Vector Analysis

Orthogonal coordinates

• The line element is
	d(Overscript[r,  ->])  = (Subscript[h, 1]) d(Subscript[q, 1]) (Subscript[((Overscript[e, ⋀])), 1]) + (Subscript[h, 2]) d(Subscript[q, 2]) (Subscript[((Overscript[e, ⋀])), 2]) + (Subscript[h, 3]) d(Subscript[q, 3]) (Subscript[((Overscript[e, ⋀])), 3]), or
	d(s^2) = ((((Subscript[h, 1]) (Subscript[dq, 1])))^2) + ((((Subscript[h, 2]) (Subscript[dq, 2])))^2) + ((((Subscript[h, 3]) (Subscript[dq, 3])))^2).
• The area element is
	d(Subscript[σ, ij]) = d(Subscript[s, i]) d(Subscript[s, j]) = (Subscript[h, i]) (Subscript[h, j]) d(Subscript[q, i]) d(Subscript[q, j]),   or
	d(Overscript[σ,  ->])  = d(Subscript[s, 2]) d(Subscript[s, 3]) (Subscript[((Overscript[e, ⋀])), 1]) + d(Subscript[s, 3]) d(Subscript[s, 1]) (Subscript[((Overscript[e, ⋀])), 2]) + d(Subscript[s, 1]) d(Subscript[s, 2]) (Subscript[((Overscript[e, ⋀])), 3])
• The volume element is
	dτ = d(Subscript[s, 1]) d(Subscript[s, 2]) d(Subscript[s, 3]) = (Subscript[h, 1]) (Subscript[h, 2]) (Subscript[h, 3]) d(Subscript[q, 1]) d(Subscript[q, 2]) d(Subscript[q, 3])
• The (Overscript[∇,  ->])  operations are
	(Overscript[∇,  ->]) ψ = (1/(Subscript[h, 1]))∂(ψ/∂)(Subscript[q, 1])(Subscript[((Overscript[e, ⋀])), 1]) + (1/(Subscript[h, 2]))∂(ψ/∂)(Subscript[q, 2])(Subscript[((Overscript[e, ⋀])), 2]) + (1/(Subscript[h, 3]))∂(ψ/∂)(Subscript[q, 3])(Subscript[((Overscript[e, ⋀])), 3])
	(Overscript[∇,  ->]) · (Overscript[A,  ->])  = (1/((Subscript[h, 1]) (Subscript[h, 2]) (Subscript[h, 3])))[(∂/∂)(Subscript[q, 1])((Subscript[h, 2]) (Subscript[h, 3]) (Subscript[A, 1])) + (∂/∂)(Subscript[q, 2])((Subscript[h, 1]) (Subscript[h, 3]) (Subscript[A, 2])) + (∂/∂)(Subscript[q, 3])((Subscript[h, 1]) (Subscript[h, 2]) (Subscript[A, 3]))]
	(Overscript[∇,  ->])  × (Overscript[A,  ->])  = (1/((Subscript[h, 2]) (Subscript[h, 3])))[(∂/∂)(Subscript[q, 2])((Subscript[h, 3]) (Subscript[A, 3])) - (∂/∂)(Subscript[q, 3])((Subscript[h, 2]) (Subscript[A, 2]))] (Subscript[((Overscript[e, ⋀])), 1])
		   + (1/((Subscript[h, 1]) (Subscript[h, 3])))[(∂/∂)(Subscript[q, 3])((Subscript[h, 1]) (Subscript[A, 1])) - (∂/∂)(Subscript[q, 1])((Subscript[h, 3]) (Subscript[A, 3]))] (Subscript[((Overscript[e, ⋀])), 2])
		   + (1/((Subscript[h, 1]) (Subscript[h, 2])))[(∂/∂)(Subscript[q, 1])((Subscript[h, 2]) (Subscript[A, 2])) - (∂/∂)(Subscript[q, 2])((Subscript[h, 1]) (Subscript[A, 1]))] (Subscript[((Overscript[e, ⋀])), 3])
	(((Overscript[∇,  ->]) )^2)ψ = (1/((Subscript[h, 1]) (Subscript[h, 2]) (Subscript[h, 3])))[(∂/∂)(Subscript[q, 1])((((Subscript[h, 3]) (Subscript[h, 2]))/(Subscript[h, 1]))∂(ψ/∂)(Subscript[q, 1])) + (∂/∂)(Subscript[q, 2])((((Subscript[h, 3]) (Subscript[h, 1]))/(Subscript[h, 2]))∂(ψ/∂)(Subscript[q, 2])) + (∂/∂)(Subscript[q, 3])((((Subscript[h, 1]) (Subscript[h, 2]))/(Subscript[h, 3]))∂(ψ/∂)(Subscript[q, 3]))]

Rectangular Cartesian Coordinates

For rectangular cartesian coordinates, we have
	(Subscript[h, 1]) = 1,  (Subscript[h, 2]) = 1,   (Subscript[h, 3]) = 1;		(Subscript[q, 1]) = x,  (Subscript[q, 2]) = y,  (Subscript[q, 3]) = z
Thus,
	((Subscript[q, 1]), (Subscript[q, 2]), (Subscript[q, 3])) -> (x, y, z)
	((Subscript[((Overscript[e, ⋀])), 1]), (Subscript[((Overscript[e, ⋀])), 2]), (Subscript[((Overscript[e, ⋀])), 3])) -> ((Overscript[i, ⋀]), (Overscript[j, ⋀]), (Overscript[k, ⋀])) or ((Overscript[x, ⋀]), (Overscript[y, ⋀]), (Overscript[z, ⋀]))
• The line element:		d(Overscript[r,  ->])  = dx (Overscript[i, ⋀]) + dy (Overscript[j, ⋀]) + dz (Overscript[k, ⋀])
• The area element:		d(Overscript[σ,  ->])  = dy dz (Overscript[i, ⋀]) + dx dz (Overscript[j, ⋀]) + dx dy (Overscript[k, ⋀])
• The volume element:		dτ = dx dy dz
• The gradient of Ψ:		(Overscript[∇,  ->]) Ψ = ∂(Ψ/∂)x(Overscript[i, ⋀]) + ∂(Ψ/∂)y(Overscript[j, ⋀]) + ∂(Ψ/∂)z(Overscript[k, ⋀])
• The divergence of (Overscript[A,  ->]) :	(Overscript[∇,  ->]) ·(Overscript[A,  ->])  = ∂((Subscript[A, x])/∂)x + ∂((Subscript[A, y])/∂)y + ∂((Subscript[A, z])/∂)z
• The curl of (Overscript[A,  ->]) :		(Overscript[∇,  ->]) ×(Overscript[A,  ->])  = (∂((Subscript[A, z])/∂)y - ∂((Subscript[A, y])/∂)z) (Overscript[i, ⋀]) 
					+ (∂((Subscript[A, x])/∂)z - ∂((Subscript[A, z])/∂)x) (Overscript[j, ⋀])
					+ (∂((Subscript[A, y])/∂)x - ∂((Subscript[A, x])/∂)y) (Overscript[k, ⋀])
• Laplacian of Ψ:		(((Overscript[∇,  ->]) )^2)Ψ = (∂^2) (Ψ/∂)(x^2) + (∂^2) (Ψ/∂)(y^2) + (∂^2) (Ψ/∂)(z^2)

Circular Cylindrical Coordinates

For circular cylindrical coordinates, we have
	(Subscript[h, 1]) = 1,  (Subscript[h, 2]) = ρ,   (Subscript[h, 3]) = 1;		(Subscript[q, 1]) = ρ,  (Subscript[q, 2]) = ϕ,  (Subscript[q, 3]) = z
Thus,
	((Subscript[q, 1]), (Subscript[q, 2]), (Subscript[q, 3])) -> (ρ, ϕ, z)
	((Subscript[((Overscript[e, ⋀])), 1]), (Subscript[((Overscript[e, ⋀])), 2]), (Subscript[((Overscript[e, ⋀])), 3])) -> ((Subscript[((Overscript[e, ⋀])), ρ]), (Subscript[((Overscript[e, ⋀])), ϕ]), (Subscript[((Overscript[e, ⋀])), z])) or ((Overscript[ρ, ⋀]), (Overscript[ϕ, ⋀]), (Overscript[z, ⋀]))
And
	ρ = (Sqrt[(x^2)  +  (y^2)])
	ϕ = (tan^(-1))((y/x))
	z = z
	(Subscript[((Overscript[e, ⋀])), ρ]) = cos ϕ (Overscript[i, ⋀]) + sin ϕ (Overscript[j, ⋀])
	(Subscript[((Overscript[e, ⋀])), ϕ]) = - sin ϕ (Overscript[i, ⋀]) + cos ϕ (Overscript[j, ⋀])
	(Subscript[((Overscript[e, ⋀])), z]) = (Overscript[k, ⋀])
Or
	x = ρ cos ϕ
	y = ρ sin ϕ
	z = z
	(Overscript[i, ⋀]) = cos ϕ (Subscript[((Overscript[e, ⋀])), ρ]) - sin ϕ (Subscript[((Overscript[e, ⋀])), ϕ])
	(Overscript[j, ⋀]) = sin ϕ (Subscript[((Overscript[e, ⋀])), ρ]) + cos ϕ (Subscript[((Overscript[e, ⋀])), ϕ])
	(Overscript[k, ⋀]) = (Subscript[((Overscript[e, ⋀])), z])
• The line element:		d(Overscript[r,  ->])  = dρ (Subscript[((Overscript[e, ⋀])), ρ]) + ρ dϕ (Subscript[((Overscript[e, ⋀])), ϕ]) + dz (Subscript[((Overscript[e, ⋀])), z])
• The area element:		d(Overscript[σ,  ->])  = ρ dϕ dz (Subscript[((Overscript[e, ⋀])), ρ]) + dρ dz (Subscript[(Overscript[e, □]), ϕ]) + ρ dρ dϕ (Subscript[((Overscript[e, ⋀])), z])
• The volume element:		dτ = ρ dρ dϕ dz
• The gradient of Ψ:		(Overscript[∇,  ->]) Ψ = ∂(Ψ/∂)ρ(Subscript[((Overscript[e, ⋀])), ρ]) + (1/ρ)∂(Ψ/∂)ϕ(Subscript[((Overscript[e, ⋀])), ϕ]) + ∂(Ψ/∂)z(Subscript[((Overscript[e, ⋀])), z])
• The divergence of (Overscript[A,  ->]) :	(Overscript[∇,  ->]) ·(Overscript[A,  ->])  = (1/ρ)\(TraditionalForm\`∂\((ρ\ A\_ρ)\)\/∂ρ\) + (1/ρ)∂((Subscript[A, ϕ])/∂)ϕ + ∂((Subscript[A, z])/∂)z
• The curl of (Overscript[A,  ->]) :		(Overscript[∇,  ->]) ×(Overscript[A,  ->])  = ((1/ρ)∂((Subscript[A, z])/∂)ϕ - ∂((Subscript[A, ϕ])/∂)z) (Subscript[((Overscript[e, ⋀])), ρ]) 
					+ (∂((Subscript[A, ρ])/∂)z - ∂((Subscript[A, z])/∂)ρ) (Subscript[((Overscript[e, ⋀])), ϕ])
					+ (1/ρ)(\(TraditionalForm\`∂\((ρ\ A\_ϕ)\)\/∂ρ\) - ∂((Subscript[A, ρ])/∂)ϕ) (Subscript[((Overscript[e, ⋀])), z])
• Laplacian of Ψ:		(((Overscript[∇,  ->]) )^2)Ψ = (1/ρ)(∂/∂)ρ(ρ∂(Ψ/∂)ρ) + (1/(ρ^2))\(TraditionalForm\`∂\^2 Ψ\/∂ϕ\^2\) + (∂^2) (Ψ/∂)(z^2)

