Method of Applied Math
Lecture 2: Legendre’s Equation and Legendre Polynomials
Sujin Khomrutai, Ph.D.
The Gamma function
Gamma function Properties
EX 1.
EX 2.
Legendre’s eqn EX 3.
Lengendre func Legendre poly EX 4.
Rodrigues’ form EX 5.
EX 6.
Definition The Gamma function, denoted Γ(x), is the function defined by
Γ(x) =
Z ∞
0
e−ttx−1 dt for all real number x > 0.
For x < 0 such that x 6∈ {−1, −2, . . .}, it is defined by
Γ(x) = Γ(x + k)
(x + k − 1)(x + k − 2) · · · (x + 1)x
where k is a natural number taken so that x + k > 0.
The Gamma function
Gamma function Properties
EX 1.
EX 2.
Legendre’s eqn EX 3.
Lengendre func Legendre poly EX 4.
Rodrigues’ form EX 5.
EX 6.
The graph of Gamma function is as shown below
Properties of Gamma function
Gamma function Properties
EX 1.
EX 2.
Legendre’s eqn EX 3.
Lengendre func Legendre poly EX 4.
Rodrigues’ form EX 5.
EX 6.
• It is a basic exercise in calculus that Γ(x + 1) = xΓ(x)
for all x ∈ R \ {−1, −2, . . . ,}.
• If k is a natural number, then
Γ(x + k) = (x + k − 1) · · ·(x + 1)xΓ(x) for all x ∈ R \ {−1, −2, . . . ,}.
• In particular, if n is zero or a natural number, then Γ(n + 1) = n!.
Example 1
Gamma function Properties
EX 1.
EX 2.
Legendre’s eqn EX 3.
Lengendre func Legendre poly EX 4.
Rodrigues’ form EX 5.
EX 6.
EX. Find the value of Γ(5) + Γ(0).
Example 2
Gamma function Properties
EX 1.
EX 2.
Legendre’s eqn EX 3.
Lengendre func Legendre poly EX 4.
Rodrigues’ form EX 5.
EX 6.
EX. Show that Γ
1 2
= √ π.
Legendre’s equation
Gamma function Properties
EX 1.
EX 2.
Legendre’s eqn EX 3.
Lengendre func Legendre poly EX 4.
Rodrigues’ form EX 5.
EX 6.
Definition The Legendre equation is the equation of the form (1 − x2)y′′ − 2xy′ + k(k + 1)y = 0
where k is a constant.
The most important special case is when k is a natural number.
Alternative form:
(1 − x2)y′′
+ k(k + 1)y = 0
Example 3: Legendre’s equation in Electrostatics
Gamma function Properties
EX 1.
EX 2.
Legendre’s eqn EX 3.
Lengendre func Legendre poly EX 4.
Rodrigues’ form EX 5.
EX 6.
Example. A two metallic spherical caps are placed so that the upper part stayed at a constant potential 110 V and the lower part is grounded. Finding the electrostatic potential at any point in the space is reduced to solve the Legendre’s equation.
Series solutions of Legendre’s equation
Gamma function Properties
EX 1.
EX 2.
Legendre’s eqn EX 3.
Lengendre func Legendre poly EX 4.
Rodrigues’ form EX 5.
EX 6.
Power series method. a = 0 is an ordinary point for the Legendre’s equation.
So we set solution y to the Legendre’s equation as y =
∞
X
n=0
bnxn
y′ =
∞
X
n=0
bnnxn−1 =
∞
X
n=1
bnnxn−1
y′′ =
∞
X
n=0
bnn(n − 1)xn−2 =
∞
X
n=2
bnn(n − 1)xn−2.
Series solutions of Legendre’s equation
Gamma function Properties
EX 1.
EX 2.
Legendre’s eqn EX 3.
Lengendre func Legendre poly EX 4.
Rodrigues’ form EX 5.
EX 6.
Substituting y, y′, y′′ in the Legendre’s equation by with the above power series expansions, we get
∞
X
n=0
(n + 1)(n + 2)bn+2 + (k − n)(k + n + 1)bn
xn = 0
Then we have the recurrence equation:
bn+2 = −(k − n)(k + n + 1)
(n + 1)(n + 2) bn (∗)
holds for n = 0, 1,2, . . .
Series solutions of Legendre’s equation
Gamma function Properties
EX 1.
EX 2.
Legendre’s eqn EX 3.
Lengendre func Legendre poly EX 4.
