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Lecture 1 Sujin Khomrutai – 1 / 28

Method of Applied MathLecture 2: Legendre’s Equation and Gamma

function

Sujin Khomrutai, Ph.D.

Legendre’s equation

Legendre’s eqn

Series Sol

Legendre func

Rodrigues’ form

Gamma function

Properties

✔ Legendre’s equation.✔ Gamma function✔ Applications

Legendre’s equation

Legendre’s eqn

Series Sol

Legendre func

Rodrigues’ form

Gamma function

Properties

Definition For a number n, the equation of the form

(1− x2)y′′ − 2xy′ + n(n+ 1)y = 0

is called the Legendre equation of order n.

• n = 0: (1− x2)y′′ − 2xy′ = 0

• n = 1: (1− x2)y′′ − 2xy′ + 2y = 0

• n = 2: (1− x2)y′′ − 2xy′ + 6y = 0

• n = 3: (1− x2)y′′ − 2xy′ + 12y = 0

In most phenomena, n is a non-negative integer. So we restrict tothat n ∈ {0, 1, 2, . . . , }.

Example 1: (Electrostatics)

Legendre’s eqn

Series Sol

Legendre func

Rodrigues’ form

Gamma function

Properties

Example. A two metallic spherical caps are placed so that theupper part stayed at a constant potential 110 V and the lowerpart is grounded. The electrostatic potential at any point in thespace can be calculated by solving the Legendre’s equation.

Series solutions of Legendre’s equation

Legendre’s eqn

Series Sol

Legendre func

Rodrigues’ form

Gamma function

Properties

Power series method. a = 0 is an ordinary point. Set solutiony to the Legendre’s equation

(1− x2)y′′ − 2xy′ + n(n+ 1)y = 0,

y =∞∑

y′ =∞∑

bk+1(k + 1)xk

y′′ =∞∑

bk+2(k + 2)(k + 1)xk.

Legendre’s eqn

Series Sol

Legendre func

Rodrigues’ form

Gamma function

Properties

The first term

(1− x2)y′′ = y′′ − x2y′′

=∞∑

bk+2(k + 2)(k + 1)xk −∞∑

bk+2(k + 2)(k + 1)xk+2

=∞∑

bk+2(k + 2)(k + 1)xk −∞∑

bjj(j − 1)xj

= b2 · 2 · 1 + b3 · 3 · 2x

+∞∑

[bk+2(k + 2)(k + 1)− bkk(k − 1)]xk

where we have use shifting and splitting.

Legendre’s eqn

Series Sol

Legendre func

Rodrigues’ form

Gamma function

Properties

The second term

2xy′ = 2x∞∑

bk+1(k + 1)xk

=∞∑

2bk+1(k + 1)xk+1

=∞∑

2bkkxk

Third term

n(n+ 1)y = n(n+ 1)∞∑

bkxk =

∞∑

n(n+ 1)bkxk

Legendre’s eqn

Series Sol

Legendre func

Rodrigues’ form

Gamma function

Properties

b2 · 2 · 1 + b3 · 3 · 2x

+∞∑

[bk+2(k + 2)(k + 1)− bkk(k − 1)]xk

−∞∑

2bkkxk +

∞∑

n(n + 1)bkxk = 0

∴ [2b2 + n(n+ 1)b0] + [6b3 − 2b1 + n(n+ 1)b1]x

+∞∑

[bk+2(k + 2)(k + 1)− bkk(k − 1)

− 2bkk + n(n+ 1)bk]xk = 0

Legendre’s eqn

Series Sol

Legendre func

Rodrigues’ form

Gamma function

Properties

Finally, we get

[2b2 + n(n+ 1)b0] + [6b3 + (n− 1)(n+ 2)b1]x

+∞∑

[(k + 2)(k + 1)bk+2 − (n− k)(n+ k + 1)bk]xk

Apply the fact:∑

k=0 ck(x− a)k = 0 ⇔ ck = 0 for all k:

b2 = −n(n+ 1)

b3 = −(n− 1)(n+ 2)

bk+2 = −(n− k)(n+ k + 1)

(k + 2)(k + 1)bk ∀ k = 2, 3, . . . .