Spherical Polar Coordinates

For circular cylindrical coordinates, we have
	(Subscript[h, 1]) = 1,  (Subscript[h, 2]) = r,   (Subscript[h, 3]) = r sin θ;		(Subscript[q, 1]) = r,  (Subscript[q, 2]) = θ,  (Subscript[q, 3]) = ϕ
Thus,
	((Subscript[q, 1]), (Subscript[q, 2]), (Subscript[q, 3])) -> (r, θ, ϕ)
	((Subscript[((Overscript[e, ⋀])), 1]), (Subscript[((Overscript[e, ⋀])), 2]), (Subscript[((Overscript[e, ⋀])), 3])) -> ((Subscript[((Overscript[e, ⋀])), r]), (Subscript[((Overscript[e, ⋀])), θ]), (Subscript[((Overscript[e, ⋀])), ϕ])) or ((Overscript[r, ⋀]), (Overscript[θ, ⋀]), (Overscript[ϕ, ⋀]))
And
	r = (Sqrt[(x^2)  +  (y^2)   +  (z^2)])
	θ = (cos^(-1))((z/(Sqrt[(x^2)  +  (y^2)  +  (z^2)])))
	ϕ = (tan^(-1))((y/x))
	(Subscript[((Overscript[e, ⋀])), r]) = sin θ cos ϕ (Overscript[i, ⋀]) + sin θ sin ϕ (Overscript[j, ⋀]) + cos θ (Overscript[k, ⋀])
	(Subscript[((Overscript[e, ⋀])), θ]) = cos θ cos ϕ (Overscript[i, ⋀]) + cos θ sin ϕ (Overscript[j, ⋀]) - sin θ (Overscript[k, ⋀])
	(Subscript[((Overscript[e, ⋀])), ϕ]) = - sin ϕ (Overscript[i, ⋀]) + cos ϕ (Overscript[j, ⋀])
Or
	x = r sin θ cos ϕ
	y = r sin θ sin ϕ
	z = r cos θ
	(Overscript[i, ⋀]) = sin θ cos ϕ (Subscript[((Overscript[e, ⋀])), ρ]) + cos θ cos ϕ (Subscript[((Overscript[e, ⋀])), θ]) - sin ϕ (Subscript[((Overscript[e, ⋀])), ϕ])
	(Overscript[j, ⋀]) = sin θ sin ϕ (Subscript[((Overscript[e, ⋀])), ρ]) + cos θ sin ϕ (Subscript[((Overscript[e, ⋀])), θ]) + cos ϕ (Subscript[((Overscript[e, ⋀])), ϕ])
	(Overscript[k, ⋀]) = cos θ (Subscript[((Overscript[e, ⋀])), r]) - sin θ (Subscript[((Overscript[e, ⋀])), θ])
• The line element:		d(Overscript[r,  ->])  = dr (Subscript[((Overscript[e, ⋀])), r]) + r dθ (Subscript[((Overscript[e, ⋀])), θ]) + r sin θ dϕ (Subscript[((Overscript[e, ⋀])), ϕ])
• The area element:		d(Overscript[σ,  ->])  = (r^2) sin θ dθ dϕ (Subscript[((Overscript[e, ⋀])), r]) + r sin θ dr dϕ (Subscript[((Overscript[e, ⋀])), θ]) + r dr dθ (Subscript[((Overscript[e, ⋀])), ϕ])
• The volume element:		dτ = (r^2) sin θ dr dθ dϕ
• The gradient of Ψ:		(Overscript[∇,  ->]) Ψ = ∂(Ψ/∂)r(Subscript[((Overscript[e, ⋀])), r]) + (1/r)∂(Ψ/∂)θ(Subscript[((Overscript[e, ⋀])), θ]) + (1/(r sin θ))∂(Ψ/∂)ϕ(Subscript[((Overscript[e, ⋀])), ϕ])
• The divergence of (Overscript[A,  ->]) :	(Overscript[∇,  ->]) ·(Overscript[A,  ->])  = (1/(r^2))∂((((r^2) (Subscript[A, r])))/∂)r + (1/(r sin θ))\(TraditionalForm\`∂\((sin\ θ\ A\_θ)\)\/∂θ\) + (1/(r sin θ))∂((Subscript[A, ϕ])/∂)ϕ
• The curl of (Overscript[A,  ->]) :		(Overscript[∇,  ->]) ×(Overscript[A,  ->])  = (1/(r sin θ))(\(TraditionalForm\`∂\((sin\ θ\ A\_ϕ)\)\/∂θ\) - ∂((Subscript[A, θ])/∂)ϕ) (Subscript[((Overscript[e, ⋀])), r]) 
					+ ((1/(r sin θ))∂((Subscript[A, r])/∂)ϕ - (1/r)∂(((r (Subscript[A, ϕ])))/∂)r) (Subscript[((Overscript[e, ⋀])), θ])
					+ (1/r)(∂(((r (Subscript[A, θ])))/∂)r - ∂((Subscript[A, r])/∂)θ) (Subscript[((Overscript[e, ⋀])), ϕ])
• Laplacian of Ψ:		(((Overscript[∇,  ->]) )^2)Ψ = (1/(r^2))(∂/∂)r((r^2)∂(Ψ/∂)r) 
					+ (1/((r^2) sin θ))(∂/∂)θ(sin θ∂(Ψ/∂)ϕ) 
					+ (1/((r^2) (sin^2) θ))\(TraditionalForm\`∂\^2 Ψ\/∂ϕ\^2\)

Formulas of Vector Analysis and Vector Calculus

(Overscript[∇,  ->])  · ((Overscript[A,  ->])  × (Overscript[B,  ->]) ) = (Overscript[B,  ->])  · (Overscript[∇,  ->])  × (Overscript[A,  ->])  - (Overscript[A,  ->])  · (Overscript[∇,  ->])  × (Overscript[B,  ->]) 
	(Overscript[∇,  ->])  ((Overscript[A,  ->])  · (Overscript[B,  ->]) ) = ((Overscript[A,  ->])  · (Overscript[∇,  ->]) ) (Overscript[B,  ->])  + ((Overscript[B,  ->])  · (Overscript[∇,  ->]) ) (Overscript[A,  ->])  + (Overscript[A,  ->])  × ((Overscript[∇,  ->])  × (Overscript[B,  ->]) ) + (Overscript[B,  ->])  × ((Overscript[∇,  ->])  × (Overscript[A,  ->]) )
	(Overscript[∇,  ->])  · ((Overscript[∇,  ->])  × (Overscript[A,  ->]) ) = 0
	(Overscript[∇,  ->])  × ((Overscript[∇,  ->])  × (Overscript[A,  ->]) ) = (Overscript[∇,  ->])  ((Overscript[∇,  ->])  · (Overscript[A,  ->]) ) - ((Overscript[∇,  ->])  · (Overscript[∇,  ->]) ) (Overscript[A,  ->]) 
	(Overscript[∇,  ->])  × ((Overscript[A,  ->])  × (Overscript[B,  ->]) ) = ((Overscript[B,  ->])  · (Overscript[∇,  ->]) ) (Overscript[A,  ->])  + (Overscript[A,  ->])  ((Overscript[∇,  ->])  · (Overscript[B,  ->]) ) - ((Overscript[A,  ->])  · (Overscript[∇,  ->]) ) (Overscript[B,  ->])  - (Overscript[B,  ->])  ((Overscript[∇,  ->])  · (Overscript[A,  ->]) )
	(Overscript[A,  ->])  × ((Overscript[B,  ->])  × (Overscript[C,  ->]) ) = (Overscript[B,  ->])  ((Overscript[A,  ->])  · (Overscript[C,  ->]) ) - (Overscript[C,  ->])  ((Overscript[A,  ->])  · (Overscript[B,  ->]) )
	(Overscript[A,  ->])  × ((Overscript[B,  ->])  × (Overscript[C,  ->]) ) + (Overscript[B,  ->])  × ((Overscript[C,  ->])  × (Overscript[A,  ->]) ) + (Overscript[C,  ->])  × ((Overscript[A,  ->])  × (Overscript[B,  ->]) ) = 0
	(Overscript[A,  ->])  · (Overscript[B,  ->])  × (Overscript[C,  ->])  = (({{(Subscript[A, x]), (Subscript[A, y]), (Subscript[A, z])}, {(Subscript[B, x]), (Subscript[B, y]), (Subscript[B, z])}, {(Subscript[C, x]), (Subscript[C, y]), (Subscript[C, z])}}))
	(Overscript[A,  ->])  · (Overscript[B,  ->])  × (Overscript[C,  ->])  = (Overscript[B,  ->])  · (Overscript[C,  ->])  × (Overscript[A,  ->])  = (Overscript[C,  ->])  · (Overscript[A,  ->])  × (Overscript[B,  ->]) 
	((Overscript[A,  ->])  × (Overscript[B,  ->]) ) · ((Overscript[C,  ->])  × (Overscript[D,  ->]) ) = ((Overscript[A,  ->])  · (Overscript[C,  ->]) ) ((Overscript[B,  ->])  · (Overscript[D,  ->]) ) - ((Overscript[A,  ->])  · (Overscript[D,  ->]) ) ((Overscript[B,  ->])  · (Overscript[C,  ->]) )
	((Overscript[A,  ->])  × (Overscript[B,  ->]) ) × ((Overscript[C,  ->])  × (Overscript[D,  ->]) ) = ((Overscript[D,  ->])  · (Overscript[A,  ->])  × (Overscript[B,  ->]) ) (Overscript[C,  ->])  - ((Overscript[C,  ->])  · (Overscript[A,  ->])  × (Overscript[B,  ->]) ) (Overscript[D,  ->]) 
	(Overscript[∇,  ->]) (ϕ ψ) = ϕ (Overscript[∇,  ->]) ψ + ψ (Overscript[∇,  ->]) ϕ
	(Overscript[∇,  ->])  · (ϕ(Overscript[A,  ->]) ) = (Overscript[A,  ->])  · (Overscript[∇,  ->]) ϕ + ϕ (Overscript[∇,  ->])  · (Overscript[A,  ->]) 
	(Overscript[∇,  ->])  × (ϕ(Overscript[A,  ->]) ) = ϕ (Overscript[∇,  ->])  × (Overscript[A,  ->])  - (Overscript[A,  ->])  × (Overscript[∇,  ->]) ϕ
	
