From this form it can be shown that
Qn(μ)＝12 Pn(μ) log μ＋1μ−1−Wn−1(μ),
where Wn−1(μ) is a rational integral function of degree n−1 in μ; it can be shown that this form is in agreement with the definition of Qn(μ) by series, for the case mod μ>1. In case mod μ<1 it is convenient to use the symbol Qn(μ) for
12Pn(μ) log 1＋μ1−μ−Wn−1(μ),
which is real when μ is real and between ±1, the function Q„(m) in this case is not the analytical continuation of the function Qt.(m) for mod m> x . but differs from it by an imaginary multiple of P„(m). It will be observed that Q n (i), Q n ( — i) are infinite, and Qn(x) =o. The function Wn-i(ju) has been expressed by Christoffel in the form
2 "-?P„_ lW + - 2 "-
•w+|^ p — w+-
I . n ^' . ' 3 . n — I
and it can also be expressed in the form
~P (p.)P^ 1 (p.)+~P l (p)?^. 2 (p.) + . . . +P„-,(m)Po(m).
It can easily be shown that the formula (28) is equivalent to
which is analogous to Rodrigue's expression for PnM- Another expression of a similar character is
It can be shown that under the condition mod \u — V (u- — 1)\ >mod \ti — V (v?— 1)|, the function il(n — u) can be expanded in the form 2(2»-|-i)P«(tt)Q„(tt) ; this expansion is connected with the definite integral formula for Q n (v) which was used by F. Neumann as a definition of the function Q»(p.), this is
which holds for all values of \i which are not real and between ± I. From Neumann's integral can be deduced the formula
d4>
••00 Qn(M) = J ,
iM+vc^-o.cosh^r'
which holds for all values of p. which are not real and between ="= 1 , provided the sign of V (/* 2 — 1 ) is properly chosen ; when p is real and greater than 1, V (p 2 — 1) has its positive value. By means of the substitution.
Jm + V(m 2 — i).cosh \fr\\ii — V(m 2 — l).cosh x! = i,
the above integral becomes
Q„0) = J *V~V (m 2 - 1) cosh xNx. where xa = ^og c j~. '
This formula gives a simple means of calculating Q„(p) for small values of n; thus
- w-/>-->*e±j.
If, in Legendre's equation, we differentiate m times, we find
. . n d m **u , . . d M+1 u , , ., , , .d m u
• c 11 1 d m u , m , , ,v m d m u
it follows that u = ^'hence «„ = 0* 2 -i)- - a -^-
2m'"
The complete solution of (26) is therefore
when m is real and lies between ± 1 , the two functions
U-M - ) 4u'" ' U_M - ) C^" 1 are called Legendre's associated functions of degree <n, and ordur m, of the fhrt and second kinds respectively. When /j is not real and between =*=i, the same names are given to the two functions
Qi(p) =/»X«-V (m 2 - I) -sinh Xo = M-^oSj^~ I.
Neumann's integral affords a means of establishing a relation between successive Q functions, thus
»Q»-(2rt-i)/«Q»-i + (n-l)(?»_2
"1 «P,(tt) + (n- i)P„- 2 (h.) - (2w- p — u
pP n -i{u)du
-ITJ
= -|J _,(2»- l)P»-l(") =0.
Again, it may similarly be proved that
19. Legendre Associated Functions. — Returning to the equation (26) satisfied by «™ the factor in the normal forms „_, ™ mtfr.u™,
we shall consider the case in which n, m are positive integers, and
n^m. Let « = (m 2 — i)* m v, then it will be found that v satisfies the equation
(m 2 -0
dp'" '
(V-i)
U d m Q*W
dp m
in either case the functions may be denoted by P n (m), Qn (m). It can be shown that, when p is real and between ± 1
p:m =5^11 (S3 im ^io.-i)"^o.+i)-i
2»(n-m)l\l+p) dp" 1 ^ V " +l ' <• In the same case, we find
Pr +2 (cos 8)-2(m + i) cot $ Pâ„¢ +1 (cos 0)
+ (n — m)(n+m + l)P™(cos 8)=o, (ra-«+2)Pr +2 (cos 0) - (2n+3) M Pr + i(cos 0)
+ (»+m + l)Pr(cos0)=o.
20. Bessel's Functions.— -If we take for three orthogonal systems of surfaces a system of parallel planes, a system of co-axial circular cylinders perpendicular to the planes, and a system of planes through the axis of the cylinders, the parameters are a, p, $, the cylindrical co-ordinates; in that case Hi = i, Hi = i, H 3 = i/p, and the equation (25) becomes
d"v yy 1 av 1 a-v d~ 2 + <v +p dp v a<^> 2 ~ '
To find the normal functions which satisfy this equation, we put V = ZRΦ, when Z is a function of z only, R of p only, and Φ of φ, the equation then becomes
I ^?j_± /^P:j_I ^E\ j.1 I 4!?_ Z dz- + K Up 2 + p dp/ V* d^~°-
1 2 Z That this may be satisfied we must have y~r^ constant, say =k',
tt yj conetant, say = — m-, and R, for which we write u, must satisfy the differential equation d 2 u , I dti
dp'
. 1 du /,„ pi-\
it follows that the normal forms arc e ^ z '-m<p.u(kp), where u(p) satisfies the equation
",7/ T n ii t n'"\
(29)
d 2 u , I du , / n7 ! \
d?+pd7 + { l -7) u=0 -
This is known as Bessel's equation of order m\ the particular case
d"-u , I du
d9 + pT P + u=0 '
(30)
corresponding to m = o. is known as Bessel's equation.
If we solve the equation (29) in series, we find by the usual process that it is satisfied by the series
+0
2.2OT+2 r 2.4.2/W-(-2.2W-|-4
the expression
P" \, P 2 ,
2"*II(m) ( 2.2m+2 2.4.2m+2.2m+4
...| s
or
(— i)V +2n
/ |) 2"' +2 "n(m+»)n(w' l
is denoted by J m (p). """i When m=o, the solution
i _pJ + _A_...
2 2 ' 2 2 . 4 2
of the equation (30) is denoted by J0(ρ) or by J(ρ).
