Page:EB1911 - Volume 25.djvu/675

Rh and let each side operate on $$1/r$$, then in virtue of (10), we have

which is known as the addition theorem for the function $$P_n$$.

It has incidentally been proved that

which is an expression for $$P_n^m(\cos\theta)$$ alternative to (4).

10. Legendre's Coefficients.—The reciprocal of the distance of a point $$(r, \theta, \phi)$$ from a point on the z axis distant r' from the origin is

which satisfies Laplace's equation, $$\mu$$ denoting $$\cos\theta$$. Writing this expression in the forms

it is seen that when $$r < r'$$, the expression can be expanded in a convergent series of powers of $$r/r'$$, and when $$r' < r$$ in a convergent series of powers of $$r'/r$$. We have, when $$h^2(2\mu - h)^2 < 1$$

and since the series is absolutely convergent, it may be rearranged as a series of powers of h, the coefficient of $$h^n$$ is then found to be

this is the expression we have already denoted by $$P_n(\mu)$$; thus

the function $$P_n(\mu)$$ may thus be defined as the coefficient of $$h^n$$ in this expansion, and from this point of view is called the Legendre's coefficient or Legendre's function of degree n, and is identical with the zonal harmonic. It may be shown that the expansion is valid for all real and complex values of h and $$\mu$$, such that mod. h is less than the smaller of the two numbers mod. $$(\mu\pm\sqrt{\mu^2 - 1})$$. We now see that

is expressible in the form

when $$r < r'$$, or

when $$r' < r$$; it follows that the two expressions $$r^nP_n(\mu)$$, $$r^{-n - 1}P_n(\mu)$$ are solutions of Laplace's equation.

The values of the first few Legendre's coefficients are

We find also

according as n is odd or even; these values may be at once obtained from the expansion (13), by putting $$\mu = 1, 0, -1$$.

11. Additional Expressions for Legendre's Coefficients.—The expression (3) for $$P_n(\mu)$$ may be written in the form

with the usual notation for hypergeometric series. On writing this series in the reverse order

or

according as n is even or odd.

From the identity

it can be shown that

By (13), or by the formula

which is known as Rodrigue's formula, we may prove that

Also that

By means of the identity

it may be shown that

Laplace's definite integral expression (6) may be transformed into the expression

by means of the relation

Two definite integral expressions for $$P_n(\mu)$$ given by Dirichlet have been put by Mehler into the forms

When n is large, and $$\theta$$ is not nearly equal to 0 or to $$\pi$$, an approximate value of $$P_n(\cos\theta)$$ is $$\{2/n\pi\sin\theta\}^{\frac{1}{2}}\sin\left\{(n + \frac{1}{2})\theta + \frac{1}{4}\pi\right\}$$.

12. Relations between successive Legendre's Coefficients and their Derivatives.—If $$(1 - 2h\mu + h^2)^{-\frac{1}{2}}$$ be denoted by u, we find

on substituting $$\sum h^nP_n$$ for u, and equating to zero the coefficient of $$h^n$$, we obtain the relation

From Laplace's definite integral, or otherwise, we find

We may also show that

the last term being $$3P_1$$ or $$P_0$$ according as n is even or odd.

13. Integral Properties of Legendre's Coefficients.—It may be shown that if $$P_n(\mu)$$ be multiplied by any one of the numbers 1, $$\mu$$, $$\mu^2$$, ... $$\mu^{n - 1}$$ and the product be integrated between the limits 1, -1 with respect to $$\mu$$, the result is zero, thus

To prove this theorem we have