	(Overscript[∇,  ->])  · (Overscript[r,  ->])  = 3
	(Overscript[∇,  ->])  × (Overscript[r,  ->])  = 0
	(Overscript[∇,  ->])  r = (((Overscript[r,  ->]) )/r)
	(Overscript[∇,  ->]) ((1/r)) = - (((Overscript[r,  ->]) )/(r^3))
	(Overscript[∇,  ->])  · ((((Overscript[r,  ->]) )/(r^3))) = - (((Overscript[∇,  ->]) )^2)((1/r)) = 4π δ((Overscript[r,  ->]) )
	(Overscript[∇,  ->])  · ((((Overscript[A,  ->]) )/r)) = (Overscript[A,  ->])  · (Overscript[∇,  ->]) ((1/r)) = - \(TraditionalForm\`\(\(A\& -> \)\ ·\(r\& -> \)\)\/\(\(\ \)\(r\^3\)\)\)
	(Overscript[∇,  ->])  × [(Overscript[A,  ->])  × ((((Overscript[r,  ->]) )/(r^3)))] = - (Overscript[∇,  ->])  ((Overscript[A,  ->])  · (((Overscript[r,  ->]) )/(r^3))),  for r ≠ 0
	(((Overscript[∇,  ->]) )^2) (((Overscript[A,  ->]) )/r) = (Overscript[A,  ->])  (((Overscript[∇,  ->]) )^2)((1/r)) = 0,  for r ≠ 0
	((((Overscript[∇,  ->]) )^2) + (k^2)) ((e^ikr)/r) = ((((Overscript[∇,  ->]) )^2) + (k^2)) ((cos kr)/r) = - 4π δ((Overscript[r,  ->]) )

Properties of Integrals over Vector Functions

• The line integral:	\(TraditionalForm\`∫\_\(\(r\& -> \)\_1\)\%\(\(r\& -> \)\_2\)\ \(A\& -> \)\ ·d\(l\& -> \)\) = - \(TraditionalForm\`∫\_\(\(r\& -> \)\_2\)\%\(\(r\& -> \)\_1\)\ \(A\& -> \)\ ·d\(l\& -> \)\)
   Note that if (Overscript[A,  ->])  = (Overscript[∇,  ->]) ϕ, or (Overscript[∇,  ->])  × (Overscript[A,  ->])  = 0, then the line integral
   			\(TraditionalForm\`∫\_\(\(r\& -> \)\_1\)\%\(\(r\& -> \)\_2\)\ \(A\& -> \)\ ·d\(l\& -> \)\) = ϕ ((Subscript[((Overscript[r,  ->]) ), 2])) - ϕ ((Subscript[((Overscript[r,  ->]) ), 1]))
   is independent of the path of integration from (Subscript[((Overscript[r,  ->]) ), 1]) to (Subscript[((Overscript[r,  ->]) ), 2]), and
   			∮ (Overscript[A,  ->])  · d (Overscript[l,  ->])  = ∮ (Overscript[∇,  ->]) ϕ · d (Overscript[l,  ->])  = 0
• Gauss's Theorem:	\(TraditionalForm\`∫\_V\ \(∇\& -> \)\(\(·\)\(A\& -> \)\) dV\) = \(TraditionalForm\`∫\_S\ \(A\& -> \)·d\(S\& -> \)\)
   Note that V is a region of volume bounded by the surface.
• Stokes's Theorem:	\(TraditionalForm\`∫\_C\ \(A\& -> \)·d\(l\& -> \)\) = \(TraditionalForm\`∫\_S\ \((\(∇\& -> \)\(\(×\)\(A\& -> \)\))\)·d\(S\& -> \)\)

Fourier Transform

Frequently in mathematical physics we encounter pairs of functions related by an expression of the form
	g(α) = (∫_a^b) f(t) K(α, 
            t) dt.
The function g(α) is called the (integral) transform of f(t) by the kernel K(α,t). The operation may also be described as mapping a function f(t) in t-space into another function g(α) in α-space. This interpretation takes on physical significance in the time-frequency or the coordinate space-momentum space relations. One of the most useful of the infinite number of possible transformations is the Fourier transform.

Let us define g(k), the Fourier transform of the function f(x), by
	g(k) = (1/(Sqrt[2 π]))(∫_(-∞)^∞) f(
            x) (e^ikx) dx
We have the inverse relation, the inverse Fourier transform of g(k):
	f(x) = (1/(Sqrt[2 π]))(∫_(-∞)^∞) g(
            k) (e^(-ikx)) dk
When we move the Fourier transform pair to three-dimensional space, the pair becomes
	g((Overscript[k,  ->]) ) = (1/(((2 π))^(3/2)))\(TraditionalForm\`∫\(f(\(r\& -> \))\) \(e\^\(i \( k\& -> \)·\(r\& -> \)\)\) d\(r\& -> \)\)
	f((Overscript[r,  ->]) ) = (1/(((2 π))^(3/2)))\(TraditionalForm\`∫\(g(\(k\& -> \))\) \(e\^\(\(-i\) \(k\& -> \)·\(r\& -> \)\)\) d\(k\& -> \)\)

We can use Mathematica function FourierTransform[f(x),x,k] to find g(k) given f(x) or InverseFourierTransform[g(k),k,x] to find f(x) given g(k). You will need to load the symbolic Fourier transform package before using the functions. For example, let's find the Fourier transform of a Gaussian function peaked at the origin and with a width proportional to 1/α: f(x) = (e^(-(α^2) (x^2)/2)). The Fourier transform is (1/α)\(TraditionalForm\`e\^\(\(-k\^2\)/\((2  α\^2)\)\)\) which is again a Gaussian but with an inverse width.

<< Calculus`FourierTransform`
FourierTransform[(E^(-(α^2) (x^2)/2)), x, 
      k]
InverseFourierTransform[%, k, x]

)Airy Functions

The Airy functions Ai(x) and Bi(x) are two independent solutions to the Airy differential equation:
	(((d^2) f(f))/(d (x^2))) = x f(x)
The Airy functions are related to modified Bessel functions with 1/3 orders. The Airy functions often appear as the solutions to boundary value problems in electromagnetic theory and quantum mechanics. Ai(x) tends to zero while Bi(x) increases unboundedly for large positive x. Ai(x) and Bi(x) are plotted in the figure below using Mathematica built-in functions AiryAi[x] and AiryBi[x].

Plot[{AiryAi[x], AiryBi[x]}, {x, -10, 4}, 
        PlotRange -> {-0.5, 1.5}, 
        PlotStyle -> {{RGBColor[0, 0, 1]}, {RGBColor[1, 0, 1]}}, 
        Epilog ->  ({Text["Ai(x)", {1.3, 0.3}], 
              Text["Bi(x)", {2.0, 1.25}]})];

The Airy function Ai(x) plotted above gives the quantum-mechanical amplitude for a particle in a potential that increases linearly from left to right.The amplitude is exponentially damped in the classically inaccessible region on the right (see Griffiths' Introduction To Quantum Mechanics, Sec. 8.3).

The integral representation of Ai(x) and Bi(x) are:
	Ai(x) = (1/π)(∫_0^∞) cos(
            ((s^3)/3) + sx) ds
	Bi(x) = (1/π)\(TraditionalForm\`∫\_0\%\(\(∞\)\(\ \)\)\([e\^\(\(-s\^3\)/3\  + \ sx\)\  + \ 
              sin(s\^3\/3 + sx)]\) ds\).
For large values of  |x|, they can be expanded  as follows:
 	Ai(x) ~ (1/(2 (Sqrt[π]) (x^(1/4))))(e^(-(2/3) (x^(3/2)))),    x≫0
 	Bi(x) ~ (1/((Sqrt[π]) (x^(1/4))))(e^((2/3) (x^(3/2)))),   x≫0
 	Ai(x) ~ \(TraditionalForm\`1\/\(\(\ \)\(\(\@π\) \((\(-x\))\)\^\(1/4\)\)\)\)sin[(2/3)(((-x))^(3/2)) + π/4],   x≪0
 	Bi(x) ~ \(TraditionalForm\`1\/\(\(\ \)\(\(\@π\) \((\(-x\))\)\^\(1/4\)\)\)\)cos[(2/3)(((-x))^(3/2)) + π/4],   x≪0

Bessel Functions

The Bessel functions (Subscript[J, n])(x) and (Subscript[Y, n])(x) appear in a wide variety of physical problems with cylindrical symmetry. They are two linearly independent solutions to the differential equation
	(x^2)(((d^2) (Subscript[Z, n])(x))/(dx^2)) + x(((Subscript[dZ, n])(x))/dx)+ ((x^2) - (n^2))(Subscript[Z, n])(x) = 0
where (Subscript[Z, n])(x) stands for either (Subscript[J, n]) (x) or (Subscript[Y, n])(x). For example, the free-particle Schrödinger equation in three dimensions when written in cylindrical coordinate leads to the differential equation above. (Subscript[J, n])(x) is often called the Bessel function of the first kind, or simply the Bessel function. (Subscript[Y, n])(x) is referred to as the Bessel function of the second kind, the Weber function, or the Neumann function (written as (Subscript[N, n])(x)). For integer n, the (Subscript[J, n])(x) are regular at x = 0, while (Subscript[Y, n])(x) have a logarithmic divergence at x = 0.