Rodrigues’ form EX 5.
EX 6.
Solve the recurrence equation (∗): For n = 0, 1, 2,3, we get b2 = −k(k + 1)
2! b0, b4 = −(k − 2)(k + 3)
3 · 4 b2 = (k − 2)k(k + 1)(k + 3)
4! b0
and
b3 = −(k − 1)(k + 2) 3! b1
b5 = −(k − 3)(k + 4)
4 · 5 b3 = (k − 3)(k − 1)(k + 2)(k + 4)
5! b1
Series solutions of Legendre’s equation
Gamma function Properties
EX 1.
EX 2.
Legendre’s eqn EX 3.
Lengendre func Legendre poly EX 4.
Rodrigues’ form EX 5.
EX 6.
Legendre functions. Setting b0 = 1, b1 = 0, we get b1 = b3 = · · · = bodd = 0. Using b2, b4, . . . we obtain
y1 = 1 − k(k + 1)
2! x2 + (k − 2)k(k + 1)(k + 3)
4! x4 + · · · Setting b0 = 0, b1 = 1, we get b0 = b2 = · · · = beven = 0. Using b1, b3, . . . we obtain
y2 = x−(k − 1)(k + 2)
3! x3+(k − 3)(k − 1)(k + 2)(k + 4)
5! x5+· · · y1, y2 are called the Legendre functions.
Legendre polynomials
Gamma function Properties
EX 1.
EX 2.
Legendre’s eqn EX 3.
Lengendre func Legendre poly EX 4.
Rodrigues’ form EX 5.
EX 6.
Theorem. Let k ∈ {0, 1, 2, . . .}. The solutions of the Legendre’s equation
(1 − x2)y′′ − 2xy′ + k(k + 1)y = 0 are as follows.
• If k = 2m is an even integer, then y1 is a polynomial of degree 2m and y2 is a power series with infinitely many terms.
• If k = 2m + 1 is an odd integer, then y2 is a polynomial of degree 2m + 1 and y1 is a power series with infinitely many terms.
Example 4
Gamma function Properties
EX 1.
EX 2.
Legendre’s eqn EX 3.
Lengendre func Legendre poly EX 4.
Rodrigues’ form EX 5.
EX 6.
EX.
Even integer. For k = 0, we get y1 = 1 and y2 = x + 23x3 + 15x5 + · · ·
For k = 2, we get y1 = 1 − 3x2, y2 = x − 3!4 x3 − 245! x5 + · · · For k = 4, we get y1 = 1 − 10x2 + 353 x4
Odd integer. For k = 1, we get y2 = x, y1 = 1−x2 − 4!8 x4 − · · · For k = 3, we get y2 = x − 53x3, y1 = 1 − 122! x2 + 724! x4 + · · ·
For k = 5, we get y2 = x − 143 x3 + 215 x5
Legendre polynomials
Gamma function Properties
EX 1.
EX 2.
Legendre’s eqn EX 3.
Lengendre func Legendre poly EX 4.
Rodrigues’ form EX 5.
EX 6.
If k ∈ {0,1, 2, . . .}, we let Pk(x) =
(y1(x)/y1(1) k = 2m
y2(x)/y2(1) k = 2m + 1
Definition. Let k ∈ {0,1, 2, 3, . . .}. We call Pk(x) the Legen- dre polynomial of degree k.
Then other solution of the Legendre’s equation which is a power series with infinitely many term is denoted Qk(x) .
Thus the complete solution of the Legendre’s equation is
Rodrigues’ formula
Gamma function Properties
EX 1.
EX 2.
Legendre’s eqn EX 3.
Lengendre func Legendre poly EX 4.
Rodrigues’ form EX 5.
EX 6.
Rodrigues’ formula If k ∈ {0, 1,2, . . .} then Pk(x) = 1
2kk!
dk
dxk (x2 − 1)k
Example 5
Gamma function Properties
EX 1.
EX 2.
Legendre’s eqn EX 3.
Lengendre func Legendre poly EX 4.
Rodrigues’ form EX 5.
EX 6.
EX. Solve the Legendre’s equation (1 − x2)y′′ − 2xy′ + 6y = 0.
Example 6
Gamma function Properties
EX 1.
EX 2.
Legendre’s eqn EX 3.
Lengendre func Legendre poly EX 4.
Rodrigues’ form EX 5.
EX 6.
EX. Find the Legendre polynomial P4(x).
