/R1))):
F7:=expand(simplify(R1^7*P7(z/R1))):
F8:=expand(simplify(R1^8*P8(z/R1))):
F9:=expand(simplify(R1^9*P9(z/R1))):
#