The Hankel functions, or the Bessel functions of the third kind, defined as
	(Subscript[H, n]^((1)))(x) = (Subscript[J, n])(x) + i (Subscript[Y, n])(x)
	(Subscript[H, n]^((2)))(x) = (Subscript[J, n])(x) - i (Subscript[Y, n])(x)
give an alternative pair of solutions to the Bessel differential equation. This is exactly analogous to taking 
	(e^( ± i θ)) = cos θ ± i sin θ.
For real arguments (Subscript[H, n]^((1)))(x) and (Subscript[H, n]^((2)))(x) are complex conjugates.

The modified Bessel functions (Subscript[I, n])(x) and (Subscript[K, n])(x) are solutions to the differential equation
	(x^2)(((d^2) (Subscript[Z, n])(x))/(dx^2)) + x(((Subscript[dZ, n])(x))/dx)- ((x^2) + (n^2))(Subscript[Z, n])(x) = 0.
The (Subscript[I, n])(x) are sometimes known as the hyperbolic Bessel functions. For integer n, (Subscript[I, n])(x) are regular at x = 0, while (Subscript[K, n])(x) always have a logarithmic divergence at x = 0.

In studying systems with spherical symmetry, for example when the free-particle Schrödinger equation in three dimwnsions is separated in spherical coordinates, the spherical Bessel functions arise, defined by
	(Subscript[j, n])(x) = (Sqrt[(π/(2  x))])(Subscript[J, n + (1/2)])(x),
	(Subscript[n, n])(x) = (Sqrt[(π/(2  x))])(Subscript[Y, n + (1/2)])(x),
	(Subscript[h, n]^((1)))(x) = (Sqrt[(π/(2  x))])(Subscript[H, n + (1/2)]^((1)))(x) = (Subscript[j, n])(x) + i (Subscript[n, n])(x),
	(Subscript[h, n]^((2)))(x) = (Sqrt[(π/(2  x))])(Subscript[H, n + (1/2)]^((2)))(x) = (Subscript[j, n])(x) - i (Subscript[n, n])(x),
	(Subscript[i, n])(x) = (Sqrt[(π/(2  x))])(Subscript[I, n + (1/2)])(x),
	(Subscript[k, n])(x) = (Sqrt[(π/(2  x))])(Subscript[K, n + (1/2)])(x).

Bessel functions of the First Kind

When n is a positive integer, (Subscript[J, n])(x) can be defined by the infinite power-series
	(Subscript[J, n])(x) = \(TraditionalForm\`∑\+\(s = 0\)\%∞\((\(-1\))\)\^s\/\(\(s!\) \(\((n + s)\)!\)\)\ \((x\/2)\)\^\(n + 2  s\)\) = ((x^n)/((2^n) n!))- \(TraditionalForm\`x\^\(n + 2\)\/\(\(2\^\(n + 2\)\) \(\((n + 1)\)!\)\)\)+ ⋯
We can use Mathematica built-in function BesselJ[n,x] to generate the Bessel functions of the first kind.

TableForm[
      Table[{"n =", n, Series[BesselJ[n, x], {x, 0, 8}]}, {n, 0, 
          2}]]

Note that (Subscript[J, n])(x) are even functions of x when n is even, odd when n is odd. The Bessel functions oscillate but are not periodic except in the limit as x -> ∞. The amplitude of (Subscript[J, n])(x) is not constant but decreases asymptotically as (x^(-1/2)). Let's plot the first three Bessel functions.

\(\(Plot[{BesselJ[0, x], BesselJ[1, x], BesselJ[2, x]}, {x, \(-2\), 10}, 
        PlotRange -> \ {\(-1.2\), 1.2}, 
        PlotStyle -> {{RGBColor[0, 0, 1]}, {RGBColor[1, 0, 
                1]}, {RGBColor[0, 1, 0]}}, 
        Epilog -> \ \(({Text[\*""\!\(J\_0\)(x)"", {1.0, 1.1}], 
              Text[\*""\!\(J\_1\)(x)"", {2.2, 0.7}], 
              Text[\*""\!\(J\_2\)(x)"", {4.4, 0.5}]})\)];\)\)

Bessel functions of the Second Kind

The series form of (Subscript[Y, n])(x) is
	(Subscript[Y, 0])(x) = (2/π)(ln x + γ - ln 2) + O((x^2)),
	(Subscript[Y, n])(x) = - (n-1)! (((2/x))^n)/π + ⋯ + (2/π)((((x/2)))^n)(1/(n!))ln ((x/2)) + ⋯,   n > 0.
 where γ = 0.577 215 664 901 ⋯ is the Euler-Mascheroni constant. We can also use Mathematica built-in function BesselY[n,x] to generate the Bessel functions of the second kind.

TableForm[
      Table[{"n =", n, Series[BesselY[n, x], {x, 0, 2}] // Simplify}, {n, 
          0, 2}]]

Note that (Subscript[Y, n])(x) diverge at least logarithmically at x = 0. Any boundary condition that requires the solution to be finite at the origin (as in the vibrating circular membrance) automatically excludes (Subscript[Y, n])(x). Conversely, in the absence of such a requirement (Subscript[Y, n])(x) must be considered.  Let's plot the first three Neumann functions.

\(\(Plot[{BesselY[0, x], BesselY[1, x], BesselY[2, x]}, {x, 0, 10}, 
        PlotRange -> \ {\(-1.5\), 0.8}, 
        PlotStyle -> {{RGBColor[0, 0, 1]}, {RGBColor[1, 0, 
                1]}, {RGBColor[0, 1, 0]}}, 
        Epilog -> \ \(({Text[\*""\!\(Y\_0\)(x)"", {1.0, 0.5}], 
              Text[\*""\!\(Y\_1\)(x)"", {4.3, 0.5}], 
              Text[\*""\!\(Y\_2\)(x)"", {6.2, 0.4}]})\)];\)\)

Modified Bessel Functions

Because the modified Bessel function (Subscript[I, n])(x) is related to the Bessel function (Subscript[J, n])(x) (with the replacement x -> ix, (Subscript[I, n])(x) ≡ (i^(-n))(Subscript[J, n])(ix)) much as sinh is related to sine, (Subscript[I, n])(x) and the second solution (Subscript[K, n])(x) are sometimes referred to as hyperbolic Bessel functions. The series forms of (Subscript[I, n])(x) and (Subscript[K, n])(x) are
	(Subscript[I, n])(x) = \(TraditionalForm\`∑\+\(s = 0\)\%∞ 1\/\(\(s!\) \(\((n + s)\)!\)\)\ \((x\/2)\)\^\(n + 2  s\)\),
	(Subscript[K, 0])(x) = - ln x - γ + ln 2 + ⋯,
	(Subscript[K, n])(x) = (2^(n - 1))(n-1)! (x^(-n)) + ⋯,      n > 0.
We can also use Mathematica built-in function BesselI[n,x] and BesselK[n,x] to generate the modified Bessel functions.

TableForm[
      Table[{"n =", n, Series[BesselI[n, x], {x, 0, 8}]}, {n, 0, 
          2}]]

TableForm[
      Table[{"n =", n, Series[BesselK[n, x], {x, 0, 2}] // Simplify}, {n, 
          0, 2}]]

Let's plot the first three modified Bessel functions.

\(\(Plot[{BesselI[0, x], BesselI[1, x], BesselI[2, x], BesselK[0, x], 
          BesselK[1, x], BesselK[2, x]}, {x, 0, 3}, 
        PlotRange -> \ {0, 3}, 
        PlotStyle -> {{RGBColor[0, 0, 1]}, {RGBColor[1, 0, 
                1]}, {RGBColor[0, 1, 0]}}, 
        Epilog -> \ \(({Text[\*""\!\(I\_0\)(x)"", {2.0, 2.7}], 
              Text[\*""\!\(I\_1\)(x)"", {2.85, 2.7}], 
              Text[\*""\!\(I\_2\)(x)"", {2.9, 1.4}], 
              Text[\*""\!\(K\_0\)(x)"", {0.3, 2.5}], 
              Text[\*""\!\(K\_1\)(x)"", {0.6, 2.2}], 
              Text[\*""\!\(K\_2\)(x)"", {1.15, 2.0}]})\)];\)\)