Legendre’s eqn

Series Sol

Legendre func

Rodrigues’ form

Gamma function

Properties

Let us consider the case n = 3. Recurrence equations are

b2 = −3 · 42

b0, b3 = −2 · 53!

bk+2 = −(3− k)(3 + k + 1)

(k + 2)(k + 1)bk ∀ k ≥ 2

k = 2 : b4 = −1 · 64 · 3b2 =

1 · 6 · 3 · 44!

k = 3 : b5 = −0 · 75 · 4b3 = 0

k = 4 : b6 = −(−1) · 86 · 5 b4 = −(−1) · 8 · 1 · 6 · 3 · 4

k = 5 : b7 = −(−2) · 97 · 6 b5 = 0

Legendre’s eqn

Series Sol

Legendre func

Rodrigues’ form

Gamma function

Properties

So in the case n = 3, we obtain the solution

y = b0

1− 3 · 42!

x2 +1 · 6 · 3 · 4

4!x4 + · · ·

x− 2 · 53!

This is the general solution with fundamental solutions

y1 = 1− 3 · 42!

x2 +1 · 6 · 3 · 4

4!x4 + · · · an infinite series

y2 = x− 2 · 53!

x3 = −5

3x3 + x a polynomial degree 3

For a general n ∈ {0, 1, 2, . . .}, the Legendre equation has onepolynomial solution of degree n and an infinite series solution.

Legendre functions

Legendre’s eqn

Series Sol

Legendre func

Rodrigues’ form

Gamma function

Properties

Theorem. For n ∈ {0, 1, 2, . . .}, the general solutions to theLegendre’s equation

(1− x2)y′′ − 2xy′ + n(n+ 1)y = 0

y = C1Pn(x) + C2Qn(x)

where Pn is a polynomial of degree n, Qn an infinite series.

• Pn = the Legendre polynomial.

• Qn = the Legendre function of the second kind.

Legendre functions

Legendre’s eqn

Series Sol

Legendre func

Rodrigues’ form

Gamma function

Properties

Legendre functions

Legendre’s eqn

Series Sol

Legendre func

Rodrigues’ form

Gamma function

Properties

Rodrigues’ formula

Legendre’s eqn

Series Sol

Legendre func

Rodrigues’ form

Gamma function

Properties

The Legendre’s equation with n = 0 is

(1− x2)y′′ − 2xy′ = 0.

A polynomial of degree n = 0 is constant. It can be easily checkthat y = 1 is a solution. So we put

P0(x) = 1.

Rodrigues’ formula If n ∈ {1, 2, 3, . . .} then

Pn(x) =1

dxn(x2 − 1)n

Example 2

Legendre’s eqn

Series Sol

Legendre func

Rodrigues’ form

Gamma function

Properties

EX. Verify Rodigues’ solution formula for n = 1, 2, 3.

n = 1 : P1(x) =1

dx(x2 − 1) = x

(1− x2)x′′ − 2x · x′ + 2x = 0

(1− x2) · 0− 2x · 1 + 2x = 0 True

n = 2 : P2(x) =1

22 · 2!d2

dx2(x2 − 1)2

dx2(x4 − 2x2 + 1) =

2x2 − 1

(1− x2)(3

2x2 − 1

2)′′ − 2x(

2x2 − 1

2)′ + 6(

2x2 − 1

2) = 0

(1− x2) · 3− 2x · 3x+ (9x2 − 3) = 0

(3− 3x2)− 6x2 + (9x2 − 3) = 0 True

Example 2

Legendre’s eqn

Series Sol

Legendre func

Rodrigues’ form

Gamma function

Properties

n = 3 : P3(x) =1

23 · 3!d3

dx3(x2 − 1)3

dx3(x6 − 3x4 + 3x2 − 1)

2x3 − 3

(1− x2)(5

2x3 − 3

2x)′′ − 2x(

2x3 − 3

2x)′ + 12(

2x3 − 3

2x) = 0

(1− x2)(15x)− 2x(15

2x2 − 3

2) + (30x3 − 18x) = 0

(15x− 15x3)− (15x3 − 3x) + (30x3 − 18x) = 0 True

Example 3

Legendre’s eqn

Series Sol

Legendre func

Rodrigues’ form

Gamma function

Properties

EX. Solve the Legendre’s equation

(1− x2)y′′ − 2xy′ + 2y = 0.

