Norms and Zeros
1
Math 4038
Legendre & Bessel Functions
Pn : The Norm of the nth Legendre Polynomial
We will prove that Pn =
2
for each nonnegative integer n, leaving numerous details for the
2n + 1
reader to check.
n
1 dn u
We will use Rodriguez’s Formula1 , established in class: Pn (x) = n
, where u = x2 − 1 .
n
2 n! dx
We will apply integration by parts repeatedly.
1
1
1
1
1
2
(2n n!)
Pn2 dx =
u(n) u(n) dx = u(n) u(n−1) −
u(n+1) u(n−1) dx =
u(n+1) u(n−1) dx
−1
−1
= . . . = (−1)n (2n)!
(A)
= (2n)!
π
2
−π
2
−1
1
−1
−1
2
n
x − 1 dx = (2n)!
−1
1
−1
n
1 − x2 dx
2 (2n n!)2
cos2n+1 θ dθ = (2n)!
(2n + 1)!
(B)
where step (A) is a trigonometric substitution, and step (B) is explained as follows.
π2
π2
2
(2n n!)
2n+1
We prove by induction that
. The case n = 0 is
cos
θ dθ = 2
cos2n+1 θ dθ =
(2n + 1)!
−π
0
2
easy and is left as an exercise. Now assume that the formula is true for n − 1 and use integration by parts
to prove the validity of the formula for n.
Finally, the steps above establish the claimed norm for Pn .
2
Zeros of the Bessel Functions: Heuristic Explanation
Consider the Bessel equation for any ν ≥ 0:
x2 y + xy + x2 − ν 2 y = 0.
u
Make the substitution y = √ , where u will be the new dependent variable, replacing y, and x remains
x
the independent variable. Show that this transforms the general Bessel equation into
1
− ν2
u + 1 + 4 2
u = 0.
x
Observe that if x >> 0, which is read as x is very much bigger than zero, then this equation is quite
similar to u + u = 0. The latter equation is solved by cos x and by sin x, which are functions that
oscillate endlessly between positive and negative values. In fact, the equation tells us that u accelerates
toward negative values whenever u is positive, and vice versa. This suggests strongly that u, and also y,
must cross the x-axis infinitely often. Details of a rigorous argument can be found in Watson’s book.
3
Norm of the Bessel Function Jn , n ≥ 0
Note that n does not need to be an integer in this section. We will prove that
Jn (λmn x2 =
1 These
R2 2
J
(λmn R).
2 n+1
and other topics can be studied further in Watson’s classic treatise, Bessel Functions.
1
Norms and Zeros
Math 4038
Here λmn R = amn , the mth zero of Jn , and Jn (λmn x2 =
Legendre & Bessel Functions
R
0
Jn2 (λmn x) x dx.
Denote φn (x) = Jn (λx) for general λ ∈ R. It follows that
−n2
2
+ λ x φn (x) = 0. Now multiply by 2xφn (x) :
[xφn (x)] +
x
2 [xφn (x)]
+ λ2 x2 − n2 φ2n (x) = 0. Next integrate both sides:
R
R
2 2
λ x − n2 φ2n (x) dx. (∗)
[xφn (x)] = −
0
0
Recall the identity
x−n Jn (x) = −x−n Jn+1 (x) and proceed as follows.
−nx−n−1 Jn (x) + x−n Jn (x) = −x−n Jn+1 (x)
−nJn (x) + xJn (x) = −xJn+1 (x). Next replace x by λx :
λx
φn (x)
= nφn (x) − λxφn+1 (x).
λ
Combining the latter calculations we find from the left side of Equation (*) that
R
2 R
2
[xφn (x)] = [nφn (x) − λxφn+1 (x)] (Now set λ = λmn )
0
0
2
= λ2mn R2 Jn+1
(λmn R) since Jn (λmn R) = 0 if n ≥ 1. Integrate (*) on the right by parts:
R
R
xJn2 (λmn x) dx.
= − λ2mn x2 − n2 Jn2 (λmn x) +2λ2mn
0
0
=0
R
It follows that
0
Jn2 (λmn x) x dx =
R2 2
J
(λmn R), which concludes the proof.
2 n+1
2