Spherical Bessel functions

As stated earlier, the spherical Bessel functions are related to the standard (or cylindrical) Bessel functions for half-integral values (n = n + 1/2):
	(Subscript[j, n])(x) = (Sqrt[(π/(2  x))])(Subscript[J, n + (1/2)])(x),
	(Subscript[n, n])(x) = (Sqrt[(π/(2  x))])(Subscript[Y, n + (1/2)])(x),
	(Subscript[h, n]^((1)))(x) = (Sqrt[(π/(2  x))])(Subscript[H, n + (1/2)]^((1)))(x) = (Subscript[j, n])(x) + i (Subscript[n, n])(x),
	(Subscript[h, n]^((2)))(x) = (Sqrt[(π/(2  x))])(Subscript[H, n + (1/2)]^((2)))(x) = (Subscript[j, n])(x) - i (Subscript[n, n])(x),
	(Subscript[i, n])(x) = (Sqrt[(π/(2  x))])(Subscript[I, n + (1/2)])(x),
	(Subscript[k, n])(x) = (Sqrt[(2/π)x])(Subscript[K, n + (1/2)])(x).
The spherical Bessel functions are actually simpler in form than the standard Bessel functions, as they can be written in closed form. The first few spherical Bessel functions are given below:
	(Subscript[j, 0])(x) = ((sin x)/x)				(Subscript[n, 0])(x) = - ((cos x)/x)
	(Subscript[j, 1])(x) = ((sin x)/(x^2)) - ((cos x)/x)			(Subscript[n, 1])(x) = - ((cos x)/(x^2)) - ((sin x)/x)
	(Subscript[j, 2])(x) = ((3/(x^3)) - (1/x)) sin x - (3/(x^2))cos x	(Subscript[n, 2])(x) = - ((3/(x^3)) - (1/x)) cos x - (3/(x^2))sin x
	(Subscript[h, 0]^((1)))(x) = (e^ix)(- (i/x))			(Subscript[h, 0]^((2)))(x) = (e^(-ix))(i/x)
	(Subscript[h, 1]^((1)))(x) = (e^ix)(- (1/x) - (i/(x^2)))		(Subscript[h, 1]^((2)))(x) = (e^(-ix))(-(1/x) + (i/(x^2)))
	(Subscript[h, 2]^((1)))(x) = (e^ix)((i/x) - (3/(x^2)) - ((3  i)/(x^3)))		(Subscript[h, 2]^((2)))(x) = (e^(-ix))(- (i/x) - (3/(x^2))+ ((3  i)/(x^3)))
	(Subscript[i, 0])(x) = ((sinh x)/x)				(Subscript[k, 0])(x) = (e^(-x)) ((1/x))
	(Subscript[i, 1])(x) = - ((sinh x)/(x^2)) + ((cosh x)/x)		(Subscript[k, 1])(x) = (e^(-x)) ((1/x) + (1/(x^2)))
	(Subscript[i, 2])(x) = ((3/(x^3))+ (1/x)) sinh x - (3/(x^2))cosh x	(Subscript[k, 2])(x) = (e^(-x)) ((1/x) + (3/(x^2)) + (3/(x^3)))
	
Note that (Subscript[h, n]^((1)))(x) and (Subscript[h, n]^((2)))(x) are the complex conjugates.

The limiting values for the spherical Bessel functions  are given below:
For x ≪ 1,
	(Subscript[j, n])(x) ≈ ((x^n)/((2  n  +  1)!!))
	(Subscript[n, n])(x) ≈ - (((2  n  -  1)!!)/(x^(n + 1)))
	(Subscript[i, n])(x) ≈ ((x^n)/((2  n  +  1)!!))
	(Subscript[k, n])(x) ≈ (((2  n  -  1)!!)/(x^(n + 1)))
For x ≫ n(n+1)/2,
	(Subscript[j, n])(x) ∼ (1/x)sin (x - n(π/2))
	(Subscript[n, n])(x) ∼ (1/x)cos(x - n(π/2))
	(Subscript[i, n])(x) ∼ ((e^x)/(2  x))
	(Subscript[k, n])(x) ∼ ((e^(-x))/x)

The spherical Bessel functions can be generated using the Rayleigh formulas:
	(Subscript[j, n])(x) = (((-1))^n)(x^n)((((1/x) (d/dx)))^n)(((sin x)/x)),
	(Subscript[n, n])(x) = - (((-1))^n)(x^n)((((1/x) (d/dx)))^n)(((cos x)/x)),
	(Subscript[h, n]^((1)))(x) = -i (((-1))^n)(x^n)((((1/x) (d/dx)))^n)(((e^ix)/x)),
	(Subscript[h, n]^((2)))(x) = i (((-1))^n)(x^n)((((1/x) (d/dx)))^n)(((e^(-ix))/x)),
	(Subscript[i, n])(x) = (x^n)((((1/x) (d/dx)))^n)(((sinh x)/x)),
	(Subscript[k, n])(x) = (((-1))^n)(x^n)((((1/x) (d/dx)))^n)(((e^(-x))/x)).

For example, to generate (Subscript[j, n])(x) using rayleigh formulas:

\(TableForm[{{"n = 0", \((\(-1\))\)\^0\ x\^0\ \((1\/x)\)\^0\ D[
              Sin[x]\/x, {x, 
                0}]}, {"n = 1", \((\(-1\))\)\^1\ x\^1\ \((1\/x)\)\ D[\((1\/x)\)\^0\ D[Sin[x]\/x, {x, 0}], {x, 
                1}]}, {"n = 2", \(\((\(-1\))\)\^2\ x\^2\ \((1\/x)\)\ D[\((1\/x)\)\ D[\((1\/x)\)\^0\ D[Sin[x]\/x, {x, 0}], {x, 1}], {x, 1}] // 
              PowerExpand\) // Expand}}]\)

To generate (Subscript[n, n])(x) using rayleigh formulas:

\(TableForm[{{"n = 0", \(-\((\(-1\))\)\^0\)\ x\^0\ \((1\/x)\)\^0\ D[
              Cos[x]\/x, {x, 
                0}]}, {"n = 1", \(-\((\(-1\))\)\^1\)\ x\^1\ \((1\/x)\)\ D[\((1\/x)\)\^0\ D[Cos[x]\/x, {x, 0}], {x, 
                1}]}, {"n = 2", \(\(-\((\(-1\))\)\^2\)\ x\^2\ \((1\/x)\)\ D[\((1\/x)\)\ D[\((1\/x)\)\^0\ D[Cos[x]\/x, {x, 0}], {x, 1}], {x, 1}] // 
              PowerExpand\) // Expand}}]\)

To generate (Subscript[h, n]^((1)))(x) using rayleigh formulas:

\(TableForm[{{"n = 0", \(-I\)\ \((\(-1\))\)\^0\ x\^0\ \((1\/x)\)\^0\ D[E\^\(I\ x\)\/x, {x, 
                0}]}, {"n = 1", \(-I\)\ \((\(-1\))\)\^1\ x\^1\ \((1\/x)\)\ D[\((1\/x)\)\^0\ D[E\^\(I\ x\)\/x, {x, 0}], {x, 1}] // 
            Expand}, {"n = 2", \(\(-I\)\ \((\(-1\))\)\^2\ x\^2\ \((1\/x)\)\ D[\((1\/x)\)\ D[\((1\/x)\)\^0\ D[E\^\(I\ x\)\/x, {x, 0}], {x, 1}], {x, 1}] // PowerExpand\) // Expand}}]\)

To generate (Subscript[h, n]^((2)))(x) using rayleigh formulas:

\(TableForm[{{"n = 0", 
          I\ \((\(-1\))\)\^0\ x\^0\ \((1\/x)\)\^0\ D[E\^\(\(-I\)\ x\)\/x, {x, 0}]}, {"n = 1", 
          I\ \((\(-1\))\)\^1\ x\^1\ \((1\/x)\)\ D[\((1\/x)\)\^0\ D[E\^\(\(-I\)\ x\)\/x, {x, 0}], {x, 1}] // 
            Expand}, {"n = 2", \(I\ \((\(-1\))\)\^2\ x\^2\ \((1\/x)\)\ D[\((1\/x)\)\ D[\((1\/x)\)\^0\ D[E\^\(\(-I\)\ x\)\/x, {x, 
                            0}], {x, 1}], {x, 1}] // PowerExpand\) // 
            Expand}}]\)

To generate (Subscript[i, n])(x) using rayleigh formulas:

\(TableForm[{{"n = 0", 
          x\^0\ \((1\/x)\)\^0\ D[Sinh[x]\/x, {x, 0}]}, {"n = 1", 
          x\^1\ \((1\/x)\)\ D[\((1\/x)\)\^0\ D[Sinh[x]\/x, {x, 0}], {x, 1}] // 
            Expand}, {"n = 2", \(x\^2\ \((1\/x)\)\ D[\((1\/x)\)\ D[\((1\/x)\)\^0\ D[Sinh[x]\/x, {x, 0}], {x, 1}], {x, 1}] // PowerExpand\) // 
            Expand}}]\)

To generate (Subscript[k, n])(x) using rayleigh formulas:

\(TableForm[{{"n = 0", \((\(-1\))\)\^0\ x\^0\ \((1\/x)\)\^0\ D[E\^\(-x\)\/x, {x, 
                0}]}, {"n = 1", \((\(-1\))\)\^1\ x\^1\ \((1\/x)\)\ D[\((1\/x)\)\^0\ D[E\^\(-x\)\/x, {x, 0}], {x, 
                1}]}, {"n = 2", \(\((\(-1\))\)\^2\ x\^2\ \((1\/x)\)\ D[\((1\/x)\)\ D[\((1\/x)\)\^0\ D[E\^\(-x\)\/x, {x, 0}], {x, 
                        1}], {x, 1}] // PowerExpand\) // Expand}}]\)

We can use the Mathematica built-in Bessel functions and the definitions to generate the spherical Bessel functions as follows. For example, to generate spherical Bessel functions: (Subscript[j, 0])(x), (Subscript[j, 1])(x), and (Subscript[j, 2])(x)

TableForm[
      Table[{"n =", 
          n, (Sqrt[(π/(2  x))]) BesselJ[n + (1/2), x] // PowerExpand // 
            Expand}, {n, 0, 2}]]

To generate spherical Bessel functions: (Subscript[n, 0])(x), (Subscript[n, 1])(x), and (Subscript[n, 2])(x)

TableForm[
      Table[{"n =", 
          n, (Sqrt[(π/(2  x))]) BesselY[n + (1/2), x] // PowerExpand // 
            Expand}, {n, 0, 2}]]

To generate spherical Bessel functions: (Subscript[h, 0]^((1)))(x), (Subscript[h, 1]^((1)))(x), and (Subscript[h, 2]^((1)))(x)

TableForm[
      Table[{"n =", 
          n, (Sqrt[(π/(2  x))]) (BesselJ[n + (1/2), x] +  
                      I BesselY[n + (1/2), x]) // PowerExpand // 
              Expand // TrigToExp}, {n, 0, 2}]]

To generate spherical Bessel functions: (Subscript[h, 0]^((2)))(x), (Subscript[h, 1]^((2)))(x), and (Subscript[h, 2]^((2)))(x)

TableForm[
      Table[{"n =", 
          n, (Sqrt[(π/(2  x))]) (BesselJ[n + (1/2), x] -  
                      I BesselY[n + (1/2), x]) // PowerExpand // 
              Expand // TrigToExp}, {n, 0, 2}]]

To generate spherical Bessel functions: (Subscript[i, 0])(x), (Subscript[i, 1])(x), and (Subscript[i, 2])(x)

TableForm[
      Table[{"n =", 
          n, (Sqrt[(π/(2  x))]) BesselI[n + (1/2), x] // PowerExpand // 
            Expand}, {n, 0, 2}]]

To generate spherical Bessel functions: (Subscript[k, 0])(x), (Subscript[k, 1])(x), and (Subscript[k, 2])(x)

TableForm[
      Table[{"n =", 
          n, (Sqrt[(2/(π x))]) BesselK[n + (1/2), x] // PowerExpand // 
            Expand}, {n, 0, 2}]]

Let's plot the first three spherical Bessel functions.