Sol. We get a solution P1(x) = x. Use the reduction of order

Q1(x) = P1(x)

(P1(x))2e−

∫−2x

1−x2dxdx

x2e− ln(1−x2)dx

x2(1− x2)dx =

1− x

∴ y = C1x+ C2

1− x

The Gamma function

Legendre’s eqn

Series Sol

Legendre func

Rodrigues’ form

Gamma function

Properties

The factorials n! are

0! = 1! = 1

2! = 2 · 1 = 2

3! = 3 · 2 · 1 = 6,

n! = n(n− 1)(n− 2) · · · 3 · 2 · 1

Observe that∫

e−tdt =

e−ttdt = 1,

e−tt2dt = 2 = 2!,

e−tt3dt = 6 = 3!.

The Gamma function

Legendre’s eqn

Series Sol

Legendre func

Rodrigues’ form

Gamma function

Properties

Definition For x > 0, the integral

e−ttx−1 dt is called the Gamma function denoted Γ(x).

If x < 0 and x 6∈ {−1,−2, . . .} we put

Γ(x) =Γ(x+ n)

(x+ n− 1)(x+ n− 2) · · · (x+ 1)x

where n ∈ {1, 2, . . .} satisfies x+ n > 0.

Consecutive product formula

Γ(x+ n) = (x+ n− 1)(x+ n− 2) · · · (x+ 1)xΓ(x).

The Gamma function

Legendre’s eqn

Series Sol

Legendre func

Rodrigues’ form

Gamma function

Properties

The graph of Gamma function is as shown below

Properties of Gamma function

Legendre’s eqn

Series Sol

Legendre func

Rodrigues’ form

Gamma function

Properties

(1) Γ(1) = 1. For x > 0,

Γ(x+ 1) = xΓ(x)

(2) Generally, for n ∈ {1, 2, . . .} we get

Γ(x+ n) = (x+ n− 1) · · · (x+ 1)xΓ(x).

(3) In particular, if n ∈ {0, 1, 2, . . .} then

Γ(n+ 1) = n!.

Legendre’s eqn

Series Sol

Legendre func

Rodrigues’ form

Gamma function

Properties

Proof. (1) For Γ(1), consider

Γ(1) =

e−tt1−1dt =

e−tdt

= (−e−t)∣

t=0= 0− (−e0) = 1.

Next, for x > 0 consider

Γ(x+ 1) =

e−tt(x+1)−1dt =

e−ttxdt

= (−e−ttx)∣

t=0−∫

(−e−txtx−1)dt (by parts)

e−ttx−1dt = xΓ(x).

Legendre’s eqn

Series Sol

Legendre func

Rodrigues’ form

Gamma function

Properties

(2) For n = 2, we use (1) (x → x+ 1):

Γ(x+ 2) = Γ((x+ 1) + 1) = (x+ 1)Γ(x+ 1),

and then use (1) one more time:

Γ(x+ 1) = xΓ(x).

Γ(x+ 2) = (x+ 1)xΓ(x)

The argument can be extended to n = 3, n = 4, . . ..

(3) Use x = 1 in (2) and that Γ(1) = 1.

Example 4

Legendre’s eqn

Series Sol

Legendre func

Rodrigues’ form

Gamma function

Properties

EX. Evaluate

Γ(1) + Γ(4),Γ(2.8)

Γ(0.8).

Example 5

Legendre’s eqn

Series Sol

Legendre func

Rodrigues’ form

Gamma function

Properties

EX. Given that Γ(0.5) =√π. Evaluate

Γ(−2.5).

Example 6

Legendre’s eqn

Series Sol

Legendre func

Rodrigues’ form

Gamma function

Properties

EX. Express

(ν + 1)(ν + 2)(ν + 3)

as a ratio of the Gamma function.

Example 7

Legendre’s eqn

Series Sol

Legendre func

Rodrigues’ form

Gamma function

Properties

EX. Evaluate the integral

t1.35e−tdt.

Express the value in terms of the Gamma function.

method of applied mathpioneer.netserv.chula.ac.th/~ksujin/slide02(ise,317).pdflecture1...

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