\(\(Plot[{\(\@\(π\/\(2  x\)\)\) 
            BesselJ[1\/2, x], \(\@\(π\/\(2  x\)\)\) 
            BesselJ[3\/2, x], \(\@\(π\/\(2  x\)\)\) 
            BesselJ[5\/2, x]}, {x, 0, 14}, 
        PlotRange -> \ {\(-0.3\), 0.8}, 
        PlotStyle -> {{RGBColor[0, 0, 1]}, {RGBColor[1, 0, 
                1]}, {RGBColor[0, 1, 0]}}, 
        Epilog -> \ \(({Text[\*""\!\(j\_0\)(x)"", {2.4, 0.7}], 
              Text[\*""\!\(j\_1\)(x)"", {3.4, 0.45}], 
              Text[\*""\!\(j\_2\)(x)"", {5.4, 
                  0.3}]})\)];\)\)
\(\(Plot[{\(\@\(π\/\(2  x\)\)\) 
            BesselY[1\/2, x], \(\@\(π\/\(2  x\)\)\) 
            BesselY[3\/2, x], \(\@\(π\/\(2  x\)\)\) 
            BesselY[5\/2, x]}, {x, 0, 14}, 
        PlotRange -> \ {\(-0.4\), 0.4}, 
        PlotStyle -> {{RGBColor[0, 0, 1]}, {RGBColor[1, 0, 
                1]}, {RGBColor[0, 1, 0]}}, 
        Epilog -> \ \(({Text[\*""\!\(n\_0\)(x)"", {3.5, 0.37}], 
              Text[\*""\!\(n\_1\)(x)"", {5.4, 0.26}], 
              Text[\*""\!\(n\_2\)(x)"", {7.3, 
                  0.18}]})\)];\)\)
\(\(Plot[{\(\@\(π\/\(2  x\)\)\) 
            BesselI[1\/2, x], \(\@\(π\/\(2  x\)\)\) 
            BesselI[3\/2, x], \(\@\(π\/\(2  x\)\)\) 
            BesselI[5\/2, x], \(\@\(2\/\(π\ x\)\)\) 
            BesselK[1\/2, x], \(\@\(2\/\(π\ x\)\)\) 
            BesselK[3\/2, x], \(\@\(2\/\(π\ x\)\)\) 
            BesselK[5\/2, x]}, {x, 0, 5}, PlotRange -> \ {0, 8}, 
        PlotStyle -> {{RGBColor[0, 0, 1]}, {RGBColor[1, 0, 
                1]}, {RGBColor[0, 1, 0]}}, 
        Epilog -> \ \(({Text[\*""\!\(i\_0\)(x)"", {0.48, 7.5}], 
              Text[\*""\!\(i\_1\)(x)"", {0.7, 7.0}], 
              Text[\*""\!\(i\_2\)(x)"", {1.15, 6.5}], 
              Text[\*""\!\(k\_0\)(x)"", {3.8, 7.5}], 
              Text[\*""\!\(k\_1\)(x)"", {4.7, 7.0}], 
              Text[\*""\!\(k\_2\)(x)"", {5.2, 6.5}]})\)];\)\)

Legendre Functions

When Helmholtz's equation (((Overscript[∇,  ->]) )^2)ψ ± (k^2) ψ = 0 is separated in spherical polar coordinate, one of the separated ordinary differential equations (with x = cos θ) is the associated Legendre equation:
	(1 - (x^2))(((d^2) Θ(x))/(dx^2)) - 2 x ((d Θ(x))/dx) + [n(n+1) - ((m^2)/(1  -  (x^2)))Θ(x) = 0.
The two linearly independent solutions are the associated Legendre function of the first kind, (Subscript[P, n]^m)(x), and the associated Legendre function of the second kind, (Subscript[Q, n]^m)(x). With m = 0, the associated Legendre equation reduces to the Legendre equation and the associated Legendre functions reduce to the Legendre functions (Subscript[P, n])(x) and (Subscript[Q, n])(x). The Legendre functions arise in studites of quantum mechanical scattering processes. When n and m are integers, the Legendre functions reduce to Legendre polynomials.

Legendre Polynomials

The Legendre polynomials can be generated using the Rodriger's formula
	(Subscript[P, n])(x) = (1/((2^n) n!))((d^n)/(dx^n))((((x^2)  -  1))^n)

legPoly[n_Integer?NonNegative, 
          x_Symbol] := ((1/((2^n) n!)) D[((((x^2)  -  1))^n), {x, 
                  n}]) // Simplify;
legPoly[n_Integer?NonNegative, 
          x_Real] := ((1/((2^n) n!)) D[((((x^2)  -  1))^n), {y, 
                    n}] /. y -> x) // 
          Simplify;
TableForm[Table[{"n =", n, legPoly[n, x]}, {n, 0, 5}]]

We can also use Mathematica built-in function LegendreP[n,x] to generate the Legendre polynomials.

TableForm[
      Table[{"n =", n, LegendreP[n, x]}, {n, 0, 5}] //  
        Factor]

The Legendre polynomials are defined over the interval (-1, +1) and satisfy the orthonormal condition
	(∫_(-1)^1) (Subscript[P, m])(x) (Subscript[P, n])(
            x) dx = ((2  (Subscript[δ, m, n]))/(2  n  +  1)).
With x = cos θ, the orthonormal condition can also be written as
	(∫_0^π) (Subscript[P, m])(
            cos θ) (Subscript[P, n])(cos θ) 
          sin θ dθ = ((2  (Subscript[δ, m, n]))/(2  n  +  1)).
For example,

(∫_(-1)^1) LegendreP[3, x] LegendreP[4, 
          x] \[DifferentialD]x
(∫_(-1)^1) LegendreP[3, x] LegendreP[3, 
          x] \[DifferentialD]x

Let's plot a few Legendre polynomials, (Subscript[P, 2])(x), (Subscript[P, 3])(x), (Subscript[P, 4])(x), and (Subscript[P, 5])(x). Of course, we can plot the Legendre polynomials as a function of θ (from 0 to π) rather than a function of x (from -1 to 1).

\(\(Plot[{LegendreP[2, x], LegendreP[3, x], LegendreP[4, x], 
          LegendreP[5, x]}, {x, \(-1\), 1}, PlotRange -> \ {\(-1\), 1}, 
        PlotStyle -> {{RGBColor[0, 0, 1]}, {RGBColor[1, 0, 
                1]}, {RGBColor[0, 1, 0]}, {RGBColor[1, 0, 0]}}, 
        Epilog -> \ \(({Text[\*""\!\(P\_2\)(x)"", {\(-0.75\), 
                  0.75}], Text[\*""\!\(P\_3\)(x)"", {\(-0.45\), 0.55}], 
              Text[\*""\!\(P\_4\)(x)"", {0.15, 0.5}], 
              Text[\*""\!\(P\_5\)(x)"", {0.5, 0.35}]})\)];\)\)

Note that high-degree Legendre polynomials oscillate rapidly. For example, let's plot (Subscript[P, 20])(x):

Plot[LegendreP[20, x], {x, -1, 1},PlotPoints->50];

Associated Legendre Polynomials

The associated Legendre polynomials are defined by
	(Subscript[P, n]^m)(x) = (((1  -  (x^2)))^(m/2))((d^m)/(dx^m))[(Subscript[P, n])(x)].
Since the highest power of x in (Subscript[P, n])(x) is (x^n) (see table for (Subscript[P, n])(x) above), we must have m ≤ n. In quantum mechanics the requirement than m ≤ n has the physical interpretation that the expectation value of the square of the z-component of the angular momentum ((Subscript[L, z])) is less than or equal to the expectation value of the square of the angular momentum vctor (Overscript[L,  ->]) .

From the definition for the associated Legendre polynomials given above, one might expect m to be non-negative since differentiating a negative number of times is not defined. However, if (Subscript[P, n])(x) is expressed by Rodrigues's formula (see Legendre polynomials above), this limitation on m is relaxed and we have -n ≤ m ≤ n, negative as well as positive values of m being permitted. Note that (Subscript[P, n]^(-m))(x) and (Subscript[P, n]^m)(x) are related by
	(Subscript[P, n]^(-m))(x)  = (((-1))^m)(((n  -  m)!)/((n  +  m)!))(Subscript[P, n]^m)(x).

The associated Legendre polynomials can be generated using the Rodriger's formula
	(Subscript[P, n]^m)(x) = (((-1))^m)((((1  -  (x^2)))^(m/2))/((2^n) n!))((d^(n + m))/(dx^(n + m)))((((x^2)  -  1))^n).
Note that the associated Legendre polynomials contain pieces proportional to (Sqrt[1  -  (x^2)]).

assoLegPoly[n_Integer?NonNegative, m_Integer?NonNegative, 
          x_Symbol] := ((((-1))^m) ((((1 - (x^2)))^(m/2))/((2^n) n!)) D[((((x^2)  -  1))^n), {x, (n + m)}]) // 
          Simplify;
assoLegPoly[n_Integer?NonNegative, m_Integer?NonNegative, 
          x_Real] := ((((-1))^m) ((((1  -  (x^2)))^(m/2))/((2^n) n!)) D[((((x^2)  -  1))^n), {y, (n + m)}] /. y -> x) // 
          Simplify;
TableForm[
      Table[{"n = 0,    m =", m, assoLegPoly[0, m, x]}, {m, 0, 
          0}]]
TableForm[
      Table[{"n = 1,    m =", m, assoLegPoly[1, m, x]}, {m, 0, 
          1}]]
TableForm[Table[{"n = 2,    m =", m, assoLegPoly[2, m, x]}, {m, 
              0, 2}] // Simplify // Together]
TableForm[
      Table[{"n = 3,    m =", m, assoLegPoly[3, m, x]}, {m, 0, 
          3}]]

We can also use Mathematica built-in function LegendreP[n,m,x] to generate the associated Legendre polynomials.

TableForm[
      Table[{"n = 0,    m =", m, LegendreP[0, m, x]}, {m, 0, 0}] // 
        Factor]
TableForm[
      Table[{"n = 1,    m =", m, LegendreP[1, m, x]}, {m, 0, 1}] // 
        Factor]
TableForm[
      Table[{"n = 2,    m =", m, LegendreP[2, m, x]}, {m, 0, 2}] // 
        Together]
TableForm[Table[{"n = 3,    m =", m, LegendreP[3, m, x]}, {m, 0, 
              3}] // Simplify // Together]

The associated Legendre polynomials are defined over the interval (-1, +1) and satisfy the orthonormal condition
	(∫_(-1)^1) (Subscript[P, p]^m)(
            x) (Subscript[P, q]^m)(x) dx = (2/(2  q  +  1))(((q  +  m)!)/((q  -  m)!))(Subscript[δ, p, q]).
With x = cos θ, the orthonormal condition can also be written as
	(∫_0^π) (Subscript[P, p]^m)(
            cos θ) (Subscript[P, q]^m)(cos θ) 
          sin θ dθ = (2/(2  q  +  1))(((q  +  m)!)/((q  -  m)!))(Subscript[δ, p, q]).
For example,

(∫_(-1)^1) LegendreP[3, 2, x] LegendreP[4, 2, 
          x] \[DifferentialD]x
(∫_(-1)^1) LegendreP[3, 2, x] LegendreP[3, 2, 
          x] \[DifferentialD]x

Spherical Harmonics

When Helmholtz's equation (((Overscript[∇,  ->]) )^2)ψ ± (k^2) ψ = 0 is separated in spherical polar coordinate, one of the separated ordinary differential equations (with x = cos θ) is the associated Legendre equation. Its solutions are already discussed. The other separated ordinary differential equation is the separated azimuthal equation:
	(1/(Φ (ϕ)))\(TraditionalForm\`\(\(d\^2\) Φ\ \((ϕ)\)\)\/\(d\ ϕ\^2\)\) = - (m^2).
The (normalized) solution is
	(Subscript[Φ, m])(ϕ) = (1/(Sqrt[2 π]))(e^im)ϕ.
The spherical harmonics are related to the associated Legendre polynomials and are defined as the product of (Subscript[P, n]^m)(θ) and (Subscript[Φ, m])(ϕ):
	(Subscript[Y, n]^m)(θ,ϕ) = (((-1))^m)\(TraditionalForm\`\@\(\(2  n\  + \ 1\)\/\(4\ π\)\ \(\((n\  - \ m)\)!\)\/\(\((n\  + \ m)\)!\)\)\)(Subscript[P, n]^m)(θ) (e^im)ϕ.

We can use Mathematica built-in function SphericalHarmonicsY[n,m,θ,ϕ] to generate the spherical harmonics.

TableForm[
      Table[{"n = 0,    m =", m, 
            SphericalHarmonicY[0, m, θ, ϕ]}, {m, 0, 0}] // 
        Factor]
TableForm[
      Table[{"n = 1,    m =", m, 
            SphericalHarmonicY[1, m, θ, ϕ]}, {m, -1, 1}] // 
        Factor]
TableForm[
      Table[{"n = 2,    m =", m, 
            SphericalHarmonicY[2, m, θ, ϕ]}, {m, -2, 2}] // 
        Together]
TableForm[Table[{"n = 3,    m =", m, 
              SphericalHarmonicY[3, m, θ, ϕ]}, {m, -3, 3}] // 
          Simplify // Together]

The spherical harmonics are defined over the surface of (a unit) sphere with θ the polar angle and ϕ the azimuth and satisfy the orthonormal condition
	(∫_(ϕ = 0)^(2 π))\(TraditionalForm\`∫\_\(θ = 0\)\%π\ \ \(\(Y\_\(n\_1\)\%\(m\_1\)\^*\)(θ, ϕ)\) \(\(Y\_\(n\_2\)\%\(m\_2\)\)(θ, ϕ)\) 
          sin\ θ dθ dϕ\) = (Subscript[δ, (Subscript[n, 1]), (Subscript[n, 2])])(Subscript[δ, (Subscript[m, 1]), (Subscript[m, 2])]).
For example,

Y42[θ_, ϕ_] :=  
        SphericalHarmonicY[4, 
          2, θ, ϕ];
Y33[θ_, ϕ_] :=  
        SphericalHarmonicY[3, 3, θ, ϕ];
  
Y33star[θ_, ϕ_] :=  (Y33[θ, ϕ] /. 
            Complex[p_, q_]  ->  
              Complex[p, -q]);
(∫_0^(2  π))((∫_0^π) Evaluate[
                Y33star[θ, ϕ] Y42[θ, ϕ] Sin[θ]] \[DifferentialD]θ) \[DifferentialD]ϕ
(∫_0^(2  π))((∫_0^π) Evaluate[
                Y33star[θ, ϕ] Y33[θ, ϕ] Sin[θ]] \[DifferentialD]θ) \[DifferentialD]ϕ

Hermite Polynomials

Hermite polynomials arise as the quantum mechanical wave functions for a harmonic oscillator. They satisfy the differential equation
	(((d^2) (Subscript[H, n])(x))/(dx^2))- 2x(((Subscript[dH, n])(x))/dx)+ 2n(Subscript[H, n])(x) = 0.

The Hermite polynomials can be generated using the Rodriger's formula
	(Subscript[H, n])(x) = (((-1))^n)(e^( (x^2)))((d^n)/(dx^n))((e^(-(x^2))))

hermitePoly[n_Integer?NonNegative, 
          x_Symbol] := ((((-1))^n) Exp[(x^2)] D[
                Exp[-(x^2)], {x, n}]) // 
          Simplify;
hermitePoly[n_Integer?NonNegative, 
          x_Real] := ((((-1))^n) Exp[(y^2)] D[
                  Exp[-(y^2)], {y, n}] /. y -> x) // 
          Simplify;
TableForm[
      Table[{"n =", n, hermitePoly[n, x] // Expand}, {n, 0, 
          5}]]

We can also use Mathematica built-in function HermiteH[n,x] to generate the Hermite polynomials.

TableForm[Table[{"n =", n, HermiteH[n, x]}, {n, 0, 5}]]

The Hermite polynomials are defined over the interval (-∞, +∞) and satisfy the orthonormal condition
	(∫_(-∞)^∞) 
                (Subscript[H, m])(x) (Subscript[H, n])(
            x) (e^(-(x^2))) dx = {({{(2^n) n! (Sqrt[π]),  m = n}, {0 ,  m ≠ n}})
For example,

(∫_(-∞)^∞) HermiteH[3, x] HermiteH[4, 
          x] Exp[-(x^2)] \[DifferentialD]x
(∫_(-∞)^∞) HermiteH[3, x] HermiteH[3, 
          x] Exp[-(x^2)] \[DifferentialD]x

Let's plot the first three Hermite polynomials.

\(\(Plot[{HermiteH[0, x], HermiteH[1, x], HermiteH[2, x]}, {x, 0, 2}, 
        PlotRange -> \ {\(-3\), 10}, 
        PlotStyle -> {{RGBColor[0, 0, 1]}, {RGBColor[1, 0, 
                1]}, {RGBColor[0, 1, 0]}}, 
        Epilog -> \ \(({Text[\*""\!\(H\_0\)(x)"", {1.75, 1.5}], 
              Text[\*""\!\(H\_1\)(x)"", {1.75, 4.2}], 
              Text[\*""\!\(H\_2\)(x)"", {1.45, 8.5}]})\)];\)\)

Actually, it would be more interesting to plot ((Subscript[H, n])(x) (e^(-(x^2)/2)))\(TraditionalForm\`\^2\) which gives the density for a quantum mechanical harmonic oscillator. Let's plot the first four ((Subscript[H, n])(x) (e^(-(x^2)/2)))\(TraditionalForm\`\^2\).

n0 = 
      Plot[(((HermiteH[0, x] (E^(-((x^2)/2)))))^2), {x, -4, 4}, DisplayFunction -> Identity, 
        PlotRange -> All]; 
n1 = 
      Plot[(((HermiteH[1, x] (E^(-((x^2)/2)))))^2), {x, -4, 4}, DisplayFunction -> Identity, 
        PlotRange -> All]; 
n2 = 
      Plot[(((HermiteH[2, x] (E^(-((x^2)/2)))))^2), {x, -4, 4}, DisplayFunction -> Identity, 
        PlotRange -> All]; 
n3 = 
      Plot[(((HermiteH[3, x] (E^(-((x^2)/2)))))^2), {x, -4, 4}, DisplayFunction -> Identity, 
        PlotRange -> All]; 
Show[GraphicsArray[{{n0, n1}, {n2, n3}}]];

For large n, the average of the wiggles is roughly the classical physics result.

Plot[(((HermiteH[40, x] (E^(-((x^2)/2)))))^2), {x, -10, 10}, PlotRange -> All, PlotPoints -> 200];

Euler Gamma Function

The gamma function appears occasionally in physical problems such as the normalization of coulomb wave wave functions and the computation of probabilities in statistical mechanics.

Using integration by parts techniques, it is easy to derive, for integer n, the following:
	(∫_0^∞) 
              (x^n) (e^(-x)) dx = n(n-1)(n-2)⋯3·2·1 ≡ n!
where n! is read as "n-factorial". The integral can be generalized to non-integer values of n (written as z), and the gamma function is defined (also frequently called Euler's form) as:
	Γ(z) = (∫_0^∞) 
              (x^(z - 
                1)) (e^(-x)) dx   for z ≠ 0, -1, -2, -3, ⋯
The -1 of the z-1 exponent in the integrand is a continual nuisance. Accordingly, the expression above is rewritten as
	(∫_0^∞) 
              (x^z) (e^(-x)) dx ≡ z!
which defines the factorial function z!. Occasionally we may still encounter ∏(z), the Gauss's notation, for the factorial function
	Π(z) = z!.
The factorial function is, of course, related to the gamma function by
	Γ(z) = (z-1)!
or
	Γ(z+1) = z!.
If z = n, a positive integer, then
	z! = n! = 1·2·3⋯n,
the familiar factorial.

Note that the gamma function satisfies
	Γ(z+1) = z Γ(z)
	Γ(z) Γ(1-z) = (π/(sin zπ)) (see Arfken & Weber, pp. 359-360 for a proof).
Setting z = 1/2, we obtain
	Γ((1/2)) = (∫_0^∞) 
              (x^(-1/
                2)) (e^(-x)) dx = 2\(TraditionalForm\`∫\_0\%∞\( e\^\(-t\^2\)\) dt\) = (Sqrt[π]).
Note that the last integral is just the Gauss error function.

In many problems of mathematical physics, particularly in connection with Legendre polynomials, we encounter products of the odd positive integers and products of the even positive integers. For convenience these are given special labels as double factorials:
	1·3·5⋯ (2n+1) = (2n+1)!!
	    2·4·6⋯ (2n) = (2n)!!.
Clearly, these are related to the regular factorial functions by
	(2n)!! = (2^n)n!
	(2n+1)!! = (((2  n + 1)!)/((2^n) n!)).

For integer n, we have
	Γ(n + (1/2)) = \(TraditionalForm\`\(1·3·5  ⋯\ \((2  n - 1)\)\)\/2\^n\)(Sqrt[π]) = (((2  n - 1)!!)/(2^n))(Sqrt[π]).
Finally, we note that Stirling's series can be used to estimate the value of the factorial function for large argument, namely,
	Γ(z+1) = z! ∼ (Sqrt[2  π])(z^(z + (1/2))) (e^(-z))(1 + (1/(12  z)) + (1/(288  (z^2))) - (139/(51840  (z^3))) + ⋯).

We can also use Mathematica built-in function Gamma[z] to compute the gamma function. For example,.

TableForm[
      Table[{{"Γ(1) (or 0!) =", 
            Gamma[1]}, {"Γ(2) (or 1!) =", 
            Gamma[2]}, {"Γ(3) (or 2!) =", 
            Gamma[3]}, {"Γ(4) (or 3!) =", 
            Gamma[4]}, {"Γ(1/2) =", 
            Gamma[1/2]}, {"Γ(3/2) =", 
            Gamma[3/2]}, {"Γ(5/2) =", 
            Gamma[5/2]}}]]

Γ(z+1) = z! is plotted below for z ranging from -1 to 4:

Plot[Gamma[z + 1], {z, -1, 4}, 
        Epilog ->  ({Text["z!", {3.7, 20.0}]})];

Laguerre Functions

Laguerre Polynomials

Laguerre's differential equation is
	x(((d^2) (Subscript[L, n])(x))/(dx^2)) + (1-x)((d (Subscript[L, n])(x))/dx) + n (Subscript[L, n])(x) = 0.

The Laguerre polynomials can be generated using the Rodriger's formula
	(Subscript[L, n])(x) = ((e^( x))/(n !))((d^n)/(dx^n))((x^( n))(e^(-x)))

laguerrePoly[n_Integer?NonNegative, 
          x_Symbol] := (Exp[x]/n! D[(x^n) Exp[-x], {x, n}]) // 
          Simplify;
laguerrePoly[n_Integer?NonNegative, 
          x_Real] := (Exp[y]/n! D[(y^n) Exp[-y], {y, n}] /. 
              y -> x) // Simplify;
TableForm[
      Table[{"n =", n, laguerrePoly[n, x] // Expand}, {n, 0, 
          5}]]

We can also use Mathematica built-in function LaguerreL[n,x] to generate the Laguerre polynomials.

TableForm[
      Table[{"n =", n, LaguerreL[n, x] // Expand}, {n, 0, 5}]]

The orthonormal condition for the Laguerre polynomials is
	(∫_0^∞) (e^(-x)) (Subscript[L, m])(
            x) (Subscript[L, n])(x) dx = (Subscript[δ, m, n])
For example,

(∫_0^∞) Exp[-x] LaguerreL[3, x] LaguerreL[4, 
          x] \[DifferentialD]x
(∫_0^∞) Exp[-x] LaguerreL[3, x] LaguerreL[3, 
          x] \[DifferentialD]x

Let's plot the first three Laguerre polynomials.

\(\(Plot[{LaguerreL[0, x], LaguerreL[1, x], LaguerreL[2, x]}, {x, 0, 4}, 
        PlotRange -> \ {\(-4\), 2}, 
        PlotStyle -> {{RGBColor[0, 0, 1]}, {RGBColor[1, 0, 
                1]}, {RGBColor[0, 1, 0]}}, 
        Epilog -> \ \(({Text[\*""\!\(L\_0\)(x)"", {2.5, 1.2}], 
              Text[\*""\!\(L\_1\)(x)"", {3.5, \(-0.5\)}], 
              Text[\*""\!\(L\_2\)(x)"", {3.5, \(-2.0\)}]})\)];\)\)

Associated Laguerre Polynomials

In many applications, particularly in quantum mechanics, we need the associated Laguerre polynomials defined by
	(Subscript[L, n]^k)(x) = (((-1))^k)((d^k)/(dx^k))[(Subscript[L, n + k])(x)]
The differential equation satisfies by the associated Laguerre polynomials is
	x(((d^2) (Subscript[L, n]^k)(x))/(dx^2)) + (k+1-x)((d (Subscript[L, n]^k)(x))/dx) + n (Subscript[L, n]^k)(x) = 0.

The associated Laguerre polynomials can be generated using the Rodriger's formula
	(Subscript[L, n]^k)(x) = (((e^( x)) (x^(-k)))/(n !))((d^n)/(dx^n))((x^( n + k))(e^(-x)))

assoLaguerrePoly[n_Integer?NonNegative, k_Integer?NonNegative, 
          x_Symbol] := ((Exp[x] (x^(-k)))/n! D[
                (x^(n + k)) Exp[-x], {x, n}]) // 
          Simplify;
assoLaguerrePoly[n_Integer?NonNegative, k_Integer?NonInteger, 
          x_Real] := ((Exp[y] (y^(-k)))/n! D[
                  (y^(n + k)) Exp[-y], {y, n}] /. y -> x) // 
          Simplify;
\(TableForm[
      Table[{n, "0", assoLaguerrePoly[n, 0, x] // Expand}, {n, 0, 2}], 
      TableHeadings -> \ {None, {"n", "k", \*""(Subscript[L, n]^k)(x)""}}]\)
\(TableForm[
      Table[{n, "1", assoLaguerrePoly[n, 1, x] // Expand}, {n, 0, 2}], 
      TableHeadings -> \ {None, {"n", "k", \*""(Subscript[L, n]^k)(x)""}}]\)
\(TableForm[
      Table[{n, "2", assoLaguerrePoly[n, 2, x] // Expand}, {n, 0, 2}], 
      TableHeadings -> \ {None, {"n", "k", \*""(Subscript[L, n]^k)(x)""}}]\)
\(TableForm[
      Table[{n, "3", assoLaguerrePoly[n, 3, x] // Expand}, {n, 0, 2}], 
      TableHeadings -> \ {None, {"n", "k", \*""(Subscript[L, n]^k)(x)""}}]\)

We can also use Mathematica built-in function LaguerreL[n,k,x] to generate the associated Laguerre polynomials. For example, for k=3 and n ranges from 0 to 2, we have

TableForm[
      Table[{"n =", n, LaguerreL[n, 3, x] // Expand}, {n, 0, 
          2}]]

The orthonormal condition for the associated Laguerre polynomials is
	(∫_0^∞) 
              (e^(-x)) (x^k) (Subscript[L, m]^k)(x) (Subscript[L, n]^k)(
            x) dx = (((n + k)!)/(n!))(Subscript[δ, m, n])
For example,

(∫_0^∞) Exp[-x] (x^3) LaguerreL[2, 3, 
          x] LaguerreL[3, 3, 
          x] \[DifferentialD]x
(∫_0^∞) Exp[-x] (x^3) LaguerreL[2, 3, 
          x] LaguerreL[2, 3, x] \[DifferentialD]x

Dirac Delta Function

In many practical situations it is convenient to consider limits in which a fixed amount of something is concentrated into an infinitesimal region. For example, let's plot the function
	f(x) = (Sqrt[(n/π)]) (e^(-n (x^2)))
for three values of n (1, 10, 100):

Plot[
        Evaluate[(Sqrt[(n/π)]) Exp[-n (x^2)] /. 
            n ->  {1, 10, 100}], {x, -2, 2}, 
        PlotRange ->  All];

Note that although the function becomes progresively more concentrated as n gets larger, ordinary mathematical funcions of the kind normally encountered in calculus can not readily represent such limits. However, it is possible to introduce generalized functions which can represent such limits. The Dirac delta function is such a generalized function and is defined by the relation
	(∫_(-∞)^∞) f(
            x) δ(x - (Subscript[x, 0])) dx = f((Subscript[x, 0]))
The integral picks out only the first term in the Taylor series expansion of the function f(x) around (Subscript[x, 0]), and this relation must hold for any function. One can also define the delta function as:
	δ(x-(Subscript[x, 0])) = 0, if x ≠ (Subscript[x, 0])
	            = ∞, if x = (Subscript[x, 0]).
The delta function is normalized to unity as can be seen from the integral above if we choose f(x) = 1. That is
	\(TraditionalForm\`∫\_\(-∞\)\%∞\ \(δ(x - x\_0)\) dx\) = 1.
Let's list some of the properties of the delta function:
	∫f(x) δ(
            x) dx = f(0)
	\(TraditionalForm\`\(δ\^*\)\)(x) = δ(x)			i.e., it is real
	δ(x) = δ(-x)			i.e., it is an even function
	δ(ax) = \(TraditionalForm\`1\/\(\(|\)\(a\)\(|\)\)\)δ(x)
	∫f(
            x) δ' (x) dx = -f '(0)	where δ '(x) = (d/(d x))δ(x)
	δ '(-x) = - δ '(x)
	x δ(x) = 0
	δ((x^2)-(a^2)) = \(TraditionalForm\`1\/\(2\  | a | \)\)[δ(x-a) + δ(x-a)]
	f(x) δ(x-a) = f(a) δ(x-a)
	∫δ(x - b) δ(
            a - x) dx = δ(